Mosaic tile model to compute gravitational field for

Mosaic tile model to compute gravitational field for infinitely thin non axisymmetric objects and its application to preliminary analysis of gravitational field of M74

Journal: Manuscript ID Manuscript type: Date Submitted by the Author: Complete List of Authors:

Keywords:

Monthly Notices of the Royal Astronomical Society MN-16-0549-MJ.R1 Main Journal n/a Fukushima, Toshio; National Astronomical Observatory of Japan, Public Relations Center gravitation < Physical Data and Processes, methods: analytical < Astronomical instrumentation, methods, and techniques, celestial mechanics < Astrometry and celestial mechanics, galaxies: individual:... < Galaxies, galaxies: spiral < Galaxies

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MNRAS 000, 1–?? (2016)

Preprint 25 March 2016

Compiled using MNRAS LATEX style file v3.0

Mosaic tile model to compute gravitational field for infinitely thin non axisymmetric objects and its application to preliminary analysis of gravitational field of M74 Toshio Fukushima1⋆ 1

National Astronomical Observatory of Japan, Ohsawa, Mitaka, Tokyo 181-8588, Japan

Accepted. Received; in original form

ABSTRACT

Using the analytical expressions of the Newtonian gravitational potential and the associated acceleration vector for an infinitely thin uniform rectangular plate, we developed a method to compute the gravitational field of a general infinitely thin object without assuming its axial symmetry when its surface mass density is known at evenly spaced rectangular grid points. We utilized the method in evaluating the gravitational field of the HI gas, dust, red stars, and blue stars components of M74 from its THINGS, 2MASS, PDSS1, and GALEX data. The non axisymmetric feature of M74 including an asymmetric spiral structure is seen from (i) the contour maps of the determined gravitational potential, (ii) the vector maps of the associated acceleration vector, and (iii) the cross section views of the gravitational field and the surface mass density along different directions. An x-mark pattern in the gravitational field is detected at the core of M74 from the analysis of its dust and red stars components. Meanwhile, along the east-west direction in the central region of the angular size of 1′, the rotation curve derived from the radial component of the acceleration vector caused by the red stars component matches well with that observed by the VENGA project. Thus the method will be useful in studying the dynamics of particles and fluids near and inside spiral galaxies with known photometry data. Electronically available are the table of the determined gravitational fields of M74 on its galactic plane as well as the Fortran 90 programs to produce them. Key words: celestial mechanics – gravitation – galaxies: individual (M74) – galaxies: spiral – methods: analytical

1 INTRODUCTION 1.1

Gravitational field of infinitely thin axisymmetric objects

The computation of the gravitational field for objects of general shape and arbitrary mass density distribution has been a fundamental issue in astronomy from the days of Newton. The commentary by Chandrasekhar (1995, chapters 1, 15, and 16) focuses on Newton’s genius in discovering the complete analytical solution for general spherically symmetric objects. Nevertheless, it remains to be a complicated problem for flattened objects such as spiral galaxies (Binney & Merrifield 1998). This is because unemployable is Newton’s theorem on the gravitational field computation of spherically symmetric bodies (Kellogg 1929). Refer to Binney & Tremaine (2008, chapter 2) for the long history of astronomers’ endeavors on this difficult issue. Recently, we provided a definitive solution of this problem for general infinitely thin axisymmetric objects (Fukushima 2016). It is a numerical method to compute the gravitational potential and the

⋆

E-mail: [email protected]

© 2016 The Authors

associated acceleration vector in the three-dimensional space by (i) splitting the interval of a line integral of the ring potential at its blowup singularity, (ii) adopting the double exponential quadrature rule (Takahashi & Mori 1973) in its numerical integration, and (iii) conducting the numerical differentiation of the numerically integrated potential by means of Ridder’s method (Ridder 1982). However, this recipe is effective only when the object is axially symmetric. Namely, it is not applicable to significantly non axisymmetric objects including grand design spiral galaxies as M74.

1.2 Face-on spiral galaxy M74 Before going further, let us present a real example of non axially symmetric object: M74 or NGC628 (Bertin & Lin 1996). It is a typical grand design spiral galaxy (i) categorized to be of the Hubble type SA(s)c, (ii) classified as the arm class 9, and (iii) with a normal size as the isochronal radius being R25 = 11.6 kpc (Elmegreen & Elmegreen 1984, Table 1). The galaxy is almost face-on. Indeed, its inclination angle is estimated as small as 7–9◦ (Tamburro et al. 2008; Blanc et al. 2013). This is convenient in an-

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Contour Map of Σ: M74, HI

Contour Map of Σ: M74, K 3

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-∆α (arcmin) Figure 1. Contour map of surface mass density: M74, gas component. Shown is the contour map of the surface mass density transformed from the 1024 × 1024 pixels HI gas map of M74 observed by the THINGS program (Walter et al. 2008). The angular distance of 1 arc minute corresponds to a linear distance of 2.1 kpc on the galactic plane if the distance to the galaxy is 7.3 Mpc (Karachentsev et al. 2004). The contour levels are linearly set as every 5% of the peak value. Clearly seen are a spiral structure and a central hole.

alyzing the structure on the galactic plane because the differential effect of the interstellar extinction may be negligibly small. Beyond controversy, the galaxy is so famous that it has been almost always included in the catalogs of the nearby galaxies in a variety of photometric survey programs (Kent 1994; Jarrett et al. 2003; Krauss et al. 2005; Martin et al. 2005; Walter et al. 2008). Also, many articles were published on three supernovae associated to the galaxy. Nevertheless, rather limited are the studies of its structure itself (Guidoni et al. 1981; Chen et al. 1992; Cornett et al. 1994; Blanc et al. 2013; Sanchez-Gil et al. 2015) Refer to Figs 1–4 showing the contour maps of the surface mass densities of its HI gas, dust, red stars, and blue stars components. They are transformed from (i) the HI (21 cm) radio intensity map obtained by the “The HI Nearby Galaxy Survey” (THINGS) program (Walter et al. 2008) using the NRAO Very Large Array, (ii) the K-band (2.2 µm) infrared mosaic image delivered by the “Two Micron All Sky Survey” (2MASS) project (Jarrett et al. 2003), (iii) the Kodak 103aE (645 nm) photographic plate taken at the Palomar 48-inch Schmidt telescope and digitized by the first generation Palomar Digitized Sky Survey (PDSS1) (Kent 1994), and (iv) the Far Ultra Violet (FUV) (151.6 nm) observation obtained by the “GALaxy Evolution eXplorer” (GALEX) (Martin et al. 2005), a science mission launched by NASA. The HI gas component map reveals its significant non axisymmetric feature while the dust component map is almost axisymmetric. Also, the red (or old) stars component map is dominated by an

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-∆α (arcmin) Figure 2. Contour map of surface mass density: M74, dust component. Same as Fig. 1 but transformed from the K-band infrared image of M74 obtained by the 2MASS program (Jarrett et al. 2003). We used the central 144 × 144 pixels. In order to exhibit the manner of its central concentration clearly, the contour levels are set logarithmically, namely 80, 40, 20, 10, and 5 % of the peak value. Easily visible is an almost axisymmetric feature.

axially symmetric part decaying almost exponentially. Nevertheless, detectable is a weak non axial symmetry. Meanwhile, the blue (or young) stars component map shows trains of spotty feature along spiral arms. 1.3 Difficluties with non axisymmetric objects Return to the issue of the gravitational field computation. There are three major approaches to treat the gravitational field of non axisymmetric discs (Binney & Tremaine 2008, section 2.6): (i) the expansion in the cylindrical polar coordinates using Bessel functions (Toomre 1963), (ii) a sort of Fourier expansion using the base functions expressing logarithmic spirals (Kalnajs 1972), and (iii) an extreme case of the expansion in the oblate spheroidal coordinates (Hunter 1963). Nonetheless, they are not satisfactory solutions. For example, the first method faces with a numerical difficulty in evaluating the integrals including Bessel functions which are heavily oscillating if its degree is high and/or the integration interval is long. Also, the second option is limited to the computation inside the disc, and therefore, can not evaluate the three-dimensional field. In the contrast, the third device is based on the assumption that an object is represented by a number of finite size spheroids compressed in the z-direction such that it is not applicable to infinitely extended objects. As a matter of course, the masspoint approximation is always a powerful tool in considering the gravitational field caused by such complicated mass distribution (Binney & Tremaine 2008, section MNRAS 000, 1–?? (2016)

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Contour Map of Σ: M74, R 6 4 ∆δ (arcmin)

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-∆α (arcmin) Figure 3. Contour map of surface mass density: M74, red (old) stars component. Same as Fig. 1 but transformed from the red color image of M74 retrieved from the PDSS1 database (Kent 1994). We used only the central 256 × 256 pixels data without excluding any spotty feature. Again, the contour levels are set logarithmically but with different values, 100, 50, 25, 12.5, and 6.25% of the peak value. Interesting is the feature that the central area is not a sharp peak but a plateau of the size of 0.2′ × 0.2′ . Apart from an exponential increase toward the galactic center, a spiral feature is vaguely noticeable.

2.9). However, the approximation is good only for the far field computation, namely when the distance between the evaluation point and a mass element is sufficiently large as will be shown later. Thus, this approach becomes erroneous for the evaluation points inside the object. The origin of this difficulty has the same root as the non integrable singularities encountered in integrating the internal gravitational field. Refer to the discussion of the softening of the masspoint potential in Aarseth (2003).

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-∆α (arcmin) Figure 4. Contour map of surface mass density: M74, blue (young) stars component. Same as Fig. 1 but transformed from the FUV image of M74 acquired by the GALEX mission (Martin et al. 2005). We used only the central 512 × 512 pixels. The contour levels are every 2% of the peak value. Prominent is several trains of spots forming a spiral structure.

dron model (Werner 1994; Werner & Scheeres 1996). The modified method simplifies the computation process after transforming the volume integral into multiple surface integrals and line integrals by means of the Gauss-Ostrogradsky divergence theorem (Lass 1950) under the assumption of constant density. In the above cases, essential is the uniformity of the object. In order to overcome this restriction, the original polyhedron models were recently extended to cover the case when the volume mass density is linearly varying (Hansen 1999) or expressed as a cubic polynomial of rectangular coordinates (Garcia-Abdeslem 2005). Refer to D’Urso (2014) and Conway (2015) for a concise summary of the existing works.

Polyhedron models

An exemplar exists in computing the gravitational field of a three-dimensional object of general shape: the polyhedron models (Waldvogel 1976, 1979). We suggest reading the classic work of MacMillan (1930). Traditionally, this kind of finite element methods have been developed in geodesy and geophysics by assuming the uniformity of the target objects (Nagy 1966; Paul 1974; Coggon 1976). Indeed, this approach dates back to Everest (Corbato 1966), who studied the deflection of a plumb line affected by the gravitation of nearby high mountains, including Mt. Everest beyond doubt, under the condition that their volume mass density is constant (Everest 1830, p. 94–97). In celestial mechanics and dynamical astronomy, in order to investigate the external gravitational field of peculiar-shaped asteroids and comet nuclei, developed was a modification of the polyheMNRAS 000, 1–?? (2016)

1.5 Problems in developing polygonal plate models The two-dimensional counterpart of a polyhedron is an infinitely thin polygonal plate. Despite an extensive search in the existing literature, we failed to find the mathematical description of its threedimensional gravitational field. Thus, we tried an adaptation of the polyhedron models to the two-dimensional case. A typical polyhedron suitable for the gravitational field computation is a right parallelepiped. MacMillan (1930) explicitly provided its Newtonian gravitational potential in section 43, and stated a general recipe for computing the associated acceleration vector in sections 44 and 45. He did not present the explicit expression of the acceleration vector. Probably, it was so trivial for him if following the recipe. We must admit that we could not translate his result into the two-dimensional solution by reducing the height of the paral-

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Contour Map of Φ: M74, HI

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-∆α (arcmin) Figure 5. Contour map of gravitational potential on galactic plane: M74, gas component. Same as Fig. 1 but for the gravitational potential computed by the method described in section 2. The contours are drawn for every 2% level of the peak value. A spiral feature of the galaxy is stressed by the steepness of slopes in the potential contours, which indicate the location of the outer edge of the spiral arms. Namely, the arms in the sense of high mass concentrations are represented by not the black spiral curves themselves but the whiter area sandwiched by the darker ones. The potential value is almost flat in the central region, say in the area of 3′ × 3′ near the galactic center. Observed there is not a deep basin but a low hill. It is caused by the strong gravitational attractions of the arms outside the region. The peak of the hill is noticeably shifted in the eastward from the coordinate origin, which was set as the barycenter of the image. This is a natural consequence of the asymmetric feature of spiral arms.

lelepiped, namely by setting the thickness in the z-direction as 0 in the three-dimensional expression. Meanwhile, he also presented the analytical expressions of the two-dimensional acceleration vector caused by an infinitely thin rectangular plate in section 18. Nevertheless, his formulas are limited to the case when the evaluation point is on the same plane as the object is located. Also, he gave neither the gravitational potential nor the z-component of the acceleration. This tendency, namely not considering an analytical expression of the gravitational field of simple two-dimensional objects such as polygonal plates in the three-dimensional space, seems to have been continuing.

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-∆α (arcmin) Figure 6. Contour map of gravitational potential on galactic plane: M74, dust component. Same as Fig. 5 but for the dust component. This time, the contours are drawn for every 1% level of the peak value. As similarly in Fig. 2, the map exhibits nearly circular shapes of the contours except a few spotty features. The central drop is so deep and the slope around it is steep. These are caused by a strong mass concentration at the center of the photo image. A knob in the south part at a distance of around 0.4′ may be an artificial effect by a foreground star.

the analytical expressions of the associated acceleration vector in the three-dimensional space. Utilizing the developed expressions, we constructed an analytical method to evaluate the gravitational field of a general infinitely thin object when known are its surface mass density data at evenly spaced rectangular grid points. We may call it the mosaic tile model since the base figure is a rectangular tile. Although the model is of approximate nature, it is sufficiently general. Indeed, it enabled us to compute the gravitational field caused by a significantly non axisymmetric profile of the surface mass density. Figs 5–8 illustrate the contour map of the gravitational potential of the HI gas, dust, red stars, and blue stars components of M74 on the galactic plane. They correspond to those of the surface mass density depicted in Figs 1–4, respectively. Below, we describe the mosaic tile model in section 2, and apply it to the analysis of the gravitational field of M74 in section 3, while explaining the analytic expression of the three-dimensional gravitational field of an infinitely thin uniform rectangular plate in Appendix A.

Outline of article

Therefore, we conducted (i) the analytical evaluation of a double integral defining the gravitational potential of an infinitely thin uniform rectangular plate for arbitrary evaluation point in the threedimensional space, and (ii) its partial differentiation with respect to the rectangular coordinates of the evaluation point so as to obtain

2 MOSAIC TILE MODEL 2.1 Assumptions Consider the evaluation of the Newtonian gravitational field of a general infinitely thin object without posing its axial symmetry. MNRAS 000, 1–?? (2016)

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Contour Map of Φ: M74, R 6 4 ∆δ (arcmin)

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Assume that its surface mass density, which may take an arbitrary value in general such as displayed in Figs 1–4, is provided at a number of pre-specified rectangular grid points. This is simply feasible for a nearly face-on object such as M74. Refer to Appendix B for a device to treat inclined object images. Denote the grid points by Qi j and express its two-dimensional ( ) rectangular coordinates as x i , y j where the indices run as i = 1, 2, . . . , I and j = 1, 2, . . . , J, respectively. Also, we assume that the grid points are evenly spaced with the fixed separation in the x- and y-directions by 2a and 2b, respectively. More specifically speaking, we write the coordinates of Qi j as x i ≡ x 0 + 2ia, y j ≡ y0 + 2 jb, (1 ≤ i ≤ I; 1 ≤ j ≤ J),

(1)

where x 0 and y0 are certain coordinate offsets. Further, we write the surface mass density at Qi j as Σi j . Uniform rectangular plate approximation

At an arbitrary point P with the three-dimensional rectangular coordinates (x, y, z), we compute the gravitational field caused by the surface mass density data, Σi j . This requires a sort of double summation in any way. Assume that (i) the grid size is sufficiently small, and (ii) the surface mass density is constant in each rectangular plate. Then, we write the gravitational potential and the associated acceleration vector at P as J I ∑ ∑

Φi j (x, y, z),

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Figure 7. Contour map of gravitational potential on galactic plane: M74, red stars component. Same as Fig. 5 but for the red stars component. This time, the contours are drawn for every 1% level of the peak value. As similarly in Fig. 3, the map exhibits nearly circular shapes of the contours except distortions caused by the clumps noticeable in the surface mass density map. Nevertheless, asymmetric two spiral arms are more visible than Fig. 3.

Φ(x, y, z) =

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(2)

Figure 8. Contour map of gravitational potential on galactic plane: M74, blue stars component. Same as Fig. 5 but for the blue stars component. The contours are drawn for every 2% level of the peak value. The spotty image is caused by strong mass concentrations at several clumps seen in Fig. 4. Despite the locally circular features around the clumps, the global map shows an asymmetric spiral structure clearly.

F(x, y, z) =

I ∑ J ∑

Fi j (x, y, z),

(3)

i=1 j=1

where Φi j (x, y, z) and Fi j (x, y, z) are the contributions from the (i, j)th rectangular plate, respectively. The analytic form of Φi j (x, y, z) and Fi j (x, y, z) are obtained from those of the unit rectangular plate as ( ) ( ) Φi j (x, y, z) = GΣi j ϕ x − x i , y − y j , z , (4) ( ) ( ) Fi j (x, y, z) = GΣi j f x − x i , y − y j , z ,

(5)

where G is Newton’s constant of universal attraction while ϕ(x, y, z) and f(x, y, z) are a scaled gravitational potential and a normalized acceleration vector evaluated at P when a uniform rectangular plate of the size, 2a × 2b, is placed on the x-y plane and centered at the coordinate origin, (0, 0, 0). 2.3 Gravitational field of uniform square plate The explicit expressions of ϕ(x, y, z) and f(x, y, z) are provided in Appendix A. Here, for the readers’ understanding, we show several sketches of them for the case a = b on the object plane where z = 0. The additional figures are found in Appendix A7. First, Fig. 9 presents the bird’s-eye view of the gravitational potential. Second, Fig. 10 shows the contour map of the potential

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Potential of Square Plate

Contour Map of φ: Square Plate 4

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As seen in Appendix A, the exact forms of ϕ(x, y, z) and f(x, y, z) are somewhat lengthy. Consequently, their computational time is not so small. Especially, the computation of ϕ(x, y, z) is time-consuming since it requires (i) 4 arctangent functions, (ii) 4 logarithms, (iii) 4 square roots, and (iv) 8 divisions, apart from around 20 operations of addition/subtraction and multiplications. On the other hand, if the distance between P and Qi j is sufficiently large and/or the maximum grid size, max(a, b), is sufficiently small, one may approximate the contribution by that of a masspoint located at Qi j such as

Fi j (x, y, z) ≈

−GMi j ri j (x, y, z) r i3j (x, y, z)

(6)

,

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x/a Figure 10. Contour map of gravitational potential: uniform square plate on object plane. Same as Fig. 9 but extracted is its contour map. The contours become more and more circular when approaching to the plate center or departing toward the infinity. Meanwhile, they tend to follow a square feature near the edges of the plate. As easily understood from this situation, the potential value is not analytic on the edges.

where Mi j is the mass of the (i, j)th plate written as Mi j ≡ 4abΣi j ,

(8)

and ri j (x, y, z) is the displacement vector of P referred to Qi j , which is expressed as ( )T ri j (x, y, z) ≡ x − x i , y − y j , z , (9) while r i j (x, y, z) is the distance between P and Qi j written as √ )2 ( r i j (x, y, z) ≡ ri j (x, y, z) = (x − x i ) 2 + y − y j + z 2 . (10)

Masspoint approximation

−GMi j , r i j (x, y, z)

-1 -3

value corresponding to Fig. 9. Third, Fig. 11 depicts the vector map of the acceleration vector on the object plane. Fourth, Fig. 12 plots the cross section along a line on the object plane of (i) the surface mass density, (ii) the gravitational potential, and (iii) the xcomponent of the acceleration vector. They are plotted as functions of x when y is fixed as y = 0. Notice that the y-component of the acceleration vector completely vanishes due to the symmetry of the plate. Also, the zcomponent is not always defined properly on the plane as explained in Appendix A2. Finally, Fig. 13 illustrates the rotation curve for two different directions, Θ ≡ atan2(y, x) = 0 and Θ = π/4.

Φi j (x, y, z) ≈

0 -2

Figure 9. Bird’s-eye view of gravitational potential: uniform square plate on object plane. Shown is a bird’s-eye view of ϕ(x, y, z), a scaled Newtonian gravitational potential of an infinitely thin uniform square plate, namely that of a rectangular plate when a = b. Presented here is a two-dimensional curve of ϕ(x, y, 0) as a function of x and y. It looks like a heated chocolate slab melting down through a square-shaped hole.

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(7)

Roughly speaking, the computational labor of these masspoint approximations is around 5% of that of the uniform rectangular plate approximations. Therefore, from a practical viewpoint, it is wise to switch conditionally the computation of Φi j (x, y, z) and Fi j (x, y, z) between (i) the rectangular plate approximations, equations (4) and (5), and (ii) the masspoint approximations, equations (6) and (7). In general, the magnitude of the difference between two approximations, the mathematical description of which is given in Appendix A6, is in proportion to 1/r i3j (x, y, z) as seen in Fig. 14. Assume that demanded is the 5 digits absolute accuracy of the total gravitational field. It may be sufficient to analyze the gravitational field from the observed surface mass density data, which are typically with 2–4 digits precision.√Then, justified is the masspoint approximation when r i j (x, y, z)/ ab exceeds a certain critical value, say 20. In other words, one may use the uniform rectangular plate approximation only for the near field computation, which covers MNRAS 000, 1–?? (2016)

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x/a Figure 11. Vector map of acceleration vector: uniform square plate on object plane. Same as Fig. 9 but provided is a vector map of the associated acceleration vector on the object plane. The magnitude of the acceleration vector increases when approaching the four corners of the plate. This is because the gravitational potential significantly changes its value there both in the x- and y-directions. As a result, the vector map resembles an open square with four horns. It may look as an x-mark when viewed far from it.

Figure 12. Cross section: uniform square plate. Plotted are (i) Σ, the surface mass density, (ii) ϕ, the gravitational potential, and (iii) f x , the x-component of the acceleration vector of an infinitely thin uniform square plate, as functions of x when y = z = 0. The y-component, f y , vanishes completely. The z-component, f z , also vanishes outside the plate, namely when |x | > 1. Meanwhile, it becomes indefinite on the edges of and inside the plate due to the discontinuity of Σ across the x-y plane. The units of the plotted values are arbitrary chosen. The curve of Σ is raised by a constant offset so as to avoid overlapping with the others. Despite their finite outlook, the spikes of f x infinitely increase or decrease. The potential ϕ is continuous everywhere but not analytic at the edges where x/a = ±1 although the non analyticity is hardly visible at this scale.

the mass elements inside a sphere centered at the evaluation point with √ the radius of 10 units in the representative scale of the plates, ab. Refer to Fig. 14 again. At any rate, it is noteworthy that the introduction of switching significantly accelerates the computation of the residual gravitational field.

( ) ∑∑ Fi j k ℓ , Fk ℓ ≡ F x k , yℓ , 0 =

2.5

where the components are written in a form of matrix product as ( ) ( ) Φi j k ℓ ≡ GΣi j ϕk−i,ℓ− j , Fi j k ℓ ≡ GΣi j fk−i,ℓ− j . (13)

Table look-up

Thus far, we have discussed the computation of the gravitational field at an arbitrary point in the three-dimensional space. It plays a key role in the orbit simulation of massless particles moving around and/or inside the object. However, this general approach may not be always required. For example, in order to obtain the rotation curve of spiral galaxies, it is enough to consider the field on the object plane only. Also, the computation of the gravitational field will be simplified if the rectangular coordinates of the evaluation point are known beforehand. This is especially true when the point is one of the ( ) data grid points, say when (x, y, z) = x k , yℓ, 0 where k and ℓ are certain integer indices in the range, 1 ≤ k ≤ I and 1 ≤ ℓ ≤ J. In such cases, the gravitational potential and the associated acceleration vector are expressed as a double summation of indexed quantities, ( ) ∑∑ Φi j k ℓ , Φ k ℓ ≡ Φ x k , yℓ , 0 = I

J

i=1 j=1

MNRAS 000, 1–?? (2016)

(11)

I

J

(12)

i=1 j=1

Here, ϕ nm and fnm are defined as ( ) (( ) ϕ nm ≡ ϕ(2na, 2mb, 0), fnm ≡ f x nm , f y

)T

,

(14)

while ( ) ( ) f x nm ≡ f x (2na, 2mb, 0), f y ≡ f y (2na, 2mb, 0). nm

(15)

nm

,0

Notice that the newly introduced indices, n and m, are in wider ranges, |n| ≤ I and |m| ≤ J. In general, the computation of Φi j k ℓ and Fi j k ℓ would be a complicated task. Fortunately, in the present case, it reduces to a matrix transform requiring two-dimensional matrices only as seen in the above. This simplicity is owing to the relative nature of the gravitational field such that its values depend not on the absolute but on the relative values of the rectangular coordinates. Once the aspect ratio of the grid size, a/b, is specified,( we) can ( ) precompute the relative quantities, ϕ nm , f x nm , and f y . nm They play the role of coefficient matrices in the matrix transform, equation (13). Notice that they depend not on the numerical values

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T. Fukushima of the surface mass density data themselves but on their geometric character only. Namely, they are specific not to the observed objects themselves but to the characteristics of the observing facilities such as the pixel size of the two-dimensional detectors employed. Therefore, it would be appropriate to prepare the coefficient matrices, store them, and reuse them. This method of table look-up will greatly reduce the actual computation time of the gravitational field on the object plane. Indeed, the computation can be executed by employing all the available cores and threads of the latest multi-core multi-thread CPUs as will be seen later.

V/V0

Circular Speed: Square Plate 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Θ=0

Θ=π/4

2.6 Coefficient matrices The symmetry of the rectangular plate and the parity of the gravitational potential and the associated acceleration vector lead to the following relations on the relative quantities:

0

0.5

1

1.5

2

2.5

3

3.5

4

ϕ nm = ϕ n,−m = ϕ−n, m = ϕ−n,−m ,

(16)

R Figure 13. Rotation curve: uniform square plate. Shown are the normalized values of V , the circular speed of hypothetical massless particles moving on a circular orbit around the center of√an infinitely thin unit square plate. They are plotted as a function of R ≡ x 2 + y 2 for a couple of azimuthal angles, Θ ≡ atan2(y, x), as Θ = 0 and π/4. The curves (i) start from the zero value, (ii) grow almost linearly with respect to R inside the plate, (iii) blow-up logarithmically at the edge of the plate, (iv) decay approximately in √ proportion to 1/ R, and (v) vanish completely at the infinity. As a result, the location of the blow-up points and the overall feature vary with the direction.

Error of Masspoint Approximation

log10 |(φ-φM)/φ0|

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 0 -1 -2 -3 -4 -5 -6 -7 -8 0.1

a=b, z=0

Θ=π/4 Θ=0

( (

) ( ) ( ) ( ) f x nm = − f x −n, m = f x n,−m = − f x −n,−m ,

fy

) nm

( ) = fy

As a result, ( ) ( ) f x 0m = f y

−n, m

n0

( ) = − fy

n,−m

= 0.

( ) = − fy

−n,−m

.

(17)

(18)

(19)

These relations greatly reduce their computational time. From a theoretical point of view, (i) a special case of the last relation written as ( ) ( ) = 0, (20) f x 00 = f y 00

and (ii) the finiteness of ϕ00 as already viewed in Fig. 12 are important results. This is because they positively resolve the singularity problem in evaluating the internal gravitational field, which is unavoidable even by the masspoint approximation. At any rate, thanks to these symmetry and anti symmetry relations, the computation of the relative quantities reduces to that over one quarter of the index domain as 0 ≤ n and 0 ≤ m. After ( ) this ( ) restriction, all negative are the sign of ϕ nm , f x nm , and f y . nm Also, if a = b as usual, we obtain additional relations as ( ) ( ) ϕ nm = ϕ mn , f x nm = f y . (21) mn

1

10

100

R/a Figure 14. Error of masspoint approximation: uniform square plate, Rdependence. Shown are the normalized absolute difference of the gravitational potential of the uniform√rectangular plate from its masspoint approximation, ϕ M (x, y, z) ≡ 4ab/ x 2 + y 2 + z 2 . The deviations are plotted as a function of R for several values of Θ as Θ = 0, π/16, π/8, 3π/16, and π/4. Hardly visible is the difference between the curves of Θ = 0 and Θ = π/16 at this scale. The azimuthal variations are significant only in a narrow range, say when 3/4 < R/a < 3/2. The sudden drop near R/a = 2/3 corresponds to an accidental coincidence of (i) the potential value of the uniform plate which is usually the larger but remains to be finite at the plate center, and (ii) that of the masspoint approximation which is the smaller almost everywhere but increases infinitely when approaching the plate center.

Thus, the computation is further reduced to that over one octant, say 0 ≤ m ≤ n. Tables 1 and 2 list the numerical values of the non-trivial components of the coefficient matrices. Namely, the tables show ( ) ( ) for the non-trivial components |ϕ nm | /a, f x nm , and f y nm with the first 10 × 10 indices when a = b. As was already seen in Fig. 14, these coefficients are sufficient to assure around 5 digit accuracy of the gravitational potential computation on the object plane if the masspoint approximation is used otherwise. 2.7 Interpolation on object plane A few warnings are necessary in the computation of the acceleration vector on the object plane where z = 0. First, one should understand that Fz , the z-component of the acceleration vector, is not uniquely defined there as explained in Appendix A2. This is because we assumed that the object is infinitely thin, and therefore, Fz is discontinuous at z = 0 with a finite size of the jump despite MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field Table 1. Numerical values of coefficient matrix components. Listed ( are ) the ( ) components of the coefficient matrices, −ϕ n m , − f x n m , and − f y , nm of the uniform square plate for the index range as 0 ≤ m ≤ n ≤ 7. We show 10 effective digits so as to guarantee at least 8 digit accuracy in their implementation. n

m

0 1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7

0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7

−ϕ n m /a

( ) − fx nm

( ) − fy

7.050988696 2.076099472 1.449394876 1.010183966 0.9021779767 0.7109544285 0.6697227404 0.6351095920 0.5565176312 0.4725171613 0.5012949147 0.4862608284 0.4481543392 0.4006751446 0.3540188092 0.4006643231 0.3928602972 0.3719265171 0.3434206555 0.3126675075 0.2830800704 0.3337181952 0.3291678268 0.3165577495 0.2984196404 0.2775735571 0.2562496392 0.2358393268 0.2859568052 0.2830781635 0.2749372454 0.2628019875 0.2482290700 0.2326267260 0.2170372222 0.2021167136

0 1.107846874 0.3831494354 0.2575200798 0.1837886484 0.08987338744 0.1126289344 0.09607965665 0.06464885860 0.03956547684 0.06298378973 0.05748872980 0.04500738136 0.03216415853 0.02218498922 0.04019882623 0.03789598778 0.03215596233 0.02531465928 0.01910456965 0.01417790545 0.02787383600 0.02674926465 0.02379158976 0.01993193960 0.01603988688 0.01261987270 0.009838116343 0.02046006947 0.01984842161 0.01818485636 0.01588169546 0.01338348909 0.01101513301 0.008945666008 0.007224633520

0 0 0.3831494354 0 0.09156980632 0.08987338744 0 0.03198907592 0.04308488459 0.03956547684 0 0.01436577220 0.02249875865 0.02412154021 0.02218498922 0 0.007577686494 0.01286076786 0.01518789284 0.01528336438 0.01417790545 0 0.004457761149 0.007929951156 0.009965538822 0.01069303594 0.01051648621 0.009838116343 0 0.002835329911 0.005195444072 0.006806237293 0.007647573956 0.007867884157 0.007667690040 0.007224633520

nm

Φ itself is continuous. Even though, effective is a practice to return the simple mean of the values evaluated just above and below the discontinuity. Namely, we recommend the interpretation written as Fz (x, y, 0) = 0,

(22)

for arbitrary value of x and y as long as we investigate the motion of test particles on the galactic plane. Caution that this practice is not suitable in studying the vertical motions of massless particles where non-zero z results a finite value of Fz . Next, when z = 0, Fx and Fy encounter with the blowing-up discontinuities at some edges of the unit rectangular plate. This is caused by the uniform plate approximation, which allows the jump of the surface mass density values plate by plate. Needless to say, desirable is a higher order approximation such as assuring the continuity of the density profile. However, its analytical computation would be fairly difficult. Instead, we recommend an interpolation of the obtained values at grid points. Assume that the value of a certain function, f (x, y), is known at the four corners of the unit rectangle. Then, a low precision MNRAS 000, 1–?? (2016)

9

Table 2. Numerical values of coefficient matrix components: continuation. Same as Table 1 but for higher values of n as n = 8, 9, and 10. n

m

−ϕ n m /a

( ) − fx nm

( ) − fy

8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10

0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10

0.2501625375 0.2482283386 0.2426842582 0.2342160896 0.2237235405 0.2120991240 0.2000835969 0.1882137588 0.1768343974 0.2223364104 0.2209751883 0.2170367850 0.2109161879 0.2031566004 0.1943336959 0.1849660738 0.1754679656 0.1661388004 0.1571753430 0.2000832603 0.1990894796 0.1961946901 0.1916385157 0.1857621085 0.1789451586 0.1715512255 0.1638922944 0.1562135198 0.1486931162 0.1414508706

0.01565544787 0.01529513269 0.01429305672 0.01284847399 0.01119791008 0.009541541428 0.008010064524 0.006667379632 0.005529691157 0.01236469662 0.01213898792 0.01150144963 0.01055559996 0.009432926021 0.008256540002 0.007119178697 0.006077807664 0.005159010051 0.004368236764 0.01001248174 0.009864033452 0.009439993252 0.008797488815 0.008012758177 0.007162599373 0.006310914375 0.005502832902 0.004765038654 0.004109443427 0.003537750067

0 0.001911827592 0.003573164695 0.004818078358 0.005598878459 0.005963415261 0.006007523853 0.005833948331 0.005529691157 0 0.001348747885 0.002555830942 0.003518482622 0.004192368006 0.004586935205 0.004746099678 0.004727173695 0.004585782982 0.004368236764 0 0.0009863895181 0.001887975145 0.002639219590 0.003205078109 0.003581279625 0.003786534571 0.003851974369 0.003812026385 0.003698497356 0.003537750067

nm

approximation of f (x, y) is computed by a bilinear interpolation as [ f (x, y) ≈ (x + a)(y + b) f (a, b) − (x + a)(y − b) f (a, −b) −(x − a)(y + b) f (−a, b) + (x − a)(y − b) f (−a, −b)

]

/(4ab), (|x| ≤ a; |y| ≤ b). (23) ( ) ( ) Regarding it as Fx x + x j , y + yk , 0 or Fy x + x j , y + yk , 0 , we may obtain their continuous approximations. This is a simple device but sufficient for the computation of the gravitational filed to be determined from the observational data.

2.8 Density splitting Sometimes, for the given set of the surface mass density at the grid points, Σi j , we may find a simple function to approximate them, ΣA (x, y). Without doubt, this is not always possible. For example, it seems difficult in the case of the HI gas and the blue stars components of M74. If ΣA (x, y) is axially symmetric, then its gravitational field can be computed precisely and quickly (Fukushima 2016). Especially, analytic solutions are known for some simple density profiles as the Kuzmin-Toomre discs (Kuzmin 1956; Toomre 1963) and an exponential disc (Freeman 1970). Refer to Evans & de Zeeuw (1992) for the detailed list of such analytically expressed pairs of the gravitational potential and the surface mass density.

Page 10 of 38

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10

T. Fukushima

Table 3. Averaged CPU time to produce 256 × 256 map of gravitational potential. Listed are the averaged CPU time of two methods to prepare a 256 × 256 map of the gravitational potential from the surface mass density data: (i) the standard N -body computation, and (ii) the new method. The used computer was a PC with Intel Core-i7 4600U CPU running at the clock 2.10–2.70 GHz. method

part

N -body new

computation preparation computation total

CPU time (sec) 19.109 0.008 1.041 1.049

Once ΣA (x, y) is determined, the remaining part of the surface mass density is obtained as ) ∫ y j +b (∫ x i +a ( ) 1 ∆Σi j ≡ Σi j − ΣA x, y dx dy, (24) 4ab y j −b x i −a where the double integration is necessary to conform with the reality of the surface photometry observation. If the gravitational potential corresponding to ΣA (x, y) is evaluated analytically or integrated numerically, then the problem is reduced to the computation of the residual gravitational field caused by ∆Σi j . It is true that this approach may be time-consuming. However, it will be useful when only a portion of the photometry data are available. Also, it will decrease the discretization errors significantly since the errors are in proportion not to Σi j , the surface density matrix itself, but to ∆Σi j , its residual. 2.9

Practical consideration

A large number of the same operations are repeated in the method presented in the previous subsections. Most of them are as simple as the matrix multiplication by the same coefficient matrix. Thus, we recommend the utilization of the "general matrix multiplication" (gemm) in the Basic Linear Algebra Subprograms (BLAS) such as implemented in the Intel Math Kernel Library (MKL) 11.3 (Intel 2015a). Notice that the matrix transform, equation (13), can be conducted in parallel with the indices i and j. Also, there are no interrelation among the computation of the gravitational potential and the rectangular components of the acceleration vector. Therefore, the transform is quite suitable for the parallel/vector computations if the coefficient matrix is prepared beforehand. Even in ordinary PCs, we can conduct a low level parallel computation by using some compilers designed to handle the automatic parallelization/vectorization such as recent Intel C++/Fortran compilers as the Intel Fortran Compiler 16.0 (Intel 2015b). Also, effective is the maximum utilization of the Single Instruction Multiple Data (SIMD) feature of recent CPUs as implemented by the second generation Advanced Vector Extension (AVX2) instruction set of recent Intel CPUs (Intel 2016). Refer to an example of the Fortran compiler options actually used in the computation of the gravitational field of M74 described later in section 3.3. Table 3 compares the averaged CPU times to prepare a 256 × 256 matrix of the gravitational potential by using the standard Nbody formulation and by using the new method. The CPU time ratio of a factor 20 shown in Table 3 comes from the difference in the computational amount of the summand in the innermost loop. In fact, the N-body formulation requires 2 additions, 2 subtractions, 2 multiplications, 1 division, and 1 square root. Meanwhile, the new

method requires 1 multiply-and-add operation, which is efficiently executed in modern computer chips enabling the Fused MultiplyAdd (FMA) operations (Intel 2016).

2.10 Error estimation Let us examine the errors of the mosaic tile model. First, there are no truncation errors in a theoretical sense. This is because the model uses the exact expressions of the gravitational field of an infinitely thin uniform rectangular plate. Also, the round-off errors are sufficiently small. This is because the model analytically evaluates the gravitational field, which is expressed in elementary functions like the logarithm, the arctangent, and the square root. Of course, the number of mass elements may be large, say 106 pieces or more. However, the computation in the IEEE745 double precision environment assures that the computational precision of the gravitational field is 13 digits or so since the precision loss due to the accumulation of the round off errors is at most 3 digits. Meanwhile, the systematic error may be introduced by an incorrect estimate of the sky background removed from the processed surface mass density data. Fortunately, a constant offset causes no trouble in this case because physically meaningful is not the potential value itself but its gradient.

2.11 Discretization error The rest possibility of the error source is the discretization effect in the original observational data. Since it is beyond the control of the post-process including the application of the mosaic tile model to extract the information on the gravitational field, we try to estimate its effect by a simulation. The theoretical estimation by using an artificial density distribution is not appropriate for the present purpose because unknown is the characteristic of the real distribution of the surface mass density. Thus, using a set of the actual data, we conduct a simulation to measure the effect of discretization below. More specifically speaking, from the 1024 × 1024 high resolution profile of the HI gas surface mass density as already shown in Fig. 1, we obtain a couple of data sets with lower resolution by averaging the 3 × 3 or 5 × 5 block data. The resulting middle and low resolution data sets are similarly processed by the method described in the previous subsections and compared with the result of the high resolution data set. Figs 15 shows the contour map obtained from the gravitational potential determined from the middle resolution profile of the surface mass density. Qualitatively, it is difficult to see its difference from the high resolution result already depicted in Fig. 5. Meanwhile, Fig. 16 indicates that the contour map of the gravitational potential determined from the low resolution data is so smoothed that the information on a small scale structure seems to be lost. Nevertheless, the global picture is not so degraded. On the other hand, Figs 17 and 18 provide the overlapped cross sections of the difference in the determined gravitational potential and the α-component of the associated acceleration vector between the middle and high resolution data sets, respectively. The plotted values are normalized by the maximum absolute value of the high resolution result. Quantitatively, they show that the maximum deviation of the middle resolution determination is 1.3% and 13% for the gravitational potential and the acceleration vector, respectively. Also, Table 4 compares the statistics of the difference between MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field

Contour Map of Φ: M74, HI 10

10

5

5

0 -5 -10

0 -5 -10

-10

-5

0

5

10

-10

-∆α (arcmin)

Table 4. Statistics of difference between lower and high resolution determinations. Listed are some statistical indices of the relative difference of the gravitational potential and the α component of the acceleration vector for the HI gas component of M74 between (i) that determined from the 1024 × 1024 high resolution data set with 1.5 arcseconds resolution and (ii) those from two sets of lower resolutions. The listed differences are normalized by the maximum absolute value of the higher resolution result. Omitted is the result of the δ component of the acceleration vector since it is mostly the same as that of the α component. item

δ F−α

average standard deviation maximum minimum average standard deviation maximum minimum

MNRAS 000, 1–?? (2016)

low resolution

middle resolution

7.5′′

4.5′′ 341 × 341 +9.73E−4 +1.56E−3 +1.32E−2 −1.99E−4 −3.59E−6 +8.59E−3 +1.21E−1 −1.04E−1

204 × 204 +1.96E−3 +3.14E−3 +2.47E−2 −2.63E−4 +6.25E−7 +1.72E−2 +2.19E−1 −1.98E−1

-5

0

5

10

-∆α (arcmin)

Figure 15. Contour map of gravitational potential on plane: M74, gas component, middle resolution. Same as Fig. 5 but determined from the middle resolution (4.5 arcseconds per pixel) profile of the surface mass density, which was obtained by averaging every 3 × 3 pixels of the original data. As a result, the number of the determined potential is reduced to 341 × 341. Despite this reduction, a comparison with the higher resolution result, Fig. 5, indicates that the global feature of the gravitational potential is not so significantly degraded.

resolution grid points δΦ

11

Contour Map of Φ: M74, HI

∆δ (arcmin)

∆δ (arcmin)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 16. Contour map of gravitational potential on plane: M74, gas component, low resolution. Same as Fig. 5 but determined from the low resolution (7.5 arcseconds per pixel) profile of the surface mass density, which was obtained by averaging every 5 × 5 pixels of the original data. As a result, the number of the determined potential is reduced to 204 × 204. Still noticeable is a spiral structure.

the middle and low resolution results and the high resolution determination. They suggest that, in the statistical sense, the gravitational field determined from the middle resolution data is of the accuracy of 3 and 2 digits for the gravitational potential and the acceleration vector, respectively. An extrapolation of the error estimate is somewhat dangerous. However, from these simulation of the data averaging and the associated gravitational field determination, we may conclude that the gravitational field determined from the 1024 × 1024 grid data of the surface mass density profile is of the accuracy of 3.5 and 2.5 digits for the gravitational potential and the acceleration vector, respectively. When considering of the observational error of the surface photometry itself, these levels of accuracy seem to be sufficient for the analytical purpose. 2.12 Effect of thickness So far, we have considered the gravitational field of a general object under the condition that it is infinitely thin and extended on a flat plane. This is a crude approximation of actual astronomical objects. For example, some disc components of the spiral galaxies have significantly finite thickness. Refer to Robin et al. (2014) showing a realistic example: the Milky Way. They conclude that (i) the vertical profile of the thick disc component of the Milky Way is not exponential but in proportion to the squared secant hyperbolic function, and (ii) the ratio of the scale height and the scale length of the thick disc component of the Milky Way is 0.47 kpc/2.3 kpc ≈ 0.2. Then, arises a natural question about the effect of the thickness in

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T. Fukushima

Discretization Error of Φ: M74, HI

Discretization Error of F-α: M74, HI

1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2

15 10 100 ∆F-α/F0

100∆Φ/Φ0

0 -5

-15 -12 -9

-6

-3

0

3

6

9

12

-12 -9

-∆α (arcmin)

the result obtained under the assumption of infinitely thin objects, especially on the x-y plane. This is indeed a difficult question. In order to answer it correctly, one must know the gravitational field of general threedimensional objects. However, its investigation is far beyond the scope of the present article. Thus, let us roughly estimate the effect by assuming a functional form of the vertical profile of the object. Imagine a pillar extended infinitely. Assume that it is located at the coordinate origin, is of a rectangular cross section of the size 2a × 2b and has the vertical profile of its volume mass density described as Σ 4h cosh2 [z/(2h)]

,

-6

-3

0

3

6

9

12

-∆α (arcmin)

Figure 17. Discretization error distribution of gravitational potential: M74, gas component, middle resolution. Plotted are the difference of Φ, the gravitational potential value, between the high (1024 × 1024) and the middle (341 × 341) resolution computations. The residuals are normalized by Φ0 , the maximum absolute value of the potential values. The lower resolution data lead to the smaller values of the magnitude of the gravitational potential. The mean value and the standard deviation of the residuals scaled by Φ0 are as small as 0.00097 and 0.00156, respectively.

ρ(z; h) =

5

-10

(25)

where Σ is a nominal constant of the dimension of the surface mass density and h is a scale height. The normalization factor 1/(4h) is selected such that, by assuming that the object is optically thin, the vertically integrated volume mass density, which is of the dimension of the surface mass density, is equal to Σ as ∫ ∞ ρ(z; h)dz = Σ. (26) −∞

Refer to Fig. 19 showing the curves of ρ(z; h) for some values of h as h = 1/4, 1/2, 1, 2, and 4. For this hypothetical pillar, consider the difference in the gravitational field on the x-y plane between (i) that caused by an infinitely thin uniform tile of the surface mass density, Σ, and (ii) that caused by the pillar of the volume mass density, ρ(z; h). As an illustration, let us consider the difference in the gravitational potential. The potential of the tile on the x-y plane is already obtained as ϕ∗ (x, y) ≡ ϕ(x, y, 0). Meanwhile, that of the pillar is expressed as

Figure 18. Discretization error distribution of acceleration vector: M74, gas component, middle resolution. Same as Fig. 17 but for F−α , the αdirection component of the acceleration vector. This time, the mean value and the standard deviation of the residuals scaled by F0 , the maximum absolute value of the acceleration vector, become 6.3 × 10−7 and 0.0086, respectively.

Vertical Profile of Model Stellar Disc

ρ/ρ0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.2 1.1 ρ(z)=sech2[z/(2h)]/(4h) 1 0.9 h=1/4 0.8 0.7 0.6 0.5 0.4 0.3 1/2 0.2 1 2 0.1 0 -10 -8 -6 -4 -2 0 2 4

4 6

8 10

z Figure 19. Vertical profile of model stellar disc. Shown is the vertical profile of the volume mass density of a model stellar disc. Plotted are several curves of squared secant hyperbolic functions for several values of the scale height h as h = 1/4, 1/2, 1,2, and 4. The normalization factor 1/4h is multiplied so as to make the areas of the integrated profiles unity. A thick curve shows the case h = 1.

an integral transform with respect to z as ∫ ∞ ϕ(x, y, z) dz. ϕ∗ (x, y; h) ≡ 2 −∞ 4h cosh [z/(2h)]

(27)

Since this integral transform is not analytically obtained in a closed form, we must evaluate it by a numerical integration. Fig. 20 compares the normalized values of ϕ∗ (x, 0; h) as funcMNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field

Effect of Thickness

Φ/|Φ0|

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1

h/(2a)=16

h=0

0

1

2

3

4

5

6

7

8

9 10

R/(2a) Figure 20. Effect of thickness. Shown is the gravitational potential of a pillar extended infinitely in the vertical direction with a square cross section of 2a × 2a. The normalized potential values along the x-axis are plotted as functions of R = x for some values of the scale heights as h/(2a) = 0, 1/4, 1/2, 1, 2, 4, 8, and 16. The case h = 0 is nothing but the infinitely thin tile.

13

galaxies (O’Brien et al. 2010a,b,c,d). The flaring is nothing but a radial increase of the scale height. The observed functional forms of the radially increasing manner are various: linear, cubic, or more rapidly growing, especially near the outer edge of the HI disc. Fig. 16 of O’Brien et al. (2010c) shows the case of UGC7321 where h increases from 1 to 4.5 kpc when R does from 8 to 11 kpc. If the functional form of the vertical profile is specified, such as the squared secant hyperbolic functions discussed in the previous subsection, it is possible to take into account the effect of flaring by replacing the tile model with an adequate pillar model. This time, nevertheless, the computational burden will be much larger because the vertical integration must be conducted for each pixel. This means that the coefficient matrices must be prepared for every different scale heights used. It would require a significant increase in the CPU time, say the multiplication of another factor, 100–1000, depending on the number of pixels with different scale heights. We have no specific idea to fulfill such a huge scale computation within a reasonable amount of CPU time. Therefore, we will leave this issue as a future problem to be solved for.

3 ANALYSIS OF GRAVITATIONAL FIELD OF M74 3.1 Surface photometry data

tions of R = x for some values of h as h/(2a) = 0, 1/4, 1/2, 1, 2, 4, 8, and 16. If we dare combine the thickness of 0.47 kpc of the thick disc component of the Milky Way with the 60 pc pixel size for the R-color photometry of M74, we obtain an estimate of thickness parameter, h/(2a) = 470/60 ≈ 7.8. The numerical integrations were conducted by Ooura’s intdei (Ooura 2006), an excellent implementation of the double exponential quadrature rule for a semi-open interval (Takahashi & Mori 1973). The case h = 0 is nothing but the infinitely thin square tile. Fig. 20 indicates that the potential curves are essentially the same when R ≥ 6a, namely except the central 3 × 3 pixels. This similarity is almost independent on the value of h. On the other hand, the magnitude of the potential values in the central region, namely within the central 3 × 3 pixels, significantly decreases when h increases. This is natural because the vertical extension of an object means a dilution of the spatial mass density distribution inside it. Therefore, it results a decrease of the gravitational attraction on the x-y plane since the total mass is kept the same. In other words, the finite thickness of a disc-like matter relaxes the singularity problem of the potential computation inside it. Since the observed difference is prominent only in a few central pixels, one may easily apply an appropriate correction of the finite thickness to the results computed under the infinitely thin condition. At any rate, the replacement of infinitely thin tiles with the corresponding vertically extended pillars significantly increases the total computation time, say 60–100 times more, in the stage of the coefficient matrix preparation. In a sense, the introduction of the pillar model means the execution of a volume integral over the vertically extended object. Although this direction is worthwhile to be examined further, we shall stop the discussion here.

2.13

Flaring of HI disc

It is well known that the HI gas component of disc galaxies are flaring (van der Kruit 2008). See some examples of edge-on disc MNRAS 000, 1–?? (2016)

In order to present a realistic example of the method described in section 2, we determined the gravitational fields of the HI gas, dust, red stars, and blue stars components of M74. More specifically speaking, we used the following image data files: (i) NGC_628_NA_MOM0_THINGS.fits quoted from the WEB site of the THINGS data archive, http://www.mpia.de/THINGS/Data.html, for the HI gas component, (ii) 2MASS_MESSIER_074_K.fits retrieved from the NASA/IPAC Extragalactic Database (NED), http://ned.ipac.caltech.edu/, for the dust component, (iii) MESSIER_074_I_103aE_dss1.fits also retrieved from the NED for the red stars component, and (iv) GI3_050001_NGC628-fdint.fits among a package data file extracted from the Mikuluski Archive for Space Telescope (MAST), https://archive.stsci.edu, for the blue stars component. We selected these data files among many candidate images of M74 found in the NED for various wavelength so as to examine four typical cases: (i) a significantly non axisymmetric object, (ii) an almost axisymmetric disc, (iii) a mixture of axisymmetric disc and a spiral feature, and (iv) bare spiral arms. The first set of the data is the 21cm radio intensity map of the galaxy acquired by the THINGS project using the NRAO Very Large Array (Walter et al. 2008). The second is the K-band picture produced by the 2MASS survey (Jarrett et al. 2003). The third is the red color photo taken by the Palomar 48-inch Schmidt telescope with a Kodak 103aE photographic plate and digitized by the PDSS1 project (Kent 1994). The last is the Far Ultra Violet (FUV) image file obtained by the GALEX mission (Martin et al. 2005). All of these data sets contain the values at evenly spaced grid points in the celestial reference frame, namely in the coordinates of the right ascension and the declination. The grid points are aligned with the same angular separations in both directions, i.e. 1.5, 2.5, 1.7, and 1.5 arcseconds for the HI gas, dust, red stars, and blue stars components, respectively. If we adopt the value of 7.3 Mpc as the distance to M33 (Karachentsev et al. 2004), these separations correspond to the linear scale of 53, 88, 60, and 53 pc, respectively.

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Blanc et al. (2013) estimated the inclination of M74 as 8.7 ± 0.7◦ . Ignoring this small angle, we transformed the observed surface photometry data into the surface density matrices. More specifically speaking, we set the x and y coordinates as those in the directions of decreasing right ascension and increasing declination. Namely, we regard x = −∆α ≡ α0 − α and y = ∆δ ≡ δ − δ0 , where the suffix 0 means the tentative center of the surface density data matrix. Let us describe the details of the transformation processes. Begin with the radio image. We first cut off the negative valued data by noting that the raw values span from −4.27 to 218.35. Next, for the infrared image data, we limited their range to the central 144 × 144 pixels covering the 3 ′ × 3 ′ area. This was in order to exclude an intense spotty feature located near the galaxy. Also, we filled zero values for a small portion of area where the data are missing. Meanwhile, the raw values of the red color image data, which are integers, range from 1073 to 12987. Thus, we first adopted a threshold value representing the sky component as 3000 and subtracted it. As we noticed earlier, an error in the threshold value has no physical effect in the determination of the gravitational field. Next, after an examination of the processed image, we selected only the central portion containing 256 × 256 pixels, which amount to roughly one quarter of the original data. This was to exclude the point sources, which are apparently not belong to the galaxy, as many as possible. Although there remain some spotty features, we did not filter out them since their point spread functions are unknown. Finally, the data set of the FUV image is arranged such that all the values are positive definite. Also, the data are fairly sparse, namely most of the pixel values are simply zero. Therefore, we selected its central portion covering 512 × 512 pixels. Consequently, the transformed sets of the surface mass density data consist of 1024 × 1024, 144 × 144, 256 × 256, and 512 × 512 elements, respectively. Notice that the precision of the data values is not so high, say 2-4 digits at most. Therefore, acceptable is the ignorance of the small inclination angle, the effect of which is at most of the order of 1% as 1 − cos 9◦ ≈ 0.012.

3.3

Scalability of CPU Time

Transformation to surface mass density

Determination of gravitational field

Noting the low precision of the data values, we determined the gravitational fields caused by the obtained surface mass density profiles without fitting any model function to them. Also, in order to avoid the unnecessary degrade in the determination process, we did not use the masspoint approximation at all. These decisions made the following computation sufficiently precise. Needless to add, the computational labor increased, especially in the preparation of the coefficient matrices. However, this was only a small increase in the total CPU time when compared with the main part, namely the matrix transform to obtain the whole gravitational field on the galactic plane. Refer to Table 3 for the case of the red stars component. The actual computation was executed at an ordinary PC under the Windows 7 OS with an Intel CPU chip named Core-i7 4600U running at the clock 2.10–2.70 GHz. All the programs are written in Fortran 90 and compiled by the Intel Visual Fortran Composer XE 2011 or ifort 12.1 for Windows OS with the following major compiler options: /O3 /Qparallel /Qpar-threshold:10 /Qvec-threshold:10

4096 1024 256 64 16 4 1 0.25 0.0625

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Φ, Fx, Fy Φ

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J Figure 21. Scalability of CPU time. Plotted are the CPU times in seconds as a function of J , the number of one-dimensional grid points. Shown are the curves of two cases: (i) the computation of Φ only, and (ii) the simultaneous computation of Φ, F x , and Fy .

/Qopt-prefetch=3 /assume:buffered_io /Qipo /Qopt-matmul /libs:static /Qmkl:parallel /QxHost

These options (i) maximize the optimization of the local instructions, (ii) activate the parallel/vector computation, (iii) set the threshold width of the parallel computation as low as 10, (iv) choose the threshold length of the vector computation as low as 10, (v) maximize the pre-fetch of long instructions, (vi) assume the buffered I/O operations, (vii) optimize the inter-procedure calls, (viii) enable the code optimization for the matrix multiplication, (ix) incorporate the library as a static image in the memory space, (x) utilize the Intel MKL, and (xi) invoke the optimal SIMD instruction set being most suitable for the CPU on board, which is the Intel AVX2 for this Haswell generation CPU including FMA3, the Intel’s 3 operand FMA operation. Fig. 21 shows the CPU times of the new method required in producing the gravitational field map of the four components of M74 as a function of J, the number of one-dimensional grid points: 144, 256, 512, and 1024. Plotted are the curves of two cases: (i) the computation of Φ only, and (ii) the simultaneous computation of Φ, Fx , and Fy . The CPU time is roughly in proportion to J 4 when J is small and J 5 when J is large. At any rate, the actual computational time is not so large as one may anticipate. In fact, if J is 256, then a few seconds is enough to produce the gravitational field map even if using an ordinary PC. This quickness is mainly achieved by the full utilization of the computational resources of the PC, namely all the cores and all the available threads of the incorporated CPU, by realizing a small-scale parallel computation. 3.4 Bird’s-eye view of gravitational potential on plane Although the resolution may be low, bird’s-eye views are helpful to have an idea of two-dimensional curves in general. Therefore, we illustrated some of them to describe roughly the computed gravitational potential of M74 as depicted in Figs 22–26. Refer to Appendix C for the tools we used in preparation of the figures. By linearly combining these potential values with a sort of weight factors, one may obtain a model of the total gravitational MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field

Gravitational Potential: M74, HI

Gravitational Potential: M74, HI

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Figure 22. Bird’s-eye view of gravitational potential on plane: M74, gas component. Presented is a bird’s-eye view of the scaled gravitational potential of the HI gas component of M74 computed by the mosaic tile model. The corresponding contour map has already been shown as Fig. 5. The central region is so complicated that it is not a simple basin but a hill surrounded by a ring shaped valley.

Figure 23. Bird’s-eye view of gravitational potential on plane: M74, gas component, central region. Same as Fig. 22 but magnified is its central region. A single bumpy valley seems to spiral downward in a clock wise manner.

potential of the galaxy. If necessary, one may add the contributions of spherical and other three-dimensional mass distributions like the dark matter halo, the central black hole, or the budge. A test particle with the zero initial velocity vector on the galactic plane will move on an orbit as if it is falling down the valleys depicted in these figures. Imagine a pinball rolling up and down a bumpy hill like that displayed in Fig. 23. Although this may be a too simplified picture, it will bring a kind of idea on the global behavior of the motion of gas and stars inside the galaxy.

Gravitational Potential: M74, K

Φ/Φ0 0 -0.5

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-1 3.5

Contour map of gravitational potential on plane

A closer examination is possible through the contour maps of the gravitational potential. Figs 5–8 have already shown the contour maps corresponding to some of the bird’s-eye views introduced in the previous subsection. Here we show Figs 28 and 29 depicting the contour maps corresponding to the closer bird’s-eye view shown in Figs 23 and 27, respectively. Additional views are given in Appendix D. Comparing with the corresponding contour maps of the surface mass density, one may notice that those of the gravitational potential are more smooth and robust against the local features as point-like clumps. Refer to Figs 1–4 with Figs 5–8 again. This is because the potential computation is, in any way, a sort of smoothing operation with a clear physical meaning. In fact, even for a significantly bounded, discrete, and discontinuous profile of the surface mass density such as that of a uniform square plate, the resulting MNRAS 000, 1–?? (2016)

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Figure 24. Bird’s-eye view of gravitational potential on plane: M74, dust component. Same as Fig. 22 but for the dust component. The corresponding contour map has already been shown as Fig. 6. The axially symmetric infundibulum is somewhat skewed by a nearby spotty dip. Notice that the normalization constant, Φ0 , is different from that used in drawing Figs 22 and 23.

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Gravitational Potential: M74, R

Gravitational Potential: M74, FUV

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Figure 25. Bird’s-eye view of gravitational potential on plane: M74, red stars component. Same as Fig. 22 but for the red stars component. The corresponding contour map has already been shown as Fig. 7. This time, the concave feature is shallower than that of the dust component. Also, a weak trench is noticeable in the south part. Again, Φ0 is different from that used in drawing Figs 22 –24.

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Figure 27. Bird’s-eye view of gravitational potential on plane: M74, blue stars component, central region. Same as Fig. 26 but magnified is its central region. Each spot is deeply dropped. Despite the central region contains no mass, the potential shapes a shallow basin there. This is due to the collective gravitation of several spotty features.

Contour Map of Φ: M74, HI

Gravitational Potential: M74, FUV

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Figure 26. Bird’s-eye view of gravitational potential on plane: M74, blue stars component. Same as Fig. 22 but for the blue stars component. The corresponding contour map has already been shown as Fig. 8. The view exhibits an asymmetric feature of trains of spots. Once again, Φ0 is different from those used in drawing Figs 22–25.

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Figure 28. Contour map of gravitational potential on galactic plane: M74, gas component in the central region. Same as Fig. 23 but extracted its contour map. The contours are drawn at every 0.5% of the peak value. Two close spiral arms spiralling inward in the clock wise sense seem to merge in the norther part of the central hill.

MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field

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Vector Map of F: M74, HI

Contour Map of Φ: M74, FUV 10

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-∆α (arcmin) Figure 29. Contour map of gravitational potential on galactic plane: M74, blue stars component in central region. Same as Fig. 28 but for the blue stars component. The single train of strong spots seems to coincide with one of the arm features viewed in the gas component contour map, Fig. 28.

gravitational potential is continuous and extended infinitely in a more smooth manner as already indicated in the previous section. It should be stressed that this physically-meaningful smoothing is effective in enhancing a global feature of the galaxy such as its spiral structure. For example, the contour maps will be useful in counting the number of spiral arms and the detection of their fine structure like the merger of spiral arms, the branching into sub arms, the flocculent structure, or the irregularities. Refer to Fig. 5 once again. This map indicates that it is difficult to judge whether (i) there exist two different arms, or (ii) a single strong arm splits into multiple sub-arms. On the other hand, Fig. 8 manifests that the massive clumps visible in the FUV band show a significantly asymmetric feature of the gravitational potential of the blue stars component. If this feature associates with the spiral arm structure, the significant arms may be formed from a branching of a single strong arm in the outer region.

3.6

5

Vector map of acceleration on plane

Let us examine the gravitational field more directly. For this purpose, we prepared Figs 30–37 showing the vector maps of the acceleration vector on the galactic plane for various components of M74. Additional maps are provided in Appendix E. These manifest that the determined gravitational fields are fairly complicated. Also, the behavior of the gravitational field is quite different component by component. This is especially true in the central region, namely a region when |x| ≤ 3 ′ and |y| ≤ 3 ′ . For example, Figs 32–35 illustrate interesting features at the galactic center: an x-mark in the dust component, and a slightly MNRAS 000, 1–?? (2016)

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-∆α (arcmin) Figure 30. Acceleration vector map: M74, gas component. Plotted is the vector map of the acceleration vector due to the HI gas component of M74 on the galactic plane. In order to show the feature clearly, we selected only a portion of the computed results, namely the vectors at 64 × 64 grid points with the 24 arcseconds separation. The vectors seem to be randomly oriented in the central region of the size of 3′ × 3′ .

rotated eight-point compass rose in the red stars component. These are not artifacts. Indeed, they correspond to a square or octagon like mass concentration near the galactic center as shown in their detailed surface mass density maps, Figs 38 and 39. See also Fig. 40 showing a bird’s-eye view corresponding to Fig. 39. Interestingly, the combination of Fig. 34 and 35 seems to suggest that the extension of the compass rose feature, which is originally of eight arms in the core region, evolves into four arms with a significant asymmetry. Meanwhile, Fig. 30 illustrates that the acceleration vector caused by the HI gas component looks randomly oriented in the central region. More specifically speaking, the vectors tend to direct toward the nearby line-like filament or point-wise clump. This is because the HI gas component of M74 lacks a central concentration (Bahe et al. 2016). Therefore, not the global but local feature of the mass distribution seems to govern the gravitational field caused by the HI gas component in the central region. On the other hand, in the outer regions, the acceleration vector of the HI gas component mainly directs toward the galactic center. This situation is almost the same as those of the other vector maps depicted in Figs 32–36. Nevertheless, except the case of the dust component shown in Fig. 32, it is also true that the magnitude of the non-radial component and the total strength of the acceleration vector are significantly perturbed by the nearest arm-like feature observed in the potential contour maps. Refer to Figs 5, 7, and 8 once again. The pattern of these vector maps rather faithfully follows that of the spiral arms.

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Central Vector Map of F: M74, HI

Core Vector Map of F: M74, K

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Figure 31. Acceleration vector map: M74, gas component, central region. Same as Fig. 30 but for the central region, |∆α | < 3′ and |∆δ | < 3′ . Detectable are many filamentary features following a few spiral arms. This may imply the existence of an internal structure of the spiral arms.

Vector Map of F: M74, K 3 2 ∆δ (arcmin)

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Figure 33. Acceleration vector map: M74, dust component, core region. Same as Fig. 32 but in the core region of the size 1′ × 1′ . A spot in the south part at a distance of 0.4′ may be a foreground star. More visible is the detailed feature of the x-mark. It seems to consist of 8 groups of the acceleration vectors: (i) 4 groups of the stronger acceleration vectors which are roughly in the directions of (2n − 1)π/4 for n = 1, 2, 3, and 4, and (ii) 4 others of the weaker acceleration vectors which are roughly in the directions of nπ/2 for n = 1, 2, 3, and 4. This is probably caused by the central mass concentration indicating square like contours, which were already noticeable in Fig. 2 and will be clearly shown in Fig. 38 later.

3.7 Cross section of gravitational field on plane

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-∆α (arcmin) Figure 32. Acceleration vector map: M74, dust component. Same as Fig. 30 but for the dust component. An x-mark feature is noticeable.

A quantitative research may be possible by examining the radial variation of the surface mass density profile and the gravitational field. As an illustration, we prepared Figs 41–44 showing their cross sections along the east-west direction. Additional cross sections along the north-south direction are found in Appendix F. At any rate, these graphs exhibit the curves along a certain straight line on the galactic plane, which is roughly passing through the galactic center. Although the plotted curves illustrate only the feature along the line of a constant δ, we confirmed that the depicted feature is more or less the same for other directions. See the similar cross section views in Appendix F. It is quite interesting that the gravitational potential due to the HI gas component exhibits a hill-like feature in the central region as seen in Fig. 41. Also, the associated curves of the acceleration vector components, especially that of Fα , are not monotonically varying but heavily oscillating. The same feature was experienced in the previous study of the gravitational field of infinitely thin axisymmetric objects when the object has a non-negligible central hole and/or the radial surface mass density is wavy (Fukushima 2016). Refer to the cross section of the uniform rectangular plate shown in Fig. 12. Based on the experience with this model case, we may regard that one oscillation, namely a pair of rise and fall, of the MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field

Vector Map of F: M74, R

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acceleration vector component curve corresponds to a similar oscillation of the surface mass density. If the wavy feature continuously evolves inward or outward with respect the azimuthal angle, it may represent a gravitationally meaningful spiral arm. Thus, counting the number of such oscillations in the acceleration vector components may result counting the number of the dynamically determined spiral arms. If this interpretation is true, we arrive at a conclusion that a heavy oscillation of Fα component of the HI gas component indicated in Fig. 41 suggests that, in the middle region of the size of 6 ′ × 6 ′ , the main spiral arms of the width of 1–2 ′ seem to contain 3-4 sub arms of narrower width of 0.5 ′ or less. At any rate, this approach is physically meaningful and more robust against the local fluctuation of the surface photometry data. We stored all the determined values of the gravitational potential and the associated acceleration vector at the grid points of the surface mass density data at an electronically-accessible archive. Therefore, one may construct a cross section view along any direction from the computed data after, if necessary, applying an interpolation process as described in section 2.7.

Direction dependent rotation curves

Once Fα and Fδ , the acceleration vector in rectangular coordinates, are known, its radial and azimuthal components are simply computed as FR = (∆αFα + ∆δFδ ) /R, FΘ = (∆δFα − ∆αFδ ) /R. MNRAS 000, 1–?? (2016)

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Figure 34. Acceleration vector map: M74, red stars component. Same as Fig. 30 but for the red stars component. A spiral structure is clearly seen. Several locally large vectors are associated with the clumps viewed in Fig. 3. Noticeable is a weak x-mark feature at the galactic center.

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Figure 35. Acceleration vector map: M74, red stars component, core region. Same as Fig. 34 but in the core region of the size 1′ ×1′ . The viewable feature is not an x-mark but an eight-point compass rose. It seems to consist of four types of the acceleration vectors: (i) 8 groups of the strong acceleration vectors aligned roughly in the directions of of (2n − 1)π/8 for n = 1, 2, . . . , and 8, (ii) 4 groups of the acceleration vectors with a medium magnitude and a narrow width oriented in the directions of (2n − 1)π/4 for n = 1, 2, 3, and 4, (iii) 2 groups of the weak acceleration vectors and of a wider width, which are along the north-south direction, and (iv) 2 groups of the acceleration vectors and of the weakest strength but of a large width along the north-south direction. Also, noticeable is an almost vanishing acceleration near the center. This may be caused by an octagon like plateau of the surface mass distribution in the core region. It was already detectable in Fig. 3 and will be easily viewed in Figs 39 and 40 later.

If FR < 0, then the circular speed is automatically obtained from FR as √ V = −RFR . (29) However, this definition is meaningful only if FR < 0, namely when the radial component of the acceleration vector on the plane is toward the coordinate center. Also, there is a practically minute but theoretically important fact that the definition of the galactic center is not unique. Indeed, not the same are the following three definitions: (i) the barycenter which can be uniquely determined but requiring the observation of the whole object and a careful removal of foreground and/or background objects not belonging to the galaxy, (ii) the point of the maximum surface mass density which is easily judged from the surface photometry if a central concentration is significant, and (iii) the location of the local maximum of the absolute value of the gravitational potential. The last point is most important in converting the determined gravitational field into the rotation curves. Indeed, without a prominent central mass concentration, it is practically erroneous to determine the local optimum of the gravitational potential near the central region of the galaxy.

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Vector Map of F: M74, FUV

Contour Map of Σ: M74, K

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Figure 38. Contour map of surface mass density: M74, dust component, core region. Same as Fig. 2 but enlarged is its core part of the size of 1′ × 1′ . A spot located at (−∆α, ∆δ) ≈ (−0.12′, −0.44′ ) may be an artifact by a foreground star. The contours are significantly different from being circular. Especially, the outermost one is rather square like.

Central Vector Map of F: M74, FUV 3 2 ∆δ (arcmin)

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3

-∆α (arcmin) Figure 37. Acceleration vector map: M74, blue stars component, central region. Same as Fig. 31 but for the blue stars component.

Bearing these general cautions in mind, we dare compute the direction-dependent rotation curves, namely the circular speed defined in equation (29), plotted as functions of R while fixing Θ. Refer to Figs 45– 48 showing such rotation curves of the HI gas, dust, red stars, and blue stars component of M74 for different directions, namely Θ = 0, and π. Additional rotation curves are displayed in Appendix G. In the case of the HI gas component, almost zero values of the curves in the central region, say within the distance R < 3 ′ , support the existence of a central lack of the HI mass. Meanwhile, in the outer region namely when 3 ′ < R < 10 ′ , the rotation curve is rather flat. This was unexpected. Of course, the curves exhibit wavy feature. The location of peaks and valleys do depend on the direction. Obviously, this is caused by the spiral arms. Especially, the two arms seem to have produced the highest peaks of the circular speed at R = 4.7 ′ when Θ = 0, and R = 5.9 ′ when Θ = π in Fig. 45. In the case of the dust and red stars components, the central concentration of the surface mass density results an initially rising feature of the rotation curves. However, the manner of the concentration is so different that the sharpness of the rising also differs. In fact, the rotation curve of the dust component soon arrives at a plateau while that of the red stars one only gradually rises. At any rate, it is surprising that the rotation curves of the dust component is rising. This means that the decreasing manner of its surface mass density profile is slower than 1/R, which results a flat rotation curve as Mestel (1963) predicted. See also Fukushima (2016, section 4). The peaks and valleys in the rotation curves correspond to the inflexion points of the gravitational potential. In other words, they MNRAS 000, 1–?? (2016)

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21

Cross Section: M74, HI

Contour Map of Σ: M74, R

8 6

0.4 ∆δ (arcmin)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4 2

0.2

0

0

-2 -4

-0.2

∆δ=0

Σ+Σ0 -Fα Fδ Φ

-6 -12 -9 -6 -3

-0.4

0

3

6

9 12

-∆α (arcmin)

-0.4 -0.2 0 0.2 -∆α (arcmin)

0.4

Figure 39. Contour map of surface mass density: M74, red stars component, core region. Same as Fig. 38 but for the red stars component. This time, the central density contours resemble polygons as an inequilateral pentagon or hexagon.

Surface Mass Density: M74, R Σ/Σ0 1.2 1 0.8 0.6 0.4

Figure 41. Cross section on plane: M74, gas. Plotted are (i) Σ, the surface mass density, (ii) Φ, the gravitational potential, and (iii) Fα and Fδ , the α- and δ-components of the acceleration vector, as functions of ∆α when ∆δ = z = 0. The curve of Σ is raised by a constant offset so as to avoid overlapping with the others. The units of the plotted values are arbitrary chosen while the same unit is adopted for two components of the acceleration vector. Significant is a heavy oscillation of Fα associated with the similar feature in Σ. This may suggest that the spiral arms may contain several filaments or sub arms. Interesting is an almost flat structure of Φ when |R | < 4′ . Rather a hill-like feature of Φ is observed near the center.

also correspond to the mass concentration such as the spiral arms and/or giant HI gas clouds. In fact, a pair of adjacent peak and valley in the acceleration components implies the mass concentration between them. Refer to Fig. 48 and compare it with Fig. 44.

3.9 Fitting to observed rotation curve

-0.4 -0.2

0 0.2 -∆α (arcmin)

0.4

0.4 0 0.2 -0.2 -0.4 ∆δ (arcmin)

Figure 40. Bird’s-eye view of surface mass density: M74, red stars component, core region. Same as Fig. 39 but for the corresponding bird’s-eye view. Clearly visible is a plateau near the galactic center.

MNRAS 000, 1–?? (2016)

Due to the small inclination angle, the radial velocity observation of M74 only weakly contributes to the rotation curve measurement. A unique exception is the observation by an integrated field spectroscopy by the “VIRUS-P Exploration of Nearby Galaxies” (VENGA) project (Blanc et al. 2013). Its data are available along the east-west direction in the core region, namely for the case ∆δ = 0 and within the distance of 1 ′ or so. Refer to Fig. 49. As clearly seen in the previous subsection, the rotation curves of the HI gas and blue stars components are less significant in the central regions. Thus, they can not be used in the comparison with the VENGA rotation curve. The rest solution is the fitting of a linear combination of V 2 of the dust and red stars components to the V 2 of the observation. Already Figs 46 and 47 have shown the corresponding rotation curves of the two components, namely those for ∆δ = 0. Notice that the characteristics of the rotation curves of these two components are significantly different in the core region: (i) a rapid increase until R ∼ 0.5 ′ and an almost flat feature after that for the dust component, and (ii) a gradual increase until R ∼ 1 ′ for the red stars component. Obviously, the latter feature is well suited to the observed

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Cross Section: M74, FUV

Cross Section: M74, K 6 5 4 3 2 1 0 -1 -2 -3 -4

5 4 3 2 1 0 -1 -2 -3 -4

∆δ=0 Σ+Σ0 -Fα Fδ Φ

-3

-2

-1

0

1

2

3

∆δ=0 Σ+Σ0 -Fα Fδ Φ

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-∆α (arcmin)

-∆α (arcmin)

Figure 42. Cross section on plane: M74, dust component. Same as Fig. 41 but for the dust component with different normalization constants. The central mass concentration is so strong that the central drop of Fα is significantly sharp.

Figure 44. Cross section on plane: M74, blue stars component. Same as Fig. 41 but for the blue stars component. The oscillatory feature of the surface mass density distribution is directly reflected to that of the acceleration vector. Interesting is the gradual decrease of the gravitational potential toward the galactic center. Unexpected is an almost flat feature of Φ in the core region where R < 1′ . This is the result of the collective gravitational attraction of the spotty features located in non axisymmetric manner but widely distributed in any way.

Cross Section: M74, R 8 6

Σ+Σ0

4

-Fα

2 0 -2

∆δ=0

Fδ Φ

rotation curve in Fig. 49. In fact, the inclusion of any portion of the dust component results a poor fitting. Therefore, we ignored its contribution and fitted the rotation curve of the red stars component only to the observed curve by adjusting a normalization constant, V0 . Fig. 49 has already shown the best fit result. We do not think that the determined normalization constant is meaningful because only one contribution, namely that of the red stars component, is taken into account. At any rate, the goodness of this fit may suggest that the galaxy has no significant mass concentration at its center1 , such as a massive black hole, a bulge, or the dark matter.

-4 4 CONCLUSION

-6 -3

-2

-1

0

1

2

3

-∆α (arcmin) Figure 43. Cross section on plane: M74, red stars component. Same as Fig. 41 but for the red stars component. This time, the central mass concentration is significantly weaker than that of the dust seen in Fig. 42. As a result, the variation of Fα is milder than that of the dust.

We analytically obtained the exact expressions of the Newtonian gravitational potential and the associated acceleration vector for an infinitely thin uniform rectangular plate. Using them, we developed an analytical method to compute the gravitational field of a general infinitely thin object without axial symmetry when its surface mass density is known at evenly spaced rectangular grid points. We applied the method to the determination of the gravitational field of the HI gas, dust, red stars, and blue stars components of M74 by using the surface photometry data acquired by the THINGS, 2MASS, PDSS1, and GALEX programs. The obtained contour maps of the gravitational potential as well as the vector 1

An intermediate-mass black hole resulting the M74 X-1 is known to be accompanied with the galaxy but located at its outskirt (Krauss et al. 2005) MNRAS 000, 1–?? (2016)

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Circular Speed: M74, HI 1.2 1

∆δ=0

1.2 Θ=0

Θ=π

∆δ=0

1

Θ=0

V/V0

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

Θ=π

0 0 1 2 3 4 5 6 7 8 9 10 11 12

0

0.5

1

R (arcmin) Figure 45. Rotation curve: M74, gas component. Plotted are the rotation curves of the HI gas component of M74 for two directions along the equal declination, Θ = 0 (solid curve) and Θ = π (dotted curve). Prominent are the lack of the circular speed in some parts of the central region, say R < 3′ . This is never an unphysical situation. It simply means that the corresponding radial component of the acceleration vector is outward such that no centrifugal force can compensate it to keep a circular orbit. The multiple peaks in the middle region, say 3′ < R < 6′ , are associated with the spiral structure as shown in Fig. 41. Interesting is an almost flat feature of the rotation curve in the far region, say 7′ < R < 12′ .

https://www.researchgate.net/profile/Toshio_Fukushima/

ACKNOWLEDGEMENTS The author appreciates the referee’s valuable advices to improve the readability of the present article.

APPENDIX A: GRAVITATIONAL FIELD OF UNIFORM RECTANGULAR PLATE Gravitational potential

Consider the Newtonian gravitational field of an infinitely thin rectangular plate of unit surface mass density. Adopt a local rectangular coordinate system such that (i) the plate is placed on the x-y plane, (ii) the plate center is located at the coordinate origin, and (iii) the edges of the plate are along the x- and y- coordinate axes, respectively. Denote (i) by A the two-dimensional area occupied by the MNRAS 000, 1–?? (2016)

2

2.5

3

Figure 46. Rotation curve: M74, dust component. Same as Fig. 45 but for the dust component. The normalization constant V0 is different from that in Fig. 45. Significant is a sharp rise until R ≈ 0.3′ and a slightly rising manner after that.

Circular Speed: M74, R ∆δ=0

1

Θ=π Θ=0

0.8 V/V0

maps of the associated acceleration vector show us a detailed feature of the gravitational field of the galaxy. Especially, we noticed that the rotation curve derived from the gravitational field of the red stars component seems to match well with that observed along the east-west direction by the VENGA project. The developed method, which may be termed the mosaic tile model, will be useful in obtaining the gravitational field of and evaluating the rotation curve of infinitely thin non axisymmetric objects such as grand design spiral galaxies from their surface photometric data. The determined gravitational fields of M74 and the Fortran 90 programs used to compute them are available at the following web site

1.5 R (arcmin)

1.2

A1

23

Circular Speed: M74, K

0.8 V/V0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

3

3.5

R (arcmin) Figure 47. Rotation curve: M74, red stars component. Same as Fig. 45 but for the red stars component. The normalization constant V0 is different from those in Figs 45 and 46. Noticeable is an initial fast rise up to R ≈ 0.5′ and then a gradual rise until R ≈ 1.5′ . After that, it jumps somewhat and keeps an almost flat feature until a pair of slow drop and sharp rise encountered when crossing a spiral arm located at R ≈ 3′ .

object, and by [−a, a] × [−b, b] its interval representation, (ii) by P an arbitrary point in the three-dimensional space, and by (x, y, z) the set of its three-dimensional rectangular coordinates, and (iii) ( ) by Q an arbitrary point inside A, and by x ′, y ′ the set of its two-dimensional rectangular coordinates satisfying the conditions, x ′ ≤ a and y ′ ≤ b. Let us directly evaluate the Newtonian gravitational potential of the object at P. Adopt the unit system such that G = 1. Then, the potential per unit surface mass density can be formally expressed

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Circular Speed: M74, FUV 1.2

∆δ=0

1

Circular Speed: M74, R 120

Θ=π

∆δ=0

100 -1

V (km s )

Θ=0

0.8 V/V0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.6 0.4

80 60 40

0.2

20

0

0

Data: Blanc et al. (2013) 0

1

2

3

4

5

6

R (arcmin)

as a double integral with respect to x ′ and y ′ in this order as ∫ b ( ) ϕ(x, y, z) ≡ ψ y ′ dy ′, (A1) −b

Notice an indefinite integration formula expressed as ∫ x′ [ ( )] dx ′′ ( ′′ ′ ) = − ln x − x ′ + r x ′, y ′ , r x ,y

(A2)

(A3)

(A4)

the correctness of which is confirmed by differentiation. Using it, ( ) we obtain an analytic expression of ψ y ′ as ( ( ) ) ( ) x − a + r a, y ′ ψ y ′ = ln ( ) . (A5) x + a + r −a, y ′ ( ) Since |x ± a| ≤ r ∓a, y ′ , the argument of the above logarithm is non negative definite. Thus, the singularities of the logarithm are ( ) the points where (x, y, z) = ±a, y ′, 0 . Except these two points, ( ′) ψ y is well defined. Of course, these singularities are blowing up. However, the ( ) integrand values increase only logarithmically. Therefore, ψ y ′ is analytically integrable. Indeed, we find another indefinite integration formula written as ∫ y′ )] [ ( ln x − x ′ + r x ′, y ′′ dy ′′ = −y ′ (( ( ) [ )( ) )] x − x′ y − y′ y − y′ − tan−1 ( ′ ′) −z tan−1 z zr x , y

0.2

0.4

0.6

0.8

1

R (arcmin)

Figure 48. Rotation curve: M74, blue stars component. Same as Fig. 45 but for the blue stars component. The normalization constant V0 is different from those in Figs 45–47. Observed is an oscillatory feature until R ≈ 3−5′ . They are caused by the spiral arms. The mass concentrations of some inner arms are so strong that the circular speed becomes imaginary in some area if R < 1′ . This is the same situation as encountered in the gas component viewed in Fig. 45.

where the inner line integral is expressed as ∫ a ( ) dx ′ ψ y′ ≡ − ( ′ ′) , r x ,y −a ( ′ ′) while r x , y is the relative distance between P and Q as √ ( ) ( ) r x ′, y ′ ≡ (x − x ′ ) 2 + y − y ′ 2 + z 2 ≥ 0.

0

Figure 49. Rotation curve: M74, red stars component. Plotted is the rotation curve of M74 along the east-west direction. The theoretical curve is derived from the acceleration vector field associated with the gravitational potential of the red stars component. It is the curve of V /V0 for the case Θ = 0 as depicted in Fig. 7 after fitting to the observational data acquired by the VENGA (Blanc et al. 2013) by tuning the normalization constant V0 appropriately.

( ) [ ( )] − x − x ′ ln y − y ′ + r x ′, y ′ ( ) [ ( )] − y − y ′ ln x − x ′ + r x ′, y ′ ,

(A6)

where the arctangent functions are assumed to return their principal values such as tan−1 t ≤ π/2. (A7) The correctness of the second formula, equation (A6), is inspected by differentiation as will be shown later in section A3. Notice that incorrect is the reduction of the number of arctangent functions by using their addition theorem ( ) ( ) ( ) −1 s 2 −1 s 1 c2 − c1 s 2 −1 s 1 tan − tan = tan . (A8) c1 c2 c1 c2 + s1 s2 This is because the sum of the principal values is not always equal to the principal value of the transformed argument obtained by the addition theorem. In fact, the domain of the summed principal values becomes twice wider than the single principal value as tan−1 t + tan−1 t ≤ π. 1 2 On the other hand, the expression of the second formula contains no blow-up singularity. Indeed, the increase due to the logarithmic singularities are suppressed by the multiplication factors, x − x ′ or y− y ′ , the magnitude of which linearly reduces when approaching the logarithmic singularities. Also, the arctangent functions causes no discontinuity in this case. In fact, the discontinuity of the principal value of the arctangent function happens when its argument expressed as t jumps between +∞ and −∞. This phenomenon occurs when the numerator of t does not change the sign while its denominator does. Fortunately, in the present case, all the denominators of the arguments of the arctangent functions become zero only when z = 0. Meanwhile, all the arctangent functions are multiplied by z. Therefore, even when MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field the denominators change the sign, the discontinuity of the function values associated with the change is diminished by the multiplication factor, z. In conclusion, the above expression, equation (A6), is continuous and finite everywhere. At any rate, utilizing this formula, equation (A6), we obtain the analytic expression of ϕ(x, y, z) as ϕ(x, y, z) ≡ −(x + a) ln s x+ − (x − a) ln s x− − (y + b) ln s y+

−(y − b) ln s y− − zu,

(A9)

where u is the sum of arctangent functions as u ≡ − tan−1 t ++ + tan−1 t +− + tan−1 t −+ − tan−1 t −−,

(A10)

while s x± , s y± , and t ±± are the auxiliary irrational functions defined as s x+ ≡

y − b + r (a, b) y + b + r (−a, −b) , s x− ≡ , y − b + r (−a, b) y + b + r (a, −b)

s y+ ≡

t ++ ≡

x + a + r (−a, −b) x − a + r (a, b) , s y− ≡ , x − a + r (a, −b) x + a + r (−a, b)

(A11)

(x + a)(y − b) (x + a)(y + b) , t +− ≡ , zr (−a, −b) zr (−a, b)

(x − a)(y + b) (x − a)(y − b) , t −− ≡ . (A12) zr (a, −b) zr (a, b) ( ) Remember that r x ′, y ′ is a function of x, y, and z although their dependence is not explicitly written for simplicity of the expressions. The above expression, equation (A9), contains logarithmic singularities. Nonetheless, they are not blowing-up. Also, the arctangent functions here cause no discontinuities. Therefore, the present form of ϕ(x, y, z) is well defined and continuous everywhere. This expression of ϕ(x, y, z) is missing in the existing literature, t −+ ≡

A2

Acceleration vector

There are two ways to obtain the analytical expressions of the acceleration vector: (i) the integration of the partially differentiated integrand of the potential as adopted by MacMillan (1930) and many other authors, and (ii) the partial differentiation of the integrated expression of the potential as conducted by Fukushima (2010) for a uniform circular disc. Adopting the second approach, we formally write the acceleration vector as ) ( ∂ϕ . (A13) f(x, y, z) ≡ − ∂(x, y, z) Since ϕ(x, y, z) is explicitly given as a function of x, y, and z in the previous subsection, its partial differentiation is complicated but ( ) automatic if keeping in mind that r x ′, y ′ is also a function of x, y, and z. At any rate, the final results are simpler than the potential expression as ( ) ∂ϕ f x (x, y, z) ≡ − = ln (s x+ s x− ) , (A14) ∂ x y, z ( f y (x, y, z) ≡ −

) ( ) ∂ϕ = ln s y+ s y− , ∂ y x, z

MNRAS 000, 1–?? (2016)

(A15)

( f z (x, y, z) ≡ −

) ∂ϕ = u. ∂z x, y

25

(A16)

These expressions coincide with those obtained by regarding the auxiliary functions, s x± , s y± , and t ±± , as if they were constants in equation (A9). The same situation was already reported in the three-dimensional case by MacMillan (1930, section 44) as ... As a matter of fact, however, it is not necessary to differentiate with respect to the coordinates in so far as these coordinates appear under the log and tan−1 symbols. It is sufficient to differentiate as though these functions were constants, and a recognition of this fact makes the differentiation a very simple matter...

See also Werner & Scheeres (1996, section 2.2.1). In the present case, however, not acceptable is the explanation of cancellation in the three-dimensional case (MacMillan 1930, section 45) since the logic MacMillan used is based on the permutation of the coordinates with respect to which the line integration is conducted. Therefore, by brute force, we found that the contributions of the partial differentiation of the auxiliary functions, which are fairly intricate, do cancel with each other after lengthy formula manipulations by hand2 . Notice that these derivatives contain algebraic and/or blowup logarithmic singularities as well as discontinuities. This is a natural consequence since the potential expression is not analytic at the edges of the rectangular plate, namely when z = 0 and x = ±a and/or y = ±b. At any rate, the derived expressions of the acceleration vector covers all the evaluation points in the threedimensional space. As far as the author knows, this result is not found in the existing literature

A3 Confirmation Let us examine the correctness of the expressions of the gravitational field derived in the previous subsections. Begin with the gravitational potential. First, we checked the validity of a couple of indefinite integration formulas, equations (A4) and (A6), by the following command sequence of Mathematica 10 (Wolfram Res. 2015): r[x_,y_,z_,p_,q_]=Sqrt[(x-p)^2+(y-q)^2+z^2]; g[x_,y_,z_,p_,q_]=Log[x-p+r[x,y,z,p,q]]; h[x_,y_,z_,p_,q_]=-z(ArcTan[(y-q)/z]-ArcTan[(x-p)(y-q)/(z r[x,y,z,p,q])]) -q-(x-p)Log[y-q+r[x,y,z,p,q]]-(y-q)Log[x-p+r[x,y,z,p,q]]; Simplify[D[g[x,y,z,p,q],p]+1/r[x,y,z,p,q]] Simplify[D[h[x,y,z,p,q],q]-g[x,y,z,p,q]]

Both of the Simplify commands return 0. This means the correctness of the integration formulas. Since the expression of ϕ(x, y, z) in equation (A9) is only a subtraction of the second integration formula, the validity of the expression is also confirmed. On the other hand, Mathematica could not prove the equality of (i) the acceleration vector expressions, equations (A14)–(A16), and (ii) the partial differentiations of the potential expression, equation (A9). Thus, we conducted its numerical examination by the following command sequence of Mathematica: sxp[x_,y_,z_,a_,b_]=(y+b+r[x,y,z,-a,-b])/(y-b+r[x,y,z,-a,b]); sxm[x_,y_,z_,a_,b_]=(y-b+r[x,y,z,a,b])/(y+b+r[x,y,z,a,-b]); syp[x_,y_,z_,a_,b_]=(x+a+r[x,y,z,-a,-b])/(x-a+r[x,y,z,a,-b]); sym[x_,y_,z_,a_,b_]=(x-a+r[x,y,z,a,b])/(x+a+r[x,y,z,-a,b]); tpp[x_,y_,z_,a_,b_]=(x+a)(y+b)/(z r[x,y,z,-a,-b]); tpm[x_,y_,z_,a_,b_]=(x+a)(y-b)/(z r[x,y,z,-a,b]);

2

Mathematica 10 could not simplify them

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tmp[x_,y_,z_,a_,b_]=(x-a)(y+b)/(z r[x,y,z,a,-b]); tmm[x_,y_,z_,a_,b_]=(x-a)(y-b)/(z r[x,y,z,a,b]); u[x_,y_,z_,a_,b_]=-ArcTan[tpp[x,y,z,a,b]]+ArcTan[tpm[x,y,z,a,b]] +ArcTan[tmp[x,y,z,a,b]]-ArcTan[tmm[x,y,z,a,b]]; phi[x_,y_,z_,a_,b_]=-(x+a)Log[sxp[x,y,z,a,b]]-(x-a)Log[sxm[x,y,z,a,b]] -(y+b)Log[syp[x,y,z,a,b]]-(y-b)Log[sym[x,y,z,a,b]]-z u[x,y,z,a,b]; fx[x_,y_,z_,a_,b_]=-D[phi[x,y,z,a,b],x]; fy[x_,y_,z_,a_,b_]=-D[phi[x,y,z,a,b],y]; fz[x_,y_,z_,a_,b_]=-D[phi[x,y,z,a,b],z]; fxA[x_,y_,z_,a_,b_]=Log[sxp[x,y,z,a,b]sxm[x,y,z,a,b]]; fyA[x_,y_,z_,a_,b_]=Log[syp[x,y,z,a,b]sym[x,y,z,a,b]]; fzA[x_,y_,z_,a_,b_]=u[x,y,z,a,b]; Plot3D[fx[x,y,1,1,1]-fxA[x,y,1,1,1],{x,-2,2},{y,-2,2}] Plot3D[fy[x,y,1,1,1]-fyA[x,y,1,1,1],{x,-2,2},{y,-2,2}] Plot3D[fz[x,y,1,1,1]-fzA[x,y,1,1,1],{x,-2,2},{y,-2,2}]

The last three commands depict the deviations of the obtained expressions from the direct differentiation of ϕ(x, y, z). The results are plotted as the two-dimensional curves with respect to x and y in the intervals, |x| ≤ 2 and |y| ≤ 2, for a non-trivial choice of other parameters as z = a = b = 1. The plotted curves, which are omitted for saving the space, are almost zero with random fluctuations of the order of the double precision machine epsilon. This process numerically confirms the correctness of the derived expressions. A4

Namely, h(x, y) is a sort of two-dimensional generalization of the Heaviside step function in the sense that it indicates whether the two-dimensional point, (x, y), is inside the plate or not. At any rate, the arctangent functions completely disappeared from the above expressions. The resulting formulas of the acceleration vectors on the x-y plane, namely equations (A18) and (A19), are exactly the same as those given in the literature (MacMillan 1930, section 18).

A5 Special values Let us show some special values of ϕ(x, y, z) and f(x, y, z). Begin with the values at the object center: ( ) (a + c) b+c ϕ0 ≡ ϕ(0, 0, 0) = −4a ln − 4b ln , (A24) a b f0 ≡ f(0, 0, 0) = 0,

where c is the radial distance to the vertices of the plate written as √ c ≡ a2 + b2 . (A26)

Expressions on object plane

The full expressions of the gravitational potential and the acceleration vector are complicated as seen in the previous subsections. When z = 0, some of them are simplified as ϕ∗ (x, y) ≡ lim ϕ(x, y, z) = −(x + a) ln s∗x+ − (x − a) ln s∗x− z→0

−(y + b) ln s∗y+ − (y − b) ln s∗y−,

(A17)

( ) f x∗ (x, y) ≡ f x (x, y, 0) = ln s∗x+ s∗x− ,

(A18)

( ) f y∗ (x, y) ≡ f y (x, y, 0) = ln s∗y+ s∗y− ,

(A19)

where s∗ s are those after substituting z = 0, namely those obtained ( ) by replacing r x ′, y ′ with √ ( ) ( ) r ∗ x ′, y ′ ≡ (x − x ′ ) 2 + y − y ′ 2 . (A20) Notice that the limit value of zu when z → 0 is zero despite that the limit value of u is not uniquely defined but finite in any way. Meanwhile, one has to take a care in understanding the limit value of f z (x, y, z) = u, the z-component of the acceleration, since it has a discontinuity at z = 0 if |x| ≤ a and |y| ≤ b. More rigorously speaking, we have two different limiting values of f z (x, y, z) when z → 0 as { −(π/2)h(x, y) (z → 0 + 0) , (A21) f z∗ (x, y) ≡ (π/2)h(x, y) (z → 0 − 0)

Next comes the potential values at the vertices of the plate, ) ( (a + c) ϕ b+c − 2b ln ϕ(a, b, 0) = −2a ln = 0, (A27) a b 2 and those at the middle points of the edges of the plate, √ b + 4a2 + b2 + ϕ(a, 0, 0) = −4a ln * 2a , √ 2a + 4a2 + b2 + * , −2b ln b , -

√ 2b + a2 + 4b2 + * . −2a ln a ,

(A22)

while sign(x) ≡ x/|x| is the sign of the argument x. It is easily shown that { 1 (|x| ≤ a; |y| ≤ b) h(x, y) = (A23) 0 (otherwise)

(A29)

When a = b, these take the following numerical values as ( √ ) ( ) ϕ0 a=b = −8 ln 1 + 2 a ≈ −7.051a, (

(A30)

( √ ) ) 1( ) ϕ ϕ(a, b, 0) a=b = −4 ln 1 + 2 a = 2 0 a=b ≈ −3.525a,

(

h(x, y) ≡ sign[(x + a)(y + b)] + sign[(x + a)(b − y)]

(A28)

√ a + a2 + 4b2 + * ϕ(0, b, 0) = −4b ln 2b , -

where

+sign[(a − x)(y + b)] + sign[(a − x)(b − y)],

(A25)

(A31)

√ ) ( ) 11 + 5 5 + ϕ(a, 0, 0) a=b = ϕ(0, b, 0) a=b = −2 ln * a 2 , ≈ −4.181a.

(A32)

The value of |ϕ0 | is more than 10% larger than 2πa ≈ 6.283a, that of a uniform circular disc of the radius a. The difference comes from the contribution of four corner areas of a square after removing its inscribed circle. MNRAS 000, 1–?? (2016)

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Mosaic tile model to compute gravitational field A6

In order to examine the feature of ϕ∗ (x, y) near the coordinate center, √ we obtain its series expansion with respect to R ≡

ϕ∗ (R cos Θ, R sin Θ) ≈ ϕ0 + ϕ∗2 (Θ)R2 + ϕ∗4 (Θ)R4,

Contour Map of φ: Rectangular Plate 4

x 2 + y 2 as

(A33)

where Θ = atan2(y, x) is the azimuth in the object plane and the coefficients are expressed as

3

ϕ∗2 (Θ) ≡ [2c/(ab)](1 − w cos 2Θ),

2

(A34)

[ ( )] [( ) ( ) ϕ∗4 (Θ) ≡ c3 / 192a3 b3 15 + 9w 2 − 36 − 4w 2 cos 2Θ (

2

+ 11 − 9w + 6w

4

)

]

cos 4Θ ,

1

(A35)

a2 − b2 w≡ 2 . a + b2

-2

(A36)

-3

If a > b and if we regard a and b the semi-major and semi-minor axes of an ellipse, then w is expressed in terms of e, the eccentricity of the ellipse, as e2 . (A37) 2 − e2 At any rate, if ignoring the fourth and higher order terms, we obtain an approximate quadratic form of the potential on the object plane around the coordinate center as ( ) 2ab x 2 y 2 ϕ∗ (x, y) ≈ ϕ0 + + . (A38) c a2 b2

-4 -4 -3 -2 -1 0

w=

This manifests that, near the coordinate center, the equipotential contour is an ellipse centered at the coordinate origin with the semiaxes a and b. On the other hand, in order to examine the asymptotic behavior of ϕ∗ (x, y) when R → ∞, we expand it with respect to 1/R as ϕ∗ (R cos Θ, R sin Θ) ∼ ϕ∗−1 R−1 + ϕ∗−3 (Θ)R−3 + ϕ∗−5 (Θ)R−5, (A39) where ϕ∗−1 ≡ −4ab,

(A40)

( ) ϕ∗−3 (Θ) ≡ − abc2 /3 (1 + 3w cos 2Θ),

(A41)

( )[ ϕ∗−5 (Θ) ≡ − abc4 /240 21 + 6w 2 + 60w cos 2Θ ( ) ] + 35 − 140w 2 cos 4Θ .

(A42)

Of course, the first term is the masspoint approximation. This expansion can be regarded as a sort of two-dimensional harmonics expansion.

In order to illustrate a more detailed feature of the gravitational field of a uniform rectangular plate on the object plane, we present below some additional graphs showing their properties. Already, Fig. 9 has presented the bird’s-eye view of the potential for the case of a square, namely when a = b. In Fig 9, the MNRAS 000, 1–?? (2016)

1

2

3

4

x/a Figure A1. Contour map of gravitational potential in x-y plane: uniform rectangular plate. Plotted is the contour map of the gravitational potential of an infinitely thin uniform rectangular plate with the aspect ration, b/a = 1.5. Although the external contours tend to be nearly circular when R → ∞, the internal ones resemble ellipses of the same aspect ratio as a/b especially when R → 0.

two-dimensional curve resembles a spherical bowl near the coordinate origin. This feature is described as a quadratic curve specified by equation (A38). In order to examine the effect of the aspect ratio of the rectangular plate, b/a, we prepared Fig. A1 for the case when b/a = 1.5. The figure shows that the contour map is adjusted such that the contours resemble a rectangular shape when approaching the edges. Indeed, as equation (A38) predicts, the internal contours become more and more elliptical when approaching the coordinate origin while keeping the same aspect ratio. Meanwhile, when going outward, the contours tend to be more and more circular. This feature is independent on the aspect ratio. Let us examine the variation of the gravitational field more carefully. Fig. A2 illustrates the R-dependence of the potential value when a = b. It is evident that the curves are almost the same except near the edges of the plate, say when 3/4 < R/a < 3/2. Also, at this scale, the masspoint approximation seems to be practically the same when R/a > 3. A8

Additional sketches

0 -1

while w is a non-dimensional factor defined as

A7

27

Series expansion

y/a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

z-dependence

In section 2, we focused on the feature of the gravitational field on the object plane. Here, we illustrate some aspects of the z-dependence of the gravitational field of the uniform rectangular plate. First, Fig. A3 gives a bird’s-eye view of the gravitational potential on the x-z plane. Its outlook is similar to that on the object plane, Fig. 9. However, an examination of the contour map reveals that the feature is significantly different as shown in Fig. A4.

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R-Profile of Potential: Square Plate 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1

Contour Map of φ: Square Plate 4 3 2

Θ=0

Θ=π/4

1 z/a

φ/|φ0|

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-1/R 0

0.5

1

0 -1

1.5

2

2.5

3

3.5

4

R/a Figure A2. Radial profile of gravitational potential: uniform square plate. Computed are the normalized values of the gravitational potential of an infinitely thin uniform square plate on the object plane when a = b. The potential values are plotted as functions of R for some values of Θ as Θ = 0, π/16, π/8, 3π/16, and π/4. The difference between the curves of Θ = 0 and Θ = π/16 is hardly visible at this scale. A thin curve shows the mass point approximation being in proportion to a/R.

-2 -3 -4 -4 -3 -2 -1 0

1

2

3

4

x/a Figure A4. Contour map of gravitational potential: uniform square plate on a vertical plane. Same as Fig. 10 but plotted is the contour map of ϕ(x, 0, z).

Potential of Square Plate φ/|φ0| 0 -0.5 -1 4 -4

-2 x/a

0

2

-2

2 0 z/a

4 -4

Figure A3. Bird’s-eye view of gravitational potential: uniform square plate when y = 0. Same as Fig. 9 but for the potential values as functions of x and z when y = 0. This resembles a melting chocolate slab dropping through a slit.

Move to the quantitative feature of the potential value. Fig. A5 plots the z-dependence of the potential value. This time, the feature of the curves are different depending on the value of x-coordinate. Namely, when |x| ≤ a, the potential has a kink at z = 0. Otherwise, it varies smoothly. This is a natural consequence since the mass density distribution behaves as Dirac’s delta function there.

Third, let us examine the errors of the masspoint approximation with respect to the z-coordinate of the evaluation point. Fig. A6 illustrates that the absolute errors are rather smoothly decay with respect to the height although their magnitude depends on the location of the foot on the object plane. Finally, we present some graphs explaining the behavior of the acceleration vector caused by the infinitely thin uniform square plate. Figs A7 and A8 show the bird’s-eye view and the corresponding contour map of the logarithm of the magnitude of f x on the x-y plane. Notice that the sign of f x is the same as that of x. These figures show clearly a logarithmic blow-up of the f x on the y-edges of the plate, namely when |x| = a and |y| ≤ a. On the other hand, Figs A9 and A10 illustrate similar graphs of f z on the x-z plane. This time, the singularity locates on the edge where |x| ≤ a and z = 0.

APPENDIX B: DATA PREPROCESSING FOR INCLINED OBJECTS In section 2, we assumed that the surface mass density data are given at the evenly spaced rectangular grid points. In general, the observed photometry data of spiral galaxies are not so. Although it is slightly out of the scope of the present article concentrated to the gravitational field computation, we shall present below a primitive two-step process by neglecting the correction of the interstellar extinction. B1 Case of rotated image First, assume that (i) the object is face on, namely its inclination angle with respect to the line of sight is 0◦ , but (ii) the coordinate MNRAS 000, 1–?? (2016)

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29

φ/|φ0|

z-Profile of Potential: Square Plate 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1

x=4a

Magnitude of fx: Square Plate log10 | fx| 1 0 -1 -2 -3

x=0 y=0 -4

-3

-2

-1

0

1

2

3

4

4

z/a

-4

-2

Figure A5. z-profile of gravitational potential: uniform square plate. Same as Fig. A2 but as a function of z when a = b and y = 0 for several values of x as x/a = 0, 1/2, 3/4, 1, 5/4, 3/2, 2, 3, and 4. A thick curve shows the case x/a = 1. When | x/a | ≤ 1, the curves have a kink at z = 0.

x/a

0

2

-2

2 0 y/a

4 -4

Figure A7. Bird’s-eye view of log10 | f x |: uniform square plate. Illustrated are the logarithm of | f x |, the magnitude of x-component of the acceleration vector, when z = 0. Notice that the sign of f x is the same as −x.

Error of Masspoint Approximation 1 0 -1 -2 -3 -4 -5 -6 -7 0.1

a=b, y=0

Contour Map of |fx|: Square Plate

x/a=4 x/a=0

4 3 2

1

10

100

z/a Figure A6. Error of masspoint approximation: uniform square plate, zdependence. Same as Fig. 14 but plotted as function of z when y = 0 for several values of x as x/a = 0, 1/2, 3/4, 1, 5/4, 3/2, 2, 3, and 4. A thick curve shows the case x/a = 1.

1 y/a

log10 |(φ-φM)/φ0|

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 -1 -2 -3

axes of its observational grid points are rotated from the desired axes by a certain position angle. Then, we suggest a following recipe: (i) to conduct the computation in the given coordinate system, (ii) to store the grid data of the computed gravitational potential and acceleration vector in that coordinate system, and (iii) to interpolate them for the desired coordinates after their reverse rotation to the given coordinate system. It is a simple device but works. This is because the mosaic tile model is a sort of discrete convolution, and therefore, its application makes the jaggy surface density profile somewhat smooth. Usually, MNRAS 000, 1–?? (2016)

-4 -4 -3 -2 -1 0 1 x/a

2

3

4

Figure A8. Contour map of log10 | f x |: uniform square plate. Same as Fig. A7 but extracted is its contour map.

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xxxxx

Contour Map of |fz|: Square Plate Magnitude of fz: Square Plate

4 3

log10 | fz| 1 0 -1 -2 -3

2 1 z/a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-1

4 -4

0

-2 x/a

0

2

-2

2 z/a

-2 -3

4 -4

Figure A9. Bird’s-eye view of log10 f z : uniform square plate. Same as Fig. A7 but for the logarithm of f z when y = 0. Notice that the sign of f z is the same as −z.

0

-4 -4 -3 -2 -1 0

1

2

3

4

x/a

Figure A10. Contour map of log10 f z : uniform square plate. Same as Fig. A9 but extracted is its contour map.

an interpolation of the smoothed data is more appropriate than the smoothing after the interpolation of ragged values. C1 File transformation B2 Treatise of inclined image Next, consider the case when the object is inclined by a known angle, θ. Then, we rotate the original two-dimensional coordinates of given grid points by the angle θ around the rotation axis of the inclination. The projections of the rotated coordinates, which are now three-dimensional, onto the plane perpendicular to the line of sight are located on not rectangular but oblique grid points. For example, the projection of a rotated square plate is transformed to a parallelogram. Nevertheless, the idea of interpolation is applicable. Namely, we may use an appropriate interpolation to obtain the values at the desired rectangular grid points on the perpendicular plane. Of course, a simple scheme is a bilinear interpolation in an oblique coordinate system. Namely, it is enough (i) to transform the desired rectangular coordinates into the corresponding oblique coordinates, and (ii) to employ the bilinear interpolation formula, equation (23), to obtain the desired value in the oblique coordinate system.

APPENDIX C: PREPARATION OF GRAPHS Below, we summarize some useful information necessary to prepare graphs from the given matrix data by using the gnuplot 5.0 (Janert 2016).

Assume that the matrix data of a scalar quantity, Ai j , are stored as a two-dimensional array, A(i,j), in the computer memory. The first thing we must do is to print out them into a sequential file of the three-record form, (i, j, A(i,j)), while inserting a blank line after the end of every group of records with the same second index, j, such as 1 1 A(1,1) 2 1 A(2,1) ... 1024 1 A(1024,1) 1 2 A(1,2) 2 2 A(2,2) ... 1024 2 A(1024,2) ... 1 1024 A(1,1024) 2 1024 A(2,1024) ... 1024 1024 A(1024,1024)

Name the file as array.dat for later use. Consider the case of vector data. Assume that the matrix data of a two-dimensional vector quantity, X i j and Yi j , are stored as a pair of two-dimensional arrays, X(i,j) and Y(i,j). Then, they must be transformed into a sequential file of the four-record form, (i, j, MNRAS 000, 1–?? (2016)

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31

X(i,j), Y(i,j)), while inserting a blank line appropriately. Name the output file as vector.dat for later use.

The modified sequence provides a monochrome contour map of the reduced data, array8.dat, such as viewed in Fig. 5.

C2

C5 Vector map

Resampling

In many cases, not suitable for plotting are the dense data files such as those containing 1024 × 1024 grid data. The typical sizes of the data files are (i) 16 × 16 or 32 × 32 for the bird’s-eye views and the vector maps, (ii) 128 × 128 or 256 × 256 for the contour maps, and (iii) 1024 × 1024 or less for the cross section views. Therefore, we suggest reducing the output file to the above appropriate sizes. There are many ways to do this task. Our preference is a single line command using awk, an interpreted programming language in UNIX (Aho et al. 1988). Its sample usage is written as follows: awk ’$1 array16.dat

This UNIX command sequence reduces the size of array.dat, the original data file, to a fraction of 1/16×1/16 = 1/256 of the original data and store them into array16.dat, The above awk command works independently on the number of records. Therefore, the same sequence can be applied to vector.dat.

C3

Bird’s-eye view

A bird’s-eye view is depicted by the following sequence of gnuplot commands:

A contour map is drawn by a slight modification of the above command sequence for the bird’s-eye view: set terminal postscript eps enhanced monochrome 24 set output ’vector-map.eps’ set border set nokey set size square set title "Vector Map of {/Times-Bold F}" set xlabel "-{/Symbol Da} (arcmin)" set ylabel "{/Symbol Dd} (arcmin)" plot "vector16.dat" u 1:2:3:4 with vectors

This sequence produces a monochrome vector map of the reduced vector data, vector16.dat, such as displayed in Fig. 30. C6 Resetting gnuplot The gnuplot user interface program keeps the effect of some of the environment setting commands even after the desired plots are done. As a result, a sequence of issuing the above gnuplot commands may cause undesired side effects. In order to avoid this phenomenon, a simple recipe we recommend is to issue a following reset command: reset session

set terminal postscript eps enhanced monochrome 24 set output ’bird-eye.eps’ set cntrparam levels auto 20 set contour set hidden3d set nokey set style data lines set surface set title "Bird’s-Eye View of {/Symbol F}" set view 30, 30, 1, 1 set xlabel "-{/Symbol Da} (arcmin)" set ylabel "{/Symbol Dd} (arcmin)" set zlabel "{/Times {/Symbol F}/{/Symbol F}_0}" splot "array16.dat" u 1:2:3 lw 1

which sets the environment back to the initial state. It is a good practice to begin with this reset command in any session to plot graphs.

This sequence produces a monochrome bird’s-eye view of the data contained in array16.dat such as seen in Fig. 22.

In order to examine the detailed structure of the determined acceleration vector of the HI gas component of M74, we present their enlarged vector maps in Figs E1–E8.

C4

Contour map

A contour map is drawn by a slight modification of the above command sequence for the bird’s-eye view: set terminal postscript eps enhanced monochrome 24 set output ’contour-map.eps’ set border set cntrparam levels auto 20 set contour set nokey set size square set style data lines set nosurface set title "Contour Map of {/Symbol F}" set view map set xlabel "-{/Symbol Da} (arcmin)" set ylabel "{/Symbol Dd} (arcmin)" splot "array8.dat" u 1:2:3 lw 1 MNRAS 000, 1–?? (2016)

APPENDIX D: DETAILED CONTOUR MAPS In order to show the detailed structure of the computed gravitational potential of the HI gas component of M74, we present their enlarged contour maps in Figs D1–D8.

APPENDIX E: ENLARGED VECTOR MAPS

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Contour Map of Φ: M74, HI

Contour Map of Φ: M74, HI

8

-4 ∆δ (arcmin)

-3

∆δ (arcmin)

9

7

-5

6

-6

5

-7

4

-8

3

-9 -9

-8

-7

-6

-5

-4

-3

-9

-8

-∆α (arcmin)

-7

-6

-5

-4

-3

-∆α (arcmin)

Figure D1. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. 28 but at an outer region where −9′ ≤ −∆α ≤ −3′ and 3′ ≤ ∆δ ≤ 9′ .

Figure D3. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where −9′ ≤ −∆α ≤ −3′ and −9′ ≤ ∆δ ≤ −3′ .

Contour Map of Φ: M74, HI

3

9

2

8

1

-1

∆δ (arcmin)

Contour Map of Φ: M74, HI

-2

4

-3

3

∆δ (arcmin)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

7

0

6 5

-9

-8

-7 -6 -5 -4 -∆α (arcmin)

-3

Figure D2. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where −9′ ≤ −∆α ≤ −3′ and |∆ |δ ≤ 3′ .

-3

-2

-1 0 1 -∆α (arcmin)

2

3

Figure D4. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where |∆α | ≤ 3′ and 3′ ≤ ∆δ ≤ 9′ .

MNRAS 000, 1–?? (2016)

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Contour Map of Φ: M74, HI

33

Contour Map of Φ: M74, HI

-4

2 ∆δ (arcmin)

3

∆δ (arcmin)

-3

-5 -6 -7

1 0

-1

-8

-2

-9

-3 -3

-2

-1

0

1

2

3

3

4

-∆α (arcmin)

5

6

7

8

9

-∆α (arcmin)

Figure D5. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where 3′ ≤ −∆α | ≤ 9′ and 3′ ≤ ∆δ ≤ 9′ .

Figure D7. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where 3′ ≤ −∆α ≤ 9′ and |∆δ | ≤ 3′ .

Contour Map of Φ: M74, HI

Contour Map of Φ: M74, HI

9

-3

8

-4

∆δ (arcmin)

7

∆δ (arcmin)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-5

6

-6

5

-7

4

-8

3

-9 3

4

5 6 7 -∆α (arcmin)

8

9

Figure D6. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where −9′ ≤ −∆α ≤ −3′ and −9′ ≤ ∆δ ≤ −3′ .

MNRAS 000, 1–?? (2016)

3

4

5 6 7 -∆α (arcmin)

8

9

Figure D8. Contour map of gravitational potential on galactic plane: M74, gas component in outer region. Same as Fig. D1 where 3′ ≤ −∆α ≤ 9′ and −9′ ≤ ∆δ ≤ −3′ .

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Outer Vector Map of F: M74, HI

9

-3

8

-4

7

-5

∆δ (arcmin)

∆δ (arcmin)

Outer Vector Map of F: M74, HI

6 5 4

-6 -7 -8

3

-9 -9

-8

-7

-6

-5

-4

-3

-9

-8

-∆α (arcmin)

-7

-6

-5

-4

-3

-∆α (arcmin)

Figure E1. Acceleration vector map in outer region: M74, gas component. Same as Fig. 31 but in an outer region where −9′ ≤ −∆α ≤ −3′ and 3′ ≤ ∆δ ≤ 9′ .

Figure E3. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but for the region −9′ ≤ −∆α ≤ −3′ and −9′ ≤ ∆δ ≤ −3′ .

Outer Vector Map of F: M74, HI

Outer Vector Map of F: M74, HI

9

3

8 ∆δ (arcmin)

2 ∆δ (arcmin)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 0 -1

7 6 5 4

-2

3

-3 -9

-8

-7

-6

-5

-4

-3

-∆α (arcmin) Figure E2. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but for the region −9′ ≤ −∆α ≤ −3′ and |∆δ | ≤ 3′ .

-3

-2

-1

0

1

2

3

-∆α (arcmin) Figure E4. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but in another outer region where |∆α | ≤ 3′ and 3′ ≤ ∆δ ≤ 9′ .

MNRAS 000, 1–?? (2016)

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Outer Vector Map of F: M74, HI

-3

3

-4

2

-5

1

∆δ (arcmin)

∆δ (arcmin)

Outer Vector Map of F: M74, HI

-6 -7 -8

0 -1 -2

-9

-3 -3

-2

-1

0

1

2

3

3

4

-∆α (arcmin)

5

6

7

8

9

-∆α (arcmin)

Figure E5. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but for the region −9′ ≤ −∆α ≤ −3′ and −3′ ≤ ∆δ ≤ 3′ .

Figure E7. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but for the region 3′ ≤ −∆α ≤ 9′ and |∆δ | ≤ 3′ .

Outer Vector Map of F: M74, HI

Outer Vector Map of F: M74, HI

9

-3

8

-4

7

-5

∆δ (arcmin)

∆δ (arcmin)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

6 5 4

-6 -7 -8

3

-9 3

4

5

6

7

8

9

-∆α (arcmin) Figure E6. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but for the region 3′ ≤ −∆α ≤ 9′ and 3′ ≤ ∆δ ≤ 9′ .

MNRAS 000, 1–?? (2016)

3

4

5

6

7

8

9

-∆α (arcmin) Figure E8. Acceleration vector map in outer region: M74, gas component. Same as Fig. E1 but for the region 3′ ≤ −∆α ≤ 9′ and −9′ ≤ ∆δ ≤ −3′ .

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Cross Section: M74, HI 8 6

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∆α=0

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∆δ (arcmin)

Figure F1. Cross section on plane: M74, gas component. Same as Fig. 41 but across the line ∆α = 0. The units are the same as those in Fig. 41.

Figure F2. Cross section on plane: M74, dust (continued). Same as Fig. 42 but across the line ∆α = 0. The units are the same as those in Fig. 42. Probably, a foreground object may cause a hump in Σ at ∆δ ≈ −0.55′ and associated bump in Φ and oscillations in the acceleration vector components.

APPENDIX F: CROSS SECTIONS ALONG NORTH-SOUTH DIRECTION In order to illustrate the detailed structure of the computed gravitational fields of various components of M74, we present their cross sections along the north-south direction in Figs F1–F4.

APPENDIX G: ROTATION CURVES ALONG NORTH-SOUTH DIRECTION In order to display the direction dependency of the rotation curves more clearly, we present the rotation curves along the north-south direction in Figs G1–G4.

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-4

0 0 1 2 3 4 5 6 7 8 9 10 11 12

-6 -3

-2

-1

0

1

2

3

∆δ (arcmin) Figure F3. Cross section on plane: M74, red stars component. Same as Fig. 43 but across the line ∆α = 0. The units are the same as those in Fig. 43.

Cross Section: M74, FUV 5 4 3 2 1 0 -1 -2 -3 -4

∆α=0 Σ+Σ0 Fδ -Fα Φ

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ∆δ (arcmin) Figure F4. Cross section on plane: M74, blue stars component. Same as Fig. 44 but across the line ∆α = 0. The units are the same as those in Fig. 44.

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R (arcmin) Figure G1. Rotation curve: M74, gas component. Same as Fig. 45 but for the other two directions along the equal right ascension, Θ = π/2 (solid curve) and Θ = 3π/2 (dotted curve). The normalization constant V0 is the same as that in Fig. 45. Again, no meaningful values of the rotation curve in the central region is due to a hill-like feature of the gravitational potential. By combining with Fig. 45, we find that the location of the peak value moves as (i) R ≈ 4.2′ when Θ = 3π/2, (ii) R ≈ 4.8′ when Θ = 0, (iii) R ≈ 5.8′ when Θ = π/2, and (iv) R ≈ 5.9′ when Θ = π. This distribution is significantly different from the standard model of grand design spiral galaxy as two logarithmic spiral arms of a constant pitch angle with the point symmetry.

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R (arcmin)

Figure G2. Rotation curve: M74, dust component. Same as Fig. G1 but for the dust component. The normalization constant V0 is the same as that of Fig. 46. Similar is a rising and bumpy feature of the rotation curve.

Figure G4. Rotation curve: M74, blue stars component. Same as Fig. G1 but for the blue stars component. The normalization constant V0 is the same as in Fig. 48. Notice that the rotation curve almost vanishes. This means that the gravitational acceleration in the north-south direction is not strongly oriented toward the galactic center but weakly or rather outward. This manifests a significant asymmetric feature of the gravitational field caused by the blue stars component of M74.

Circular Speed: M74, R 1.2

∆α=0

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Θ=3π/2 Θ=π/2

0.8 V/V0

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0.6 0.4 0.2 0 0

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R (arcmin) Figure G3. Rotation curve: M74, red stars component. Same as Fig. G1 but for the red stars component. The normalization constant V0 is the same as in Fig. 47. The appearance of the rotation curve is almost the same as that given in Fig. 47.

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