Numerical computation of gravitational field for

MNRAS 000, 1–43 (2016)

Preprint 29 May 2016

Compiled using MNRAS LATEX style file v3.0

Numerical computation of gravitational field for general axisymmetric objects Toshio Fukushima1⋆ 1

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

Accepted. Received; in original form

ABSTRACT

We developed a numerical method to compute the gravitational field of a general axisymmetric object. The method (i) numerically evaluates a double integral of the ring potential by the split quadrature method using the double exponential rules, and (ii) derives the acceleration vector by numerically differentiating the numerically integrated potential by Ridders’ algorithm. Numerical comparison with the analytical solutions for a finite uniform spheroid and an infinitely extended object of the Miyamoto-Nagai density distribution confirmed the 13 and 11 digit accuracy of the potential and the acceleration vector computed by the method, respectively. The averaged CPU time to obtain a single point value of the potential of an infinite object is 0.4–2.3 ms at an ordinary PC, say that with an Intel Core i7-4600U CPU running at 2.10 GHz clock. By using the method, we present the gravitational potential contour map and/or the rotation curve of various axisymmetric objects: (i) finite uniform objects covering rhombic spindles and circular toroids, (ii) infinitely extended spheroids including Sérsic and NavarroFrenk-White spheroids, and (iii) other axisymmetric objects such as an X/peanut-shaped object like NGC128, a power-law disc with a central hole like the protoplanetary disc of TW Hya, and a tear-drop-shaped toroid like an axisymmetric equilibrium solution of plasma charge distribution in an ITER-like tokamak. The method is directly applicable to the electrostatic field and will be easily extended for the magnetostatic field. The fortran 90 programs of the new method and some test results are electronically available. Key words: gravitation — celestial mechanics — protoplanetary discs — methods: numerical — galaxies: general

1 INTRODUCTION 1.1

Gravitational field of ellipsoidally symmetric objects

The computation of the gravitational potential and of the associated acceleration vector of a general three-dimensional object is a classic problem in physics (Laplace 1799, part 2, section 11). Beyond controversy, it is an essential problem in astronomy and astrophysics, electrostatics and magnetostatics, geodesy and geophysics, planetary science, and plasma physics. An extraordinary achievement is a set of theorems derived by Newton on the solution of a general spherically symmetric body (Chandrasekhar 1995, chapter 1). It assures that, in the case of gravitational potential, (i) the point mass approximation is exact outside an arbitrary finite body with the spherical symmetry, (ii) the acceleration vector caused by the external part of the body cancels at any internal point of the body, and therefore, (iii) the acceleration vector at an arbitrary point is the same as that caused by the point mass approximation of the internal part of the body (Kellogg 1929). Later, these theorems were generalized to ellipsoidally sym-

⋆

E-mail: [email protected]

© 2016 The Authors

metric objects step by step by a number of great mathematicians (Chandrasekhar 1969). Here we use the term ‘ellipsoidally symmetric’ in the sense that the volume mass density, which is generally a function of three rectangular coordinates, (x, y, z), accidentally depends only on a single variable defined as √ s ≡ (x/a) 2 + (y/b) 2 + (z/c) 2, (1) where a, b, and c are the semi-axes of the reference ellipsoid of an ellipsoidal coordinate system (Hobson 1931). Historically, most of the researchers seeked for the specific solution of the gravitational potential for a finite uniform oblate spheroid as an approximation of the geopotential (Heiskanen and Moritz 1967). Refer to Appendix A for its cancellation-free solution. In astronomy and astrophysics, however, important is its application to an infinitely extended object (Chandrasekhar 1969, p.59, theorem 12). The generalization of Newton’s theorems guarantees that the gravitational potential of an ellipsoidally symmetric object is expressed as a double integral of the given mass density function with no singularities (Binney & Tremaine 2008, section 2.5). Sometimes its inner integral is exactly obtained for simple density distributions (Evans & de Zeeuw 1992). In this case, the total

2

T. Fukushima while J0 (x) is the Bessel function of degree 0 (Olver et al. 2010, equation 10.2.2). Nevertheless, the numerical execution of the Hankel transforms is generally difficult since the integrand is heavily oscillating as shown in Fig. 1, and therefore its numerical integration suffers from a severe cancellation when R is large, say when R > 1. Of course, in this specific case, namely when the volume mass density profile is separable in R and z coordinates as

Integrand of Reverse Hankel Transform 1

a=1, c=0.3, R=10, z=0

0.8

I(k;R,z)

0.6 0.4

ρ(R, z) = Σ(R)ζ (z),

0.2 0 -0.2 -0.4 0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

k Figure 1. Integrand of reverse Hankel transform for biexponential object. Depicted is a heavy[ oscillation of I (k; R, z)] ≡ ) 3/2 ( ) [ ] ( J0 (Rk ) c exp(−|z |/c)k − exp(−k |z |) / a 2 k 2 + 1 c2k 2 − 1 , the integrand of the reverse Hankel transform for a bi-exponential object the volume mass density of which is expressed as ρ BE (R, z) ≡ ρ 0 exp(−R/a − |z |/c). Plotted is the curve when a = 1, c = 0.3, R = 10, and z = 0. The wavelength of the oscillation is in proportion to 1/R, and therefore, the numerical integration of the integrand becomes impractical for large value of R, say when R > 1.

integral evaluation reduces to that of a line integral of an analytic function, which is usually conducted by numerical quadrature. For example, Terzic & Sprague (2007, table 1) presented the densitypotential-force triplets of some triaxial ellipsoids to model stellar systems and dark matter haloes based on the generalized theorems. 1.2

Gravitational field of axisymmetric objects without spheroidal symmetry

On the other hand, if the object has no ellipsoidal symmetry, the problem becomes difficult. This is true even if we assume the axial symmetry of the objects. There exist a few approaches to treat the gravitational field of general axisymmetric discs (Binney & Tremaine 2008, section 2.6). A popular method is the forward and reverse Hankel transforms (Toomre 1963), which actually dates back to the work of Weber (1873) in the electrostatic theory. For example, Sackett & Sparke (1990, equation 26) extended the method of Casertano (1983) for a bi-exponential object, the volume mass density of which is expressed as ρBE (R, z) ≡ ρ0 exp(−ξ − |η|),

(2)

where ξ and η are normalized cylindrical coordinates defined as ξ ≡ R/a, η ≡ z/c.

(3)

The resulting line integral form of the gravitational potential is written as ∫ ∞ ΦBE (R, z) = −GM I (k; R, z)dk, (4) 0

where G is Newton’s constant of unversal attraction, M is the mass of the object, and the integrand I (k; R, z) is defined as [ ] J (Rk) {c exp(−|η|)}k − exp(−k |z|) , (5) I (k; R, z) ≡ 0 ( ) 3/2 ( ) a2 k 2 + 1 c2 k 2 − 1

(6)

effective is a method to superpose the potentials of infinitely thin discs (Binney & Tremaine 2008, Section 2.6.1(c)), which is an extension of the method of Cuddeford (1993). However, this approach (i) is applicable only when ρ(R, z) is written in a separable form, and (ii) becomes cumbersome if Σ(R), the separated surface mass density, is not a member of the known potential-density pairs of infinitely thin disc. Another scheme is a superposition of the known solutions of potential-density pairs. The simplest method uses that of a finite uniform spheroid (Hunter 1963). There exist many solutions for infinitely extended objects (de Zeeuw & Pfenniger 1988) including the famous Miyamoto-Nagai model (Miyamoto & Nagai 1975) and the Satoh model (Satoh 1980). Except a practical difficulty of the resolution of a given density profile into a linear combination of these basic density functions, this approach seems sufficient. 1.3 Difficulties in practical cases However, the reality is not so simple. Refer to Fig. 2 depicting the contour map of the volume mass density of a hypothetical object resembling an X/peanut-shaped galaxy, NGC 128 (D’Onofrio et al. 1999). In order to mimic the observed morphology of the galaxy, we write the volume mass density of the hypothetical object as ρXP (R, z) ≡ ρ0 exp(−s/h),

(7)

where the ellipsoidal argument s is now rewritten as √ s ≡ ξ 2 + η 2,

(8)

and h is a normalized scale height of the object defined as [ ] ( ) h ≡ 1 + h6 exp − (s − s0 ) 2 /s21 C6 ξ, η ,

(9)

while C6 (ξ, η) is an auxiliary function expressed as [ ] C6 (ξ, η) ≡ cos 6 tan−1 (η/ξ) ( ) /( )3 = ξ 6 − 15ξ 4 η 2 + 15ξ 2 η 4 − η 6 ξ2 + η2 .

(10)

We experimentally determined the functional form and the parameters from a clue found in the proposed model curve of the isophote contour of NGC 128 (Ciambur & Graham 2016). Obviously, this kind of density profile is neither ellipsoidally symmetric nor separable in R and z. Its density contours are too wavy to be compactly approximated by a linear superposition of those of the known density models. Also, not all astrophysically meaningful objects are infinitely extended or centrally concentrated. Refer to Fig 3 showing the contour map of the volume mass density of another hypothetical object. It was designed to resemble a protoplanetary disc observed around TW Hya (Andrews et al. 2016, Fig. 2). More specifically speaking, we assume the volume mass density of the object as ( ) ρPH (R, z) ≡ ρ0 ξ −d−β exp −η 2 ξ −2β /2 , (11) MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

ρ of X/Peanut-Shaped Object

ρ of Power-Law Disc with Hole 1.2

1.5 1

0.8

0.5

0.4

0

0

z

z

3

-0.5

-0.4

-1

-0.8

-1.5

-1.2 0 0.5 1 1.5 2 2.5 3 3.5 R

Figure 2. Density contour map of X/peanut-shaped object. Shown is the contour map of the volume mass density of an infinitely extended hypothetical axisymmetric object. Its volume √ mass density is expressed as ρ XP (R,[z) ≡ ρ 0 exp(−s/h) where s ≡ (R/a) 2 + (z/c) 2 , ] h ≡ 1 + h 6 exp − (s − s 0 ) 2 /s 12 C 6 (R/a, |z |/c), and C 6 (ξ, η) ≡ [ ] cos 6 tan−1 (η/ξ) . The figure shows a specific case when a = 1, c = 0.3, s 0 = 4, s 1 = 2, and h 6 = 0.1. The contours are drawn for every 5 per cent of the peak value at the coordinate center. Although the object seems to be limited in a finite region, it infinitely extends with an almost exponential damping. If the unit distance is set as 10 arcsec, the contour map mimics that of the isophote of the HST/NIC3 image of NGC 128 (Ciambur & Graham 2016, Fig. 5). The functionnal form and the parameters were experimentally determined.

where d is a power-law index of the object, which is defined by a step function as { d 1 (Rin < R < R0 ) d≡ (12) d 2 (R0 < R < Rout ) The object has a small but non negligible center hole and of a finite radius as Rin ≤ R ≤ Rout . Again, the density profile is neither ellipsoidally symmetric nor separable in R and z. Also, the profile is not radially analytic at R = R0 , the switching point of the powerlaw index. As a result, none of the aforementioned methods are effectively applicable. Further, some physically important objects have a toroidal shape. Refer to Fig 4 illustrating the contour map of the volume charge (and current) density of a hypothetical object approximating a poloidal mode equilibrium solution of the plasma current circulating in an ITER-like tokamak (Evangelias & Throumoulopoulos 2016, Fig. 4). In a practical sense, this object is limited both in radial and vertical directions. In fact, we model its volume mass density in a modified Gaussian form such that [ ] ρTD (R, z) ≡ ρ0 exp −p2 (ξ − t) 2 − q2 η 2 , (13) where p, q, and t are auxiliary functions defined as ( ) /( ) p ≡ 1 + p1 ξ + p2 ξ 2 1 + b1 t + b2 t 2 , MNRAS 000, 1–43 (2016)

0

0.4 0.8 1.2 1.6 R

2

2.4

Figure 3. Density profile of power-law disc with hole. Same as Fig. 2 but for an axisymmetric object with a central hole. The object is of a finite radial range as R in ≤ R ≤ R out . Its [ volume mass density ] is expressed as ρ PH (R, z) ≡ ρ 0 (R/a) −d−β exp −(z/c) 2 (R/a) −2β /2 where d is a step function defined as d = d 1 when R in < R < R 0 and d = d 2 when R 0 < R < R out . The figure shows a specific case when a = 0.35, c = 0.105, R in = 0.2, R 0 = 2.355, R out = 10, d 1 = 0.53, d 2 = 8.0, and β = 1.25. This time, plotted are the contours when the relative magnitude is exponentially different as e −n for n = 1, 2, . . . , 10. Despite its outlook limited in a finite region as 0.2 < R < 2.36 and |z |/R < 0.3, the object actually extends far outward as R < 10 and infinitely in the vertical direction when 0.2 < R < 10. If the unit distance is chosen as 20 au, this density profile is analogous to that of the observed protoplanetary disc of TW Hya (Andrews et al. 2016, Fig. 2). The functional form is quoted from Hogerheijde et al. (2016) while the parameters were borrowed from a single component determination of Andrews et al. (2016).

q ≡ 1 + q1 ξ + q2 ξ 2, t ≡ t 0 + t 1 η + t 2 η 2 + t 3 η 3 .

(14)

Clearly, the object is not plane symmetric and some of its density contours have a kink near the so-called X-point. Thus, the existing methods are hardly applicable to compute its electrostatic and magnetostatic field. In short, most of the existing methods are not sufficiently general to compute the gravitational, electrostatic, and/or magnetostatic field of a variety of infinite or finite axisymmetric three-dimensional objects; elliptical and lenticular galaxies, disc components of spiral galaxies, dark matter halos, accretion discs, protoplanetary discs, and toroidal plasma, even if their volume mass/charge/current density distributions are known (Márquez et al. 2001; Kormendy & Kennicutt 2004; Graham & Driver 2005; Reylé et al. 2009; Kormendy & Bender 2012; Hunter 2014; Fornasa and Green 2014; Romero-Gómez et al. 2015; Chatzopoulos et al. 2015). Huré and his colleagues published a series of papers to tackle this difficult issue to determine the gravitational field of a

4

T. Fukushima

ρ of Tear-Drop-Shaped Toroid

Double Split Quadrature 4

0.6

3 2 4

1

0

z

z

0.3

3

0

.

-1

-0.3

2

1

-2 -3

-0.6

-4 0

0.3

0.6

0.9

1.2

1.5

R Figure 4. Density contour map of tear-drop-shaped toroid. Shown is the contour map of the volume mass density of yet another hypothetical axisymmetric object, the density of which is expressed as ρ TD (R, z) ≡ [ ] ρ 0 exp −p 2 (R/a − t ) 2 − q 2 (z/c) 2 where p, q, and t are auxiliary [ ] ( ) functions defined as p ≡ 1 + p 1 (R/a) + p 2 (R/a) 2 / 1 + b1 t + b2 t 2 , q ≡ 1+ q 1 (R/a) + q 2 (R/a) 2 , and t ≡ t 0 + t 1 (z/c) + t 2 (z/c) 2 + t 3 (z/c) 3 . The figure shows a specific case when the parameters are set as a = 0.15, c = 0.25, b1 = p 1 /c = p 2 /c 2 = q 1 /a = 0.2, q 2 /a 2 = 0.5, b 2 = −2, t 0 = 1, t 1 = 0, t 2 /c 2 = −0.3, and t 3 /c 3 = −0.1. The contours are drawn for every 5 per cent of the peak value. The functional forms and the parameters are determined such that the resulting contour map resembles that of the poloidal cross-section for an equilibrium solution of the plasma current density in an ITER-like tokamak (Evangelias & Throumoulopoulos 2016, Fig. 4).

general three-dimensional object. Refer to Huré & Trova (2015); Chemin et al. (2016) and the references therein. He noticed that the algebraic singularity of the Newton kernal, i.e., the Green’s function of the Poisson’s equation expressed as 1/ x − x ′ , is the right origin of the difficulty. Then, he proposed a rewriting of the gravitational potential as a cross partial derivative of a new function denoted by ‘hyperpotential’ and defined by an integral transform with a nonsingular kernel function (Huré 2013). In the axisymmetric case, the rewriting reduces to a radial partial derivative of a simplified hyperpotential (Huré & Dieckmann 2012). This approach is sufficiently general although the achieved level of the computational precision is as low as 2.5–3 digits (Huré 2013, Figs 4 and 5).

1.4

Outline of article

Recently, we presented a method to integrate the gravitational field of an infinitely thin object with arbitrary size, shape, and surface mass density distribution (Fukushima 2016). The method numerically computes (i) the gravitational potential by evaluating its line integral expression and (ii) the acceleration vector by conducting a numerical differentiation of the numerically integrated potential.

0

1

2

3

4

5

6

7

8

R Figure 5. Sketch of double split quadrature. In the numerical integration of the gravitational potential of an axisymmetric object at its internal point, we divide the integration domain into four numbered regions radially and vertically separated at the point. Depicted is a case when the cylindrical coordinates of the internal point are R = 4 and z = 0, which is indicated by a bullet in the figure, for a finite object with a convex cross-section.

Although the method uses the classic kernel of an axisymmetric object (Durand 1953), its logarithmic singularity (Huré 2013) is properly treated by the split quadrature method using the double exponential rule (Takahashi & Mori 1973). This scheme can be easily extended to the axisymmetric objects of arbitrary vertical profile of the volume mass density by enlarging the quadrature domain from a line segment to a two-dimensional cross-section. For instance, Fig. 5 shows the manner of splitting quadrature for an axisymmetric object with a finite convex crosssection. In practice, the extended method works well. As will be seen later, it is so precise that the achieved accuracy of the potential computation amounts to around 13 digits. Using the extended method, we prepared Figs 6–9 illustrating the contour map of the computed gravitational potential of the axisymmetric objects the volume mass density of which are already shown in Figs 2–4. The distribution of the gravitational potential is much smoother than that of the density profile, and therefore, the former is sometimes significantly different from the latter. This was unexpected. In this article, we will describe the extended method in Section 2, examine its accuracy in Section 3, and provide some examples in Section 4. MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

4

Φ of Power-Law Disc with Hole 4

3

3

2

2

z

z

Φ of X/Peanut-Shaped Object

1

1

0

0 0

1

2

3

4

0

1

R

2

3

4

R

Figure 6. Potential contour map of X/peanut-shaped object. Shown is the contour map of the gravitational potential of the object the volume mass density of which is described in Fig. 2. The contours are drawn at every 5 per cent level of the bottom value, which is achieved at the coordinate center. Omitted is the lower half part where z < 0 since the object is of the x–y plane symmetry. Despite the peculiar feature of the density contours shown in Fig. 2, the potential contours illustrated here are almost elliptical although they are not genuine ellipses.

Figure 7. Potential contour map of power-law disc with hole. Same as Fig. 6 but for the power-law disc the density of which is depicted in Fig. 3. This time, the bottom value is achieved when R ≈ 0.85 and z = 0. If the spherically symmetric potential of a point mass of certain magnitude is superposed, it would result a realistic model of the gravitational potential of TW Hya.

Φ of Power-Law Disc with Hole 0.4

2 METHOD Integral expression of gravitational potential of axisymmetric object

Consider a general axisymmetric object of an infinite or finite size. Adopt the cylindrical coordinate system, (R, z). Denote the inner and outer radii of the object by Rin (≥ 0) and Rout (≤ +∞), respectively. Without losing the generality, we can assume that the volume mass density of the object, ρ(R, z), vanishes when z ≤ zlower (R) or z ≥ zupper (R) where zlower (R)(≥ −∞) and zupper (R)(≤ +∞) are certain functions of R as shown in Fig. 5. In fact, if the vertical boundary of the object is not uniquely described by functions of R, one may vertically split the object into several pieces so as to satisfy the condition. Under the above assumption, the gravitational potential of the axisymmetric object is expressed as a double integral as ∫ R out ( ) Φ(R, z) ≡ Ψ R ′ ; R, z dR ′, (15) R in

( ) Ψ R ′ ; R, z ≡ −G

∫ z upper (R′ ) z lower (R′ )

0.2 0.1 0 0

) ( ) ( g R ′, z ′ ; R, z ρ R ′, z ′ dz ′,

(16)

( ) where g R ′, z ′ ; R, z is the azimuthal integral of a normalized Newton kernel in the three-dimensional space written as ∫ 2π ( ′ ′ ) R ′ dφ g R , z ; R, z ≡ √ ( ′ ) ( ) 0 R cos φ − R 2 + R ′ sin φ 2 + (z ′ − z) 2 MNRAS 000, 1–43 (2016)

0.3 z

2.1

5

0.1

0.2 R

0.3

0.4

Figure 8. Potential contour map of power-law disc with hole: close-up. Same as Fig. 7 but focused is the cenral part. Notice that the disc exists in a limited region, R ≥ 0.2.

6

T. Fukushima

Φ of Tear-Drop-Shaped Toroid Radial Profile of Kernel Function

1

g(R’,z’;1,0)

0.5 z

0 -0.5

10 9 8 7 6 5 4 3 2 1 0

z’=0

1

z’=10 0

0.5

1

1.5

-1 0

0.5

1

1.5

2

R Figure 9. Potential contour map of tear-drop-shaped object. Same as Fig. 6 but for the tear-drop-shaped toroid. At this scale, the computed contour map of the gravitational potential looks almost plane symmetric although the density distribution described in Fig. 4 is not so.

2.5

3

3.5

4

Figure 11. Radial profile of kernel function. Same as Fig. 10 but plotted as functions of R ′ for various values of z ′ as z ′ = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The thick curves are those for z ′ = 1 and z ′ = 5. The curve of z ′ = 0 blows up logarithmically at R ′ = 1.

( ( )) ( ) = 4R ′ K m R ′, z ′ ; R, z /P R ′, z ′ ; R, z , while

Contour Map of Kernel Function 2

K (m) ≡

∫ π/2 0

dθ , √ 1 − m sin2 θ

(17)

(18)

is the complete elliptic integral of the first kind (Byrd & Friedman ( ) ( ) 1971) with the parameter m, and m R ′, z; R, z and P R ′, z ′ ; R, z are functions defined as / [( ( ) ) ( ) ] m R ′, z ′ ; R, z ≡ 4R ′ R R′ + R 2 + z ′ − z 2 , (19)

1 z’

2 R’

( ) P R ′, z ′ ; R, z ≡

0

√

(R ′ + R) 2 + (z ′ − z) 2 . (20) ( ′ ′ ) The kernel function g R , z ; R, z is, except for the multiplica( ) tion factor R ′ , equivalent with the point value evaluated at R ′, z ′ of the gravitational potential of a uniform ring located at (R, z) (Fukushima 2010). The precise and fast computation of K (m) is realized by the program ceik (Fukushima 2015). Refer to Fig. 10 ( ) showing the contour map of g R ′, z ′ ; R, z for the case R = 1 and z = 0. See also Figs 11 and 12 plotting its radial and vertical profiles, respectively.

-1 -2 0

1

2

3

4

R’ Figure 10. Contour map of kernel function. Illustrated is the contour map of the kernel function of the axisymmetric integral expression of the gravita( ) tional potential, g R ′, z ′ ; R, z , when R = 1 and z = 0. The function has a blow-up logarithmic singularity at R ′ = R and z ′ = z, which is indicated by dense concentric circles around it.

2.2 Piecewise density function Assume that ρ(R, z) is a doubly piecewise function with respect to R and z such as ( ) ρ(R, z) = ρi j (R, z), Ri < R < Ri+1 ; z i j (R) < z < z i, j+1 (R) (21) for i = 1, 2, . . . , I and j = 1, 2, . . . , Ji . Here we presume that R1 ≡ Rin , R I +1 ≡ Rout , z i1 (R) ≡ zlower (R), and z i, Ji +1 (R) ≡ zupper (R). MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

10 9 R’=1 8 7 6 5 4 0.5 3 2 1 0 0 0.5

Computational Cost of Vertical Integration 10 9 8 7 6 5 4 3 2 1 0

R’=∞

2

1

1.5

2

2.5

3

without splitting

3.5

4

split quadrature 0

0.2 0.4 0.6 0.8

z’

Then, we compute Φ(R, z) by summing the contributions of each piece as Ji I ∑ ∑ *. Φ (R, z) +/ , ij i=1 , j=1 -

where Φi j (R, z) ≡

∫ R j+1 Rj

( ) Ψi j R ′ ; R, z dR ′,

zi j

(R ′ )

(23)

( ) ( ) g R ′, z ′ ; R, z ρi j R ′, z ′ dz ′ . (24)

In principle, any kind of numerical quadrature rule can be used for the evaluation of the double integrals. Nevertheless, we adopt a group of the double exponential (DE) quadrature rules (Takahashi & Mori 1973) since it is regarded as a set of the most efficient methods to evaluate a line integral (Bailey et al. 2005). See also Press et al. (2007, section 4.5).

2.3

Vertical split quadrature

( ) As seen in Figs 10–12, the kernel function, g R ′, z ′ ; R, z , has a ( ′ ′) logarithmic blow-up singularity at R , z = (R, z). This hinders a proper convergence of the numerical integration of the inner line integral, equations (16) or (24), by almost all the existing quadrature ( ) ( ) rules. Thus, when z i j R ′ < z < z i, j+1 R ′ for a certain pair of indices, (i, j), and for a given value of R ′ , we split the vertical integration interval at z ′ = z as ∫ z ) ( ( ) ( ) g R ′, z ′ ; R, z ρi j R ′, z ′ dz ′ Ψi j R ′ ; R, z = −G z i j (R ′ )

MNRAS 000, 1–43 (2016)

2

Figure 13. Computational cost of vertical integration. Shown is Neval , the number of function calls required in the numerical integration. Plotted are the results for a line integral representing the gravitational potential of an infinitely thin uniform cylindrical surface of the radius and the half height, a = c = 1. Compared are Neval as functions of R′ , the radius of the location of the vertical line integral, of two methods using the double exponential quadrature rule with the relative error tolerance δ = 10−8 : (i) the single quadrature without interval splitting, and (ii) the sum of two quadratures for the intervals split at z ′ = z. The former method could not obtain the correct integral value when R′ ≈ a.

−G

∫ z i, j+1 (R′ )

1.2 1.4 1.6 1.8

(22)

while ( ) Ψi j R ′ ; R, z ≡ −G

1 R’

Figure 12. Vertical profile of kernel function. Same as Fig. 10 but plotted as functions of z ′ for various values of R ′ as R ′ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.2, 1.4, 1.6, 1.8, 2, 3, 4, 5, 6, 8, 10, 20, and R ′ → ∞. The thick curves show the cases when R ′ = 0.5, 1, 2, and R ′ → ∞. The curve of R ′ = 1 blows up logarithmically at z ′ = 0. The curve of R ′ → ∞ is a constant value of 2π.

Φ(R, z) =

z=0, δ=10-8

Uniform Cylindical Surface, a=c=1

Neval/102

g(R’,z’;1,0)

Vertical Profile of Kernel Function

7

∫ z i, j+1 (R′ ) z

( ) ( ) g R ′, z ′ ; R, z ρi j R ′, z ′ dz ′ .

(25)

One may think that the unconditional splitting is inefficient because the sharpness of the peak is relaxed when R ′ is sufficiently far from R. However, this anticipation is betrayed. Experimentally, we learn that any kind of singularity including the logarithmic one slows down the convergence of the DE rules independently on the distance of integration element from the singularity unless it is located at the end point of integration as Takahashi & Mori (1973) predicted. Indeed, Fig. 13 indicates that the split quadrature is not less efficient than the quadrature without splitting. This situation is generally independent on the distance from the singular point. Thus, it is a good practice to split the interval of vertical line integrals always at the foot of the singular point as z ′ = z even if R ′ , R. 2.4 Radial split quadrature Move to the outer line integral, equations (15) or (23). First of all, we notice that the singularity remains even after the vertical integration. This fact is easily understood from the fact that the integral of the logarithmic function is of a similar singularity as ∫ (ln x) dx = −x + x ln x. (26) This time, nonetheless, somewhat costly is the unconditional splitting with respect to the radial integration variable, R ′ , such as Φi j (R, z) =

∫ R Ri

( ) Ψi j R ′ ; R, Z dR ′ +

∫ R i+1 R

( ) Ψi j R ′ ; R, Z dR ′,

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-8

Uniform Cylinder, a=c=1, z=0, δ=10

Neval/104

2 unconditional 1.5 |R’-R| < 0.2 1 0.5 external

2.7 Calculation of circular speed

0 0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

R Figure 14. Computational cost of double split quadrature. Same as Fig. 13 but for the numerical evaluation of a double integral representing the gravitational potential of a finite uniform cylinder of the radius and the half height as a = c = 1. Compared are two double split quadrature methods using the DE rules: (i) that of unconditional radial splitting at R′ = R, and (ii) that with the radial splitting only when |R ′ − R | < 0.2.

(27) when Ri < R < Ri+1 . In fact, the restriction of the splitting to the cases of small distance from the singularity reduces the computational labor slightly, say 10 per cent in average. Refer to Fig. 14 for the case of the gravitational potential computation of a uniform cylinder of finite thickness. However, we do not recommend the conditional splitting from the viewpoint of the parallel computing as will be explained in the next subsection.

2.5

Parallel computing of split quadrature

The computational amount of the numerical integration by the DE rules is almost independent on the length of the interval. This is because the DE rules (i) transform the given integration interval, whether being infinite or finite, into an infinite interval, (−∞, +∞), and then (ii) compute the truncated trapezoidal sum (Press et al. 2007, section 4.5). If both the vertical and radial splitting is conducted, the gravitational potential computation is reduced to those of at least four integrals. Refer to Fig. 5 once again. Each of the split integrals can be computed independently with each other and their computational times are roughly the same. Thus, appropriate is their parallel execution. Since the number of parallel computable pieces is as small as 2 typically and 6 or 8 at most, even an ordinary PC with 2–8 cores can be used for their parallel computation, say by employing the OpenMP architecture (OpenMP 2015).

2.6

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which are denoted by K R and K z in some literature. We did not select these notations in order to avoid the confusion with the derivatives of K (m). In principle, these differentiations are not well defined at some points where the potential is not analytic. Examples are some exponential disc models of spiral galaxies on the x–y plane and/or along the z-axis. However, the non analyticity is an unphysical situation for a natural smoothly continuous body. Also, we may assign zero values in those cases if the object is of a certain symmetry such as those with respect to the x–y plane in almost all cases.

2.5

internal

by Ridder’s method (Ridder 1982) as ( ) ) ( ∂Φ(R, z) ∂Φ(R, z) FR (R, z) ≡ − , Fz (R, z) ≡ − , ∂R ∂z z R

Computation of acceleration vector

Following the recipe for general infinitely thin axisymmetric objects (Fukushima 2016), we evaluate the associated acceleration vector by numerically differentiating the numerically integrated potential

When the object is symmetric with respect to the x–y plane as usual, the z-component of the acceleration vector vanishes on the plane as Fz (R, 0) = 0. In this case, if FR (R, 0) ≤ 0 further, the circular speed is well defined and expressed as √ V (R) ≡ −RFR (R, 0). (29) As will be shown later, FR (R, 0) can be positive in some cases, and then V (R) becomes pure imaginary. This is no unphysical situation (Fukushima 2016). It simply means that the gravitational attraction from the external part is so strong that the resulting acceleration directs toward outside, and therefore no centrifugal force can keep a hypothetical particle on a circular orbit in such cases. 2.8 Implementation notes As the programs of numerical quadrature, we recommend Ooura’s efficient implementations of the DE rules, intde and intdei, the programs to obtain the integrals over a finite interval, (t 1, t 2 ), and over a semi-infinite interval, (t, ∞), respectively (Ooura 2006). In evaluating the double integrals, we prepare two sets of their copies, say intdez and intdeiz for the vertical integration and intder and intdeir for the radial integration. On the other hand, in computing the partial derivatives, we utilize dfridr, the fortran function listed in Press et al. (1992, section 5.7). Refer to Appendix B for the implementation notes on the parallel integration. In computer programming, the usage of global variables are strongly discouraged (Brooks 1997). Here a variable is global when it is viewable from and rewritable by any sub program. Its typical example is a variable in the common blocks of fortran. Global variables are disliked because, in general, (i) they make the interaction among sub programs so complicated, and as a result, (ii) they tend to generate erroneous computer codes. However, they are necessary in utilizing the ready-made programs like intde. This is because it would be cumbersome to specify all the variables/parameters explicitly as the arguments of the subprograms. In place of giving a general discussion, we present below the main part of a sample fortran code to evaluate the double integral, Φ(R, z), without the integration interval splitting: real*8 function phiint(r,z) ... common /param/rin,rout,zlower,zupper,rx,zx,rpx rx=r; zx=z call intder(psiint,rin,rout,eps,phiint,err) return; end MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object Table 1. Averaged CPU time of potential computation. Shown are the averaged CPU time to compute a single point value of the gravitational potential by the new method when the relative error tolerance is set as moderate as δ = 10−8 . The unit of the CPU time is ms at a PC with an Intel Core i7-4600U CPU running at 2.10 GHz clock while no parallel computing is employed. The results for various infinitely extended objects, the explanation of which are found in section 4 and Appendix E, are listed in the increasing order of the CPU time. object Gaussian spheroid perfect spheroid Burkert spheroid modified Hubble spheroid Plummer spheroid exponential/Gaussian object bi-exponential object Navarro-Frenk-White spheroid exponential spheroid exponential/sech2 object flared bi-exponential object modified exponential spheroid Hernquist spheroid differenced modified-exponential object boxy object Einasto spheroid (i = 2) Jaffe spheroid Einasto spheroid (i = 4) differenced Gaussian spheroid Einasto spheroid (i = 7) Gaussian oval toroid Sérsic spheroid (n = 2) de Vaucouleurs spheroid Sérsic spheroid (n = 1) X/peanut-shaped object

CPU time (ms) 0.41 0.46 0.46 0.47 0.49 0.53 0.55 0.60 0.61 0.63 0.65 0.71 0.71 0.71 0.74 0.93 0.96 0.98 1.23 1.29 1.57 1.87 2.23 2.28 2.29

real*8 function psiint(rp) ... common /param/rin,rout,zlower,zupper,rx,zx,rpx rpx=rp call intdez(fint,zlower,zupper,eps,psiint,err) return; end real*8 function fint(zp) ... common /param/rin,rout,zlower,zupper,rx,zx,rpx fint=-rho(rpx,zp)*green(rpx,zp,rx,zx) return; end where rx = R, zx = z, and rpx = R ′ are stored in a named common block and their numerical values are secretly passed by way of the block.

2016). This amount is so small that the rotation curve of such an object is typically prepared in less than a second.

3 VALIDATION 3.1 Finite uniform spheroid Let us examine the accuracy of the new method by comparing its result with a few exact solutions. We begin with a finite axisymmetric object: the finite uniform spheroid of the semi-axes, a and c. The exact solution is already displayed in Appendix A. The new method integrates the gravitational potential (i) by a single line integral if R = 0 or a < R as ∫ a ( ) ΦS (R, z) = ΨS R ′ ; R, z dR ′, (30) 0

and (ii) by a radial split quadrature when 0 < R < a as ∫ R ∫ a ( ) ( ) ΦS (R, z) = ΨS R ′ ; R, z dR ′ + ΨS R ′ ; R, z dR ′ . 0

CPU time

One may think that the new method is time-consuming. This is only partially true. Table 1 lists the averaged CPU time of a single point evaluation of the gravitational potential of various infinitely extended objects, which will be explained later in section 4 and Appendix E, by the new method when the relative error tolerance is set as moderate as δ = 10−8 . The CPU times range 0.4–2.3 ms at an ordinary PC. On the other hand, Ridder’s method requires 2– 10 evaluations of the gravitational potential in general (Fukushima MNRAS 000, 1–43 (2016)

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R

( ) Meanwhile, ΨS R ′ ; R, z is well-defined only when 0 ≤ R ′ ≤ a. Thus, it is integrated (i) by a reflected quadrature if z = 0 as ∫ √c 2 −(R/a) 2 ( ′ ) ( ) ΨS R ; R, 0 = −2G ρ0 g R ′, z ′ ; R, 0 dz ′, (32) 0

√ (ii) by a single quadrature if |z| > c2 − (R/a) 2 as ∫ √c 2 −(R/a) 2 ( ) ( ′ ) g R ′, z ′ ; R, z dz ′, ΨS R ; R, z = −G ρ0 √ − c 2 −(R/a) 2

(33)

and (iii) by a split quadrature otherwise as ∫ z ( ) ( ) ΨS R ′ ; R, z = −G ρ0 √ g R ′, z ′ ; R, z dz ′ − c 2 −(R/a) 2

−G ρ0

∫ √c 2 −(R/a) 2 z

( ) g R ′, z ′ ; R, z dz ′ .

(34)

3.2 Miyamoto-Nagai model Next, consider the case of an infinitely extended object: the Miyamoto-Nagai model (Miyamoto & Nagai 1975). Refer to Appendix C for the explicit description of its density-potentialacceleration triplet. The new method integrates the gravitational potential (i) ordinary if R = 0 as ∫ ∞ ( ) ΦMN (0, z) = ΨMN R ′ ; 0, z dR ′, (35) 0

and (ii) by a radial split quadrature otherwise as ∫ R ∫ ∞ ( ) ( ) ΦMN (R, z) = ΨMN R ′ ; R, z dR ′ + ΨMN R ′ ; R, z dR ′ . 0

2.9

9

R

(36) ( ′ ) ( ′ ′) Meanwhile, ΨMN R ; R, z is prepared from ρMN R , z , the volume mass density of the model, (i) as a reflected quadrature if z = 0 as ∫ ∞ ) ( ) ( ) ( ΨMN R ′ ; R, 0 = −2G g R ′, z ′ ; R, 0 ρMN R ′, z ′ dz ′, (37) 0

and (ii) by a split quadrature into three pieces otherwise as ∫ ∞ ) ( ) ( ) ( ΨMN R ′ ; R, z = −G g R ′, z ′ ; R, z ρMN R ′, z ′ dz ′ |z |

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Gravitational Potential Error: Uniform Spheroid

Gravitational Potential Error: Miyamoto-Nagai

-12

-12 δ=10-13, aM=1, bM=0.5

δ=10-13, a=1, c=0.5 -13 log10 |δΦ|

log10 |δΦ|

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Figure 15. Integration error of gravitational potential: finite uniform spheroid. Shown are the integration errors of the gravitational potential of finite uniform spheroid obtained by the new method. Results are expressed as the base 10 logartihm of the normalized absolute errors of the potential numerically integrated with the relative error tolerance δ = 10−13 . They are plotted as functions of R for several values of z as z = 0, 0.1, 1, and 10 while the semi-axes are fixed as a = 1 and c = 1/2. The errors of different z are overlapped since their distribution is almost the same.

−G

0

∫ ∞ 0

( ) ( ) g R ′, z ′ ; R, z ρMN R ′, z ′ dz ′

( ) ( ) g R ′, −z ′ ; R, z ρMN R ′, −z ′ dz ′ .

8

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Radial Acceleration Error: Miyamoto-Nagai δ=10-13, aM=1, bM=0.5

-10

(38)

The interval splitting at z ′ = 0 is needed since the Miyamoto-Nagai density function, ρMN (R, z), is not analytic on the x–y plane.

log10 |δFR|

−G

7

Figure 16. Integration error of gravitational potential of MiyamotoNagai model. Same as Fig. 15 but for the Miyamoto-Nagai model (Miyamoto & Nagai 1975) for its model parameters, a M = 1 and bM = 1/2.

-9 ∫ |z |

6

R

-11 -12 -13

3.3

Care for numerical differentiation

In conducting the numerical differentiation, we must take care in selecting the test displacements in Ridder’s method such that (i) the test radial arguments never become negative, and (ii) the test radial/vertical arguments do not cross the internal/external manifolds with the non-analyticity such as the surface of the finite spheroid, (R/a) 2 +(z/c) 2 = 1, or the x–y plane in the Miyamoto-Nagai model. More concretely speaking, we choose the initial test displacements in the latter case of the Miyamoto-Nagai model as ( ) ( ) R max(1, R) |z| max(1, |z|) h R = min , h = min . (39) , , z 2 2 103 103

3.4

Computational errors

Figs 15–18 show the errors of the gravitational potential and the associated acceleration vector computed by the new method when the input relative error tolerance of the numerical quadrature is set as small as δ = 10−13 . The figures plot the base 10 logarithm of the errors normalized in the sense [ ]/ δΦ(R, z) ≡ Φ(R, z) − Φ∗ (R, z) Φ∗ (R, z), (40)

-14 0

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R Figure 17. Computational error of acceleration vector of Miyamoto-Nagai model: radial component. Same as Fig. 16 but for the computational errors of the radial component acceleration.

[ ]/ δFR (R, z) ≡ FR (R, z) − FR∗ (R, z) F ∗ (R, z),

(41)

]/ [ δFz (R, z) ≡ Fz (R, z) − Fz∗ (R, z) F ∗ (R, z),

(42)

where the asterisk stands for the exact solution and √ ]2 [ ]2 [ ∗ F (R, z) ≡ FR∗ (R, z) + Fz∗ (R, z) .

(43)

Although Figs 15–18 present the case only for a limited selection of the parameters such as a, c, aM , and bM , we observed almost MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

11

ρ of Bi-Exponential Object Vertical Acceleration Error: Miyamoto-Nagai δ=10

-13

1.5

, aM=1, bM=0.5

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R Figure 18. Computational error of acceleration vector of Miyamoto-Nagai model: vertical component. Same as Fig. 17 but for the vertical component acceleration.

the same situation for other combinations. At any rate, as seen in Figs 15–18, the obtained errors of the gravitational potential and of the associated acceleration vector are roughly the same as and 100–1000 times larger than the input relative error tolerance, δ, respectively. This fact shows a guideline to select δ in the actual computation. For example, if the five digits accuracy is required in the rotation curve computation, one may set δ = 10−8 .

0

0.5 1 1.5 2 2.5 3 R

Figure 19. Density contour map of bi-exponential object. Shown is the contour map of the volume mass density of an axisymmetric object expressed as ρ BE (R, z) ≡ ρ 0 exp(−R/a − |z |/c) when the scale length parameters are chosen as a = 1 and c = 0.3. The contours are drawn at every 5 per cent level of the peak value. They are rhombi of the aspect ratio, a : c = 10 : 3. Although the contours seem to be localized in a diamond-shaped region as R + |z |/0.3 ≤ 3, the object is infinitely extended in reality.

where f (ξ) is a function describing the manner of flaring as f (ξ) ≡ 1 + kξ,

4 EXAMPLES In order to show the performance of the new method, we present below the potential contour map and the rotation curve of a variety of axisymmetric objects. In section 1, we already illustrated the contour map of the gravitational potential for fairly complicated objects; (i) an infinitely extended X/peanut-shaped object, (ii) a radially finite and vertically infinite power-law disc with a central hole, and (iii) an almost bounded tear-drop-shaped toroid with a non-analytic cross-section. On the other hand, the gravitational field of some simple axisymmetric objects are given in Appendices D and E. The former deals with some finite uniform objects, which are interesting from a theoretical view point, while the latter listed a number of infinitely extended spheroids, which will be useful in considering the gravitational field of the elliptical galaxies, the core/bulge of disc/spiral galaxies, and the visible and dark matter halos. Therefore, in this section, we focus on non-trivial but simple objects with the axial symmetry, namely (i) several infinitely extended axisymmetric objects with non-spheroidally symmetric density distribution, and (ii) a few finite axisymmetric objects with a non-uniform density distribution. Actually considered are the following objects: (i) the bi-exponential object the volume mass density of which is already described in equation (2), (ii) a flared bi-exponential object with the density profile as ρFE (R, z) ≡ ρ0 exp[−ξ − |η|/ f (ξ)], MNRAS 000, 1–43 (2016)

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(45)

(iii) a variation of the bi-exponential object the density profile of which is defined as ρES (R, z) ≡ ρ0 exp(−ξ)sech2 (η/2),

(46)

(iv) a radially exponential and vertically Gaussian object with the density profile ) ( (47) ρEG (R, z) ≡ ρ0 exp −ξ − η 2 , (v) an object with a boxy density profile defined as ( √ ) 4 4 ρBX (R, z) ≡ ρ0 exp − ξ + η ,

(48)

(vi) the difference of the Gaussian spheroids as [ ( ) ( )] ρDG (R, z) ≡ ρ0 exp −s2out − exp −s2in ,

(49)

where sout and sin are defined as √ √ 2 + η2 , s ≡ ξ2 + η2 , sout ≡ ξout in in in out

(50)

and the related normalized variables are written as ξout ≡ R/aout , η out ≡ z/cout , ξin ≡ R/ain , and η in ≡ z/cin , where aout , cout , aout , and cin are the semi-axes of the spheroids, (vii) a similar difference of the modified exponential spheroids as [ ] ρDM (R, z) ≡ ρ0 exp (−σout ) − exp (−σin ) (51)

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ρ of Exp-Gaussian Object

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ρ of Flared Bi-Exponential Object

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0.5 1 1.5 2 2.5 3 R

Figure 20. Density contour map of flared bi-exponential object. Same as Fig. 19 but for an axisymmetric object with the volume mass density profile expressed as ρ FE (R, z) ≡ ρ 0 exp[−R/a − ( |z |/c)/ f (R/a)] where f (ξ) is a function describing the normalized vertical scale height as f (ξ) = 1 + k ξ while the parameters are chosen as a = k = 1 and c = 0.3.

0

Figure 22. Density contour map of radially exponential and vertically Gaussian disc. Same as Fig. 19 but when the volume )mass density profile is ex( pressed as ρ EG (R, z) ≡ ρ 0 exp −R/a − z 2 /c 2 where a = 1 and c = 0.3.

ρ of Boxy Object

ρ of Exp-Sech2 Object

0.9

1.5

0.6

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0.3 z

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0.5 1 1.5 2 2.5 3 R

0

0 -0.3

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Figure 21. Density contour map of radially exponential and vertically secanthyperbolic-squared disc. Same as Fig. 19 but when the volume mass density profile is expressed as ρ ES (R, z) ≡ ρ 0 exp(−R/a)sech2 [−|z |/(2c)] where a = 1 and c = 0.3.

0.3 0.6 0.9 1.2 1.5 1.8 R

Figure 23. Density contour map of boxy object. Same as Fig. 19 but when the profile is expressed as ρ BX (R, z) ≡ ] [ √volume mass density ρ 0 exp − (R/a) 4 + (z/c) 4 where a = 1 and c = 0.3.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

ρ of Gaussian Oval Toroid

13

Φ of Bi-Exponential Object

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Figure 24. Density profile of Gaussian oval toroid. Same as Fig. E2 but for [a toroid with the volume mass density written as ρ GT (R, z) ≡ ] ρ 0 exp − {(R − R T ) /a }2 − {( |z |/c)/ f (R/a) }2 where f (ξ) ≡ 1 + k ξ is a linear flairing function while the parameters are set as R T = 2, a = 1, c = 0.3, and k = 0.7. We determined the functional form and the parameters to mimic the result of N -body simulation of particle swarm with a toroidal feature (Bannikova et al. 2012, Fig. 10).

Figure 25. Potential contour map of bi-exponential object. Same as Fig. 6 but for the bi-exponential object. Its volume mass density is described as ρ BE (R, z) ≡ ρ 0 exp(−R/a − |z |/c) when a = 1 and c = 0.3. This distribution resembles that of the thick stellar disc of the Milky Way if the unit distance is chosen as 4.0 kpc (Robin et al. 2003, table 3). The contours are drawn at every 5 per cent level of the bottom value. The potential is not analytic on the x–y plane and along the z-axis although the non-analyticity is hardly visible at this scale.

where σout and σin are defined as 2s2out 2s2in σout ≡ , σin ≡ , √ √ 1 + 1 + 4s2out 1 + 1 + 4s2in

(52)

and (x) a toroidal object with the Gaussian density profile as [ ( ] ) ρGT (R, z) ≡ ρ0 exp − ξ − ξT 2 − {η/ f (ξ)}2 , (53) where ξT is a normalized radius of the toroid while f (ξ) is the same flare function as given in equation (45). Refer to Figs 19–24 for the contour map of the volume mass density of these objects when a = 1, c = 0.3, and the other parameters are appropriately specified. Let us explain the background of these objects one by one. First, a bi-exponential disc is a simple but realistic model of the thick stellar disc and/or of the Inter Stellar Medium (ISM), namely the gas and dust, of the spiral galaxies (Bahcall & Soneira 1980). In fact, Fig. 19, namely the case when c/a = 0.3, approximates the thick stellar disc of the Milky Way if the unit distance is chosen as 4.0 kpc (Robin et al. 2012, table 2). Also, the bi-exponential disc when c/a = 0.03 resembles the ISM of the Milky Way if the unit distance is chosen as 4.5 kpc (Robin et al. 2003, table 3). Of course, the infinitely thin limit when c → 0 is nothing but the infinitely thin exponential disc (Freeman 1970). Secondly, the gas discs of many edge-on galaxies show a flaring (van der Kruit 2008). Although some profiles indicate a more rapid increase (O’Brien et al. 2010a,b,c,d), we assume here a linear flaring as ρFE (R, z) for simplicity. Thirdly, there exist a few variations on the vertical profile of a MNRAS 000, 1–43 (2016)

bi-exponential disc; the Gaussian decay, the exponential damping, and their linear combination. A recent investigation (Robin et al. 2014) implies that the vertical profile of the thick stellar disc component of the Milky Way seems to be in proportion to the square of the secant hyperbolic function as ρES (R, z). Fourthly, some galaxies including LEDA 074886 exhibit a boxy outlook like ρBX (R, z) (Graham et al. 2012). Fifthly, practically important is the difference of representative spheroidal distributions like (i) the differenced Gaussian spheroid as ρDG (R, z), and (ii) the differenced modified exponential spheroid as ρDM (R, z). For example, in the Besançon galaxy model, the thin disc of young and old stars are approximated by the former and the latter, respectively (Robin et al. 2003). In principle, their gravitational field can be obtained as the difference of the gravitational field of the undifferenced spheroids as shown in Appendix E. However, since the structure is so different, we shall discuss them here as special examples. Finally, as a simple model of toroidal object, we consider a Gaussian toroid with a flared scale height. Its functional form was hinted from the result of N-body simulation of a particle swarm (Bannikova et al. 2012).

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Φ of Bi-Exponential Object, c=0.03 4

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R Figure 27. Potential contour map of bi-exponential object: c = 0.03. Same as Fig. 25 but when c = 0.03. This approximates the contour map of the gravittional potential of the ISM disc of the Milky Way if the unit distance is chosen as 4.5 kpc (Robin et al. 2003, table 3). Still visible is the nonanalyticity on the x–y plane.

Φ of Bi-Exponential Object, c=3 4 3 z

Figure 26. Potential contour map of bi-exponential object: c = 0.003. Same as Fig. 25 but when c = 0.003. At this scale, observed will be no further difference in the potential contour map when c ≤ 0.003. In this sense, the contour map shown here is practically the same as that of the infinitely thin exponential disc (Freeman 1970). This time, the non-analyticity on the x–y plane is obvious.

2

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Figure 28. Potential contour map of bi-exponential object: c = 3. Same as Fig. 25 but when c = 3.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of Flared Bi-Exponential Object 4

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Figure 29. Potential contour map of bi-exponential object: c = 30. Same as Fig. 25 but when c = 30.

Figure 31. Potential contour map of flared bi-exponential object. Same as Fig. 25 but for a flared bi-exponential object. Its volume mass density is described as ρ FE (R, z) ≡ ρ 0 exp[−R/a − ( |z |/c)/ f (R/a)] where f (ξ) = 1 + k ξ. Shown is the result when a = k = 1 and c = 0.3.

Φ of Bi-Exponential Object, c=300 4

Φ of Exp-Sech2 Object 4

3 2 z

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Figure 30. Potential contour map of bi-exponential object: c = 300. Same as Fig. 25 but when c = 300. At this scale, observed will be no further difference in the potential contour map when c ≥ 300.

MNRAS 000, 1–43 (2016)

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Figure 32. Potential contour map of radially exponential and vertically secant-hyperbolic-squared object. Same as Fig. 25 but when the volume mass density profile is given as ρ ES (R, z) ≡ ρ 0 exp(−R/a)sech2 [−|z |/(2c)] when a = 1 and c = 0.3.

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Φ of Boxy Object

4

4

3

3

2

2

z

z

Φ of Exp-Gaussian Object

1

1

0

0 0

1

2

3

4

0

1

R

2

3

4

R

Figure 33. Potential contour map of radially exponential and vertically Gaussian object. Same as Fig. 25( but when the volume mass density profile ) is given as ρ EG (R, z) ≡ ρ 0 exp −R/a − z 2 /c 2 when a = 1 and c = 0.3.

Figure 34. Potential contour map of boxy object. Same as Fig. 25 but when the profile is expressed as ρ BX (R, z) ≡ [ √volume mass density ] ρ 0 exp − (R/a) 4 + (z/c) 4 when a = 1 and c = 0.3.

Using the new method, we evaluated the gravitational potential of these objects. Refer to Figs 25–36 for the contour map of the gravitational potential of these objects when a = 1, c = 0.3, and the other parameters are appropriately specified. Also, we prepared the associated rotation curves of these objects for various values of the aspect ratio, c/a, in Figs 38–46. Refer to Fig. 47 for the comparison of the rotation curves of some selected objects for the same aspect ratio, c/a = 0.3. Examine these results one by one. First, Fig. 25 illustrates that, despite the rhombic feature of the density contours shown in Fig. 19, the potential contours depicted here are almost elliptical. Of course, the contours are not genuine ellipses. In fact, they are not analytical along the z-axis and on the x–y plane although the non-analyticity is hardly visible at this scale. At any rate, impressive is a significant difference in the potential and density contour maps. Let us investigate the details of the bi-exponential objects; especially the effect of the aspect ratio. Figs 26–30 show the similar potential contour map for several values of c as c = 0.003, 0.03, 3, 30, and 300 while a is fixed as a = 1. That of the case when c = 0.3 is already shown in Fig. 25. As noticed already, the case when c/a = 0.03 well represents the gravitational potential of the ISM disc of the Milky Way if a is chosen as 4.5 kpc (Robin et al. 2003, table 3). Also, when c/a is sufficiently small, say when c/a ≤ 0.003, no qualitative difference is observed from the infinitely thin limit, namely the infinitely thin exponential disc (Freeman 1970). Another point of interest is the effect of flaring. Fig. 31 shows the contour map of the gravitational potential of the flared biexponential disc when a = 1 and c = 0.3. Its comparison with the unflared case, Fig. 25, may give an impression that the effect of flaring is small. In order to investigate its details, we prepared Fig. 40 showing the rotation curves for various values of the flaring factor, k, as k = 0 through k = 5. The comparison of the rota-

Φ of Differenced Gaussian Spheroid 4

z

3 2 1 0 0

1

2 R

3

4

Figure 35. Potential contour map of differenced Gaussian spheroid. Same as Fig. 25 but when the volume mass density profile √ is given as ρ DG (R, z) ≡ [ ( ) ( )] 2 2 2 + z 2 /c 2 , s ρ 0 exp −s out − exp −s in where s out ≡ R 2 /a out in ≡ out √ 2 + z 2 /c 2 , while a R 2 /a in out = 2, c out = 0.6, a in = 1, and c in = 0.3. in Notice that the central concentration is significantly reduced.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of Gaussian Oval Toroid 4

3

3

2

2

z

z

Φ of Differenced Mod. Exp. Spheroid 4

1

1

0

0 0

1

2

3

4

0

1

2

R Figure 36. Potential contour map of differenced modified exponential spheroid. Same as Fig. 35 but when the volume mass density profile [ ] is given as ρ DM (R, z) ≡ ρ 0 exp (−σ out exp (−σ in)) where σ out ≡ ( ) −√ √ ) ( 2 2 / 1 + 1 + 4s 2 , σ 2 2s out in ≡ 2s in / 1 + 1 + 4s in , while s out ≡ out √ √ 2 + z 2 /c 2 , s ≡ 2 + z 2 /c 2 , and a R 2 /a out R 2 /a in out = 2, c out = 0.6, in out in a in = 1, and c in = 0.3. The central concentration is further reduced.

17

3

4

R Figure 37. Potential contour map of Gaussian oval toroid. Same as Fig. 25 but when [ the volume mass density profile ]is given as ρ GT (R, z) ≡ ρ 0 exp − (R − R T ) 2 /a 2 − {(z/c)/ f (R/a) }2 while a = 1, c = 0.3, R T = 2, and k = 0.7.

tion curves reveals that, when k increases, the peak location moves outward although the outlook of the rotation curves is almost the same. The volume mass density distribution of thin discs in the vertical direction is a controversial issue. One possibility is the same exponential damping as described by the bi-exponential objects. However, this vertical profile has a kink on the x–y plane. As a result, the resulting gravitational potential is continuous but nonanalytic on the plane. This is unphysical. Other options to concur this problem are the vertical distribution described by (i) the squared secant hyperbolic function as ρES (R, z) described in equation (46), and (ii) the Gaussian distribution as ρEG (R, z) described in equation (47). Indeed, Figs 32 and 33 provide the smoother contour maps of their gravitational potential although the flattening of the contours are different. Meanwhile, the bulge components of some disc/spiral galaxies exhibit a boxy feature like that described in equation (48). In this specific case, Fig. 34 indicates that the central concentration is stronger despite the fact that the elliptical outlook of contours is almost unchanged. In some models of the stellar mass distribution, the age difference is reflected to the difference of spatial distributions. For example, the Besançon galaxy model adopted a classification of the thin stellar disc into 7 components of different ages and assigned the differenced Gaussian or modified Gaussian spheroids to each of them (Robin et al. 2003, table 3). Figs 35 and 36 present the potential contour map of these difference spheroids. Obvious is the central flatness of the potential values. This is well reflected in their MNRAS 000, 1–43 (2016)

V/V0

Circular Speed: Bi-Exponential Disc 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

δ=10-8, a=1

c=0 0.3 c=1 c=3 c=10 c=30

0

2

4

6

8

10 12 14 16 18 20 R

Figure 38. Rotation curve: bi-exponential object. Shown are the rotation curves of bi-exponential objects the volume mass density of which are expressed as ρ BE (R, z) ≡ ρ 0 exp(−R/a − |z |/c). Plotted are the circular velocities for various values of c as c = 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10, and 30 while a is fixed as a = 1. The normalization constant V0 is arranged such that the total mass of the objects are the same. As a result, all the curves approach to the same functional form, namely that of a point mass, when R → ∞. At this scale, invisible are the departure of the curves when c ≤ 0.01 from the limit of c = 0, namely the rotation curve of the infinitely thin exponential disc described by a cross product of the modified Bessel functions (Freeman 1970).

18

T. Fukushima

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

Circular Speed: Exponential/Sech-Squared Object

δ=10-8, a=k=1

c=0 0.3 c=1

V/V0

V/V0

Circular Speed: Flared Bi-Exponential Object

c=3 c=10 c=30

0

2

4

6

8

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

10 12 14 16 18 20

δ=10-8, a=1

c=0 0.1 c=0.3 c=1 c=3 c=10 c=30 0

2

4

6

8

R

R

Figure 39. Rotation curve: flared bi-exponential object. Same as Fig. 38 but for an object with the volume mass density expressed as ρ FE (R, z) ≡ [ ] ρ 0 exp −(R/a) − ( |z |/c)/ f (R/a) where f (ξ) ≡ 1 + k ξ.

Figure 41. Rotation curve: radially exponential and vertically secanthyperbolic-squared object. Same as Fig. 38 but for an object with the volume mass density expressed as ρ ES (R, z) ≡ ρ 0 exp(−R/a)sech2 [|z |/(2c)].

Circular Speed: Exponential/Gaussian Object

δ=10-8, a=1, c=0.3 k=5 1 V/V0

V/V0

Circular Speed: Flared Bi-Exponential Object 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

k=0

0

2

4

6

8

10 12 14 16 18 20

10 12 14 16 18 20

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

δ=10-8, a=1

c=0 0.3 c=1 c=3 c=10 c=30

0

2

4

R Figure 40. Rotation curve of flared bi-exponential object. Same as Fig. 39 but plotted are the curves for various values of k, the flaring factor, while the other parameters are fixed as a = 1, c = 0.3. Compared are the results for k = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5. This time, the nominal density value ρ 0 is set the same. The thick curve is the case of k = 1.

rotation curves depicted in Figs 44 and 45. These figures illustrate that the initial rise of the circular speed is not linear but quadratic with respect to R. Finally, Fig. 37 shows the contour map of the gravitational potential of a toroidal object. Although Fig. 24 indicates the density distribution in a rather limited cross-section, the potential distribution extends to the whole space. For example, the coordinate origin is a saddle point and the minimum potential value is achieved at a circular ring where R ≈ 1.6 and z = 0. As a result, Fig 46 shows that the inner part of the rotation curve is not properly defined. This

6

8

10 12 14 16 18 20 R

Figure 42. Rotation curve: radially exponential and vertically Gaussian object. Same as Fig. 38 but for[ an object with the ] volume mass density expressed as ρ(R, z) = ρ 0 exp −(R/a) − (z/c) 2 .

never means an unphysical situation. Refer to Appendix D for the similar phenomena in some uniform objects illustrated in Figs D14, D17, and D18.

5 CONCLUSION By extending the previous work on an infinitely thin axisymmetric disc (Fukushima 2016), we developed a new method to compute the gravitational potential and the associated acceleration vector of general axisymmetric objects. The method consists of (i) the numerical evaluation of the double integral transform of a given MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

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

Circular Speed: Differenced Modified Exp. Spheroid

δ=10-8, a=1

c=0

0.3 c=1

V/V0

V/V0

Circular Speed: Boxy Object

c=3 c=10 c=30 0

2

4

6

8

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

10 12 14 16 18 20

δ=10-8, a=1

c=0 0.3 c=1 c=3 c=10 c=30

0

2

4

6

8

R

Figure 45. Rotation curve: differenced modified exponential spheroid. Same as Fig. 44 but for an object with the [ volume mass density expressed as( ρ DM (R, z))}] ≡ { √ √ )} { ( 2 2 2 2 ρ 0 exp −2s out / 1 + 1 + 4s out − exp −2s in / 1 + 1 + 4s in where the auxiliary arguments √are defined as s out ≡ √ 2 2 2 2 (R/a out ) + (z/c out ) and s in ≡ (R/a in ) + (z/c in ) while the constants are set as a out = 2a, c out = 2c, a in = a, and c in = c.

δ=10-8, a=1

c=0

Circular Speed: Gaussian Oval Toroid

c=1 c=3 c=10 c=30

2

4

6

8

V/V0

V/V0

Circular Speed: Differenced Gaussian Spheroid

0

10 12 14 16 18 20 R

Figure 43. Rotation curve: radially exponential and vertically secanthyperbolic-squared disc. Same as Fig. 38 but [for√an object with the ]volume mass density expressed as ρ(R, z) = ρ 0 exp − (R/a) 4 + (z/c) 4 .

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

19

10 12 14 16 18 20 R

Figure 44. Rotation curve: differenced Gaussian spheroid. Same as Fig. 38 but for an object with the volume expressed [ (mass density ) ( )] as ρ ∆G (R, z) ≡ 2 2 ρ G (s out ) − ρ G (s in ) = ρ 0 exp −s out − exp −s in where the auxil√ 2 iary arguments are defined as s out ≡ (R/a out ) + (z/c out ) 2 and s in ≡ √ (R/a in ) 2 + (z/c in ) 2 while the constants are set as a out = 2a, c out = 2c, a in = a, and c in = c.

volume mass density by the split quadrature method employing the double exponential rules, and (ii) the numerical differentiation of the numerically integrated potential by Ridder’s algorithm. The comparison with the exact analytical solutions confirm the 13 and 11 digit accuracy of the potential and acceleration computed by the new method in the double precision environment. Although the new method requires the quadrature of double integrals, its CPU time is not so large such as 0.4–2.3 ms for a single point potential MNRAS 000, 1–43 (2016)

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

δ=10-8, a=1

c=0

0.3 c=1 c=3 c=10 c=30 0

2

4

6

8

10 12 14 16 18 20 R

Figure 46. Rotation curve: Gaussian oval toroid. Same as Fig. 38 but for the Gaussian oval toroid[ the volume mass density of which is described ] as ρ GT (R, z) ≡ ρ 0 exp − {(R − R T ) /a }2 − {( |z |/c)/ f (R/a) }2 while f (ξ) ≡ 1 + k ξ.

computation executed at an ordinary PC. Using the new method, we presented the potential contour map and/or the rotation curves of various kind of axisymmetric objects including an X/peanut-shaped galaxy like NGC 128 and finite power-law protoplanetary disc with a central hole such as associated with TW Hya. The new method is immediately applicable to the electrostatic field, and will be easily adapted to the magnetostatic field of general axisymmetric objects.

20

T. Fukushima

Density Distribution Dependence of Circular Speed

V/Vmax

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

δ=10-8, a=1, c=0.3

Boxy Object Bi-Exponential Object Differenced Mod. Exp. Spheroid Differenced Gaussian Spheroid Gaussian Oval Toroid

A2 External potential

0

0.5

1

1.5 R/Rmax

2

2.5

3

Figure 47. Density distribution dependence of of rotation curve. Compared are the rotation curves of eight axisymmetric objects: (i) the bi-exponential object, (ii) the flared bi-exponential object, (iii) the radially exponetial and the vertically secant-hyperbolic-function-squared object, (iv) the radially exponetial and the vertically Gaussian object, (v) the boxy object, (vi) the differenced modified exponential spheroid, (vii) the differenced Gaussian spheroid, and (viii) the Gaussian oval toroid. Plotted are the rotation curves when a = 1 and c = 0.3 while the velocity V and the radius R are normalized by the maximum speed Vmax and the location of maximum speed R max , respectively. Once this normalization is applied, almost invisible are the difference among the first four objects.

The fortran 90 programs of the new method are electronically available from the following author’s WEB site: https://www.researchgate.net/profile/Toshio_Fukushima/

APPENDIX A: GRAVITATIONAL FIELD OF FINITE UNIFORM SPHEROID A1

terms of the formulas given in Eric Weisstein’s World of Physics (Weisstein 2016) despite that its explanation already pointed some typos in a standard textbook. Also, the original expressions (MacMillan 1930, section 39, equations (2) and (4)) suffer from a heavy cancellation when a ≈ c. The same thing would be said for the resulting expressions of the acceleration vector, which are not found in the existing literature. In order to avoid this problem, Danby (1988, p.106) obtained the first few terms of the series expansion of the internal potential expression with respect to the deviation from the sphere. This is educative but not enough for precise computation. Therefore, we present below the complete solutions in a unified and cancellationfree form.

Introduction

Consider a uniform spheroid of the semi-axes, a and c. Its volume mass density distribution is expressed as ( ) 2 + η2 ≤ 1  ρ ξ  0 ( ) (A1) ρ(R, z) =   0 ξ2 + η2 > 1  where ξ ≡ R/a and η ≡ z/c. The gravitational potential of a uniform triaxial ellipsoid at an internal point is provided in terms of the complete elliptic integrals of the first and second kind (MacMillan 1930, section 32, equation (13)). Similarly, that at an external point is expressed by the incomplete elliptic integrals of the first and second kind (MacMillan 1930, section 36, equation (1)). In the special cases of spheroids whether being oblate or prolate, these elliptic integrals reduce to the elementary functions, namely the inverse trigonometric and/or logarithm functions (MacMillan 1930, section 39, equations (2) and (4)). See also Danby (1988, section 5.5) and Binney & Tremaine (2008, tables 2.1 and 2.2). Thus, in principle, it is automatic to obtain the gravitational potential and the acceleration vector of the spheroids. However, misprints and typos are frequently found in the existing literature. For example, incorrect are the signs of the first

Begin with the potential expression at an external point of the spheroid, namely when ξ 2 + η 2 > 1. If the spheroid is oblate such that a > c, the gravitational potential1 is expressed (MacMillan 1930, section 39, equation (2)) as ( ) √ R2 − 2z 2 −2µ ν sin−1 Φ (ext) (R, z) = √ 0 1 − 2ν α ν µ γR2 2µ z 2 − 0 2 + 0 , νγ να

(A2)

where (i) µ0 and ν are constants defined as µ0 ≡ πG ρ0 a2 c, ν ≡ a2 − c2,

(A3)

(ii) α and γ are auxiliary variables defined as √ √ α ≡ a2 + κ, γ ≡ c2 + κ,

(A4)

while (iii) κ is another auxiliary variable defined as the larger solution of an equation expressed as R2 z 2 R2 z2 + 2 = 2 + 2 = 1. 2 α γ a +κ c +κ

(A5)

In fact, κ is computed without suffering from the cancellation issue as { (B + D) /2 (B ≥ 0) κ= (A6) 2C/ (D − B) (B < 0) where

( ) B ≡ R2 + z 2 − a 2 + c 2 ,

(A7)

C ≡ c2 R2 + a2 z 2 − a2 c2,

(A8)

while √ D ≡ B2 + 4C.

(A9)

Similarly, the gravitational potential of the prolate spheroid where a < c, and therefore ν < 0, the expression (MacMillan 1930, section 39, equation (4)) reads √ ) ( R2 − 2z 2 −2µ0 −1 −ν (ext) sinh 1+ Φ (R, z) = √ 2(−ν) α −ν MacMillan (1930) has adopted the negative potential convention as V ≡ −Φ. 1

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object µ γR2 2µ z 2 + 0 2 − 0 , (−ν)γ (−ν)α

Auxiliary Function

(A10)

0.6

which is an analytical continuation of equation (A2) if noting the identity √ √ sin−1 s sinh−1 −s . (0 < |s| ≤ 1) (A11) = √ √ s −s

Φ (ext) (R, z) = S(κ)R2 + T (κ)z 2 + U (κ),

(A12)

where S(κ), T (κ), and U (κ) are the functions of κ defined as ( ν ) µ µ + 03 A 2 , (A13) S(κ) ≡ 2 0 α (α + γ) α α ( ν ) 2µ0 2µ T (κ) ≡ − 30 A 2 , αγ(α + γ) α α

0.4 0.3 0.2 0.1 0 -1 -0.8 -0.6 -0.4 -0.2

0

0.2 0.4 0.6 0.8

1

s

(A14)

−2µ0 2µ0 ν ( ν ) − U (κ) ≡ A 2 , (A15) α α3 α while A(s) is an auxiliary function described later. Equation (A12) is robust against the smallness of ν in the sense it has no small divisor containing ν. This cancellation-free form of the external gravitational potential of spheroid is not found in the existing literature. At any rate, it is obvious that the external potential expression in equation (A12) is never written as a scalar function of a homogenous quadratic polynomial of R and z such as R2 /α 2 + z 2 /γ 2 . In other words, this is a good specimen that, even if an object is spheroidally symmetric, its gravitational potential is not spheroidally symmetric in general. A3

A(s)=[F(1/2,1/2;3/2;s)-1]/s 0.5

A(s)

Unfortunately, these forms cause a catastrophic cancellation when a ≈ c, and therefore ν ≈ 0. In order to avoid this phenomenon, we rewrote them in a unified manner as

21

Figure A1. Auxiliary function A(s). Plotted is the curve of an auxiliary function, A(s) ≡ [F (1/2, 1/2; 3/2; s) − 1]/s for the standard interval, |s | ≤ 1. The function is not analytic at s = 1 despite it is finite as A(1) = π/2 − 1 ≈ 0.57.

This condition guarantees the absolute convergence of the above hypergeometric series, and therefore A(s) is well defined. As for the numerical calculation of A(s), we recommend a switching formula expressed as ( √ ) √ sin−1 s − s A(s) = , (0.1 < s < 1), √ s s √ ) √ sinh−1 −s − −s , (s < −0.1), = √ (−s) −s (

Auxiliary function

In the previous subsection, we introduced an auxiliary function A(s). It is defined as [ ( ) ] 1 1 1 3 A(s) ≡ F , ; ;s −1 , (A16) s 2 2 2 where F (a, b; c; z) is Gauss’ hypergeometric function (Olver et al. 2010, equation (15.2.1)). The function is well defined if s ≤ 1 and positive definite then. It has a few special values as A(1) = π/2 − 1 ≈ 0.570796,

(A17)

= A14 (s), (|s| ≤ 0.1),

(A22)

where A14 (s) is its Maclaurin series expansion truncated as A14 (s) ≡ +

1 3s 5s2 35s3 63s4 231s5 143s6 6435s7 + + + + + + + 6 40 112 1152 2816 13312 10240 557076

12155s8 46189s9 88179s10 676039s11 + + + 1245184 5505024 12058624 104857600

1300075s12 5014575s13 9694845s14 + + . (A23) 226492416 973078528 2080374784 This expression is of the 16 digit accuracy when |s| ≤ 0.1. +

A(0) = 1/6 ≈ 0.166667,

(A18)

( √ ) A(−1) = 1 − ln 1 + 2 ≈ 0.118626,

(A19)

lim A(−s) = 0.

s→∞

(A20)

Refer to Fig. A1 showing its monotonically increasing behavior. If the evaluation point is outside the spheroid, namely when ξ 2 + η 2 > 1, then κ > 0, and therefore, ν a 2 − c2 s≡ 2 = 2 < 1. α a +κ MNRAS 000, 1–43 (2016)

(A21)

A4 Internal potential Secondly, the potential expression at a boundary or internal point is derived from equation (A12) by simply substituting κ = 0, and therefore by rewriting α = a and γ = c. The simplified result becomes a homogeneous quadratic polynomial of R and z as Φ (int) (R, z) = S0 R2 + T0 z 2 + U0 where the suffix 0 stands for special values when κ = 0 as µ µ A S0 ≡ S(0) = 2 0 + 03 0, a (a + c) a

(A24)

(A25)

22

T. Fukushima

T0 ≡ T (0) =

U0 ≡ U (0) = while A0 ≡ A

Let us confirm this assertion. The direct differentiation of S(κ), T (κ), and U (κ) results

2µ0 2µ A − 03 0 , ac(a + c) a

(A26)

−2µ0 2µ0 ν A0 , − a a3

(A27)

( ν ) , a2

(A28)

This cancellation-free expression of the internal gravitational potential of spheroid is also missing in the existing literature.

A5

independently on the value of α. Also, in this case, √ α = γ = r ≡ R2 + z 2 .

(A29)

S0 = T0 =

2µ0 −2µ0 , U (κ) = , r 3r 3

2µ0 −2µ0 , U0 = . a 3a3

(A30)

(A31)

(A32)

Consequently, both of the above internal and external expressions reduce to the spherical solution as ( ) −4πG ρ0 a3 Φ(r) = , (r > a) 3 r =

A6

) 2πG ρ0 ( 2 r − 3a2 . (r ≤ a) 3

(A33)

Acceleration vector

(A34)

where the coefficients S0 and T0 are already provided in equations (A25) and (A26), respectively. Secondly, those of the external cases are similarly obtained as FR(ext) (R, z) = −2S(κ)R, Fz(ext) (R, z) = −2T (κ)z,

This vanishes because of the definition of κ given in equation (A5). Therefore, the assertion is proven.

When z = 0, the circular velocity is well defined as √ V (R) = R 2S0, (R ≤ a) √

µ(R)/R, (R > a)

(A38)

where µ(R) is an additional function defined as [ ( ν )] 2R (A39) µ(R) ≡ µ0 + 2A 2 . √ R R + R2 − ν ( ) If R → ∞, then A ν/R2 → A(0) = 1/6, and therefore µ(R) → (4/3) µ0 . This means that V (R) (i) linearly increase inside the spheroid, and (ii)√decreases outside in a similar manner as that of a point mass as 1/ R.

A8 Sketches For the readers’ understanding, we present Figs A2 and A3 depicting the contour map of the gravitational potential of a pair of uniform spheroids with semi-axes as (i) a = 3 and c = 1, and (ii) a = 1 and c = 3. Also, we plot the rotation curves of uniform spheroids with various aspect ratios in Fig. A4.

APPENDIX B: HINTS FOR PARALLEL INTEGRATION

Let us move to the expressions of the acceleration components. Firstly, those of the internal cases are automatically obtained as FR(int) (R, z) = −2S0 R, Fz(int) (R, z) = −2T0 z,

Thus, the common proportional coefficient of the additional terms, which should have appeared in equation (A35) if the potential is formally differentiated, becomes as ) ( ) ( ) ( dS 2 dT 2 dU −µ R2 z 2 + − 1 . (A37) R + z + = 20 dκ dκ dκ α γ α2 γ 2

=

Thus, S(κ) = T (κ) =

(A36)

A7 Rotation curve

Spherical limit

In the spherical limit, a = c, then ν = 0, and therefore ( ) A ν/α 2 = A(0) = 1/6,

dS −µ0 dT −µ dU µ = 4 , = 2 03 , = 20 . dκ dκ dκ α γ α γ α γ

The proper execution of parallel computing is not an easy task. This is true especially when each computing unit may randomly call the same subprograms in a nested manner. Here, in place of discussing a general treatment, let us focus on a specific case; the parallel quadrature of a double integral given in equations (15) and (16). For simplicity, we assume that Rin = 0, Rout = ∞, zlower (R) = −∞, and zupper (R) = ∞. Imagine a case when the total potential is split into four components as

(A35)

where S(κ) and T (κ) are already given in equations (A13) and (A14), respectively. This is a surprisingly simple result as MacMillan (1930, section 39) stressed ... Although κ is a function of x, y, and z, it is not necessary to regard it as such in forming the first partial derivatives of V with respect to x, y, and z; for it was shown in Sec. 36 that even for the general ellipsoid the derivative of the potential with respect to x, y, and z, in so far as these variables are involved implicitly through κ, vanishes...

Φ(R, z) ≡

4 ∑

Φ j (R, z),

(B1)

j=1

where each component integral is expressed as ∫ R ( ) Φ1 (R, z) ≡ Ψ1 R ′ ; R, z dR ′,

(B2)

0

Φ2 (R, z) ≡

∫ ∞ R

( ) Ψ2 R ′ ; R, z dR ′,

(B3) MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of Uniform Prolate Spheroid

4

4

3

3

2

2

z

z

Φ of Uniform Oblate Spheroid

1

1

0

0 0

1

2

3

4

0

1

2

R

Φ4 (R, z) ≡

0

∫ ∞ R

Circular Speed: Uniform Spheroid ( ) Ψ3 R ′ ; R, z dR ′, ( ) Ψ4 R ′ ; R, z dR ′ .

(B4)

(B5)

Refer to Fig. 5. We number the sub component line integrals explicitly as ( ′ ) ( ) Ψ1 R ; R, z = Ψ2 R ′ ; R, z ≡

−G

∫ z −∞

) ( ) ( g R ′, z ′ ; R, z ρ R ′, z ′ dz ′,

(B6)

−G

z

) ( ) ( g R ′, z ′ ; R, z ρ R ′, z ′ dz ′ .

a=1

c=0 0.3 c=1 c=3 c=10 c=30 0.5

1

1.5

2

2.5

3

3.5

4

R

(B7)

This discrimination of component integrals is the key of success in parallel computing. In fact, in the actual execution of the parallel integration utilizing the OpenMP architecture, we (i) prepare several sets of copies of the DE quadrature programs, intde and intdei, beforehand, and (ii) assign a differently numbered set of the DE quadrature programs to a different computing unit by using the omp sections block and multiple omp section statements. This is in order to avoid the fatal MNRAS 000, 1–43 (2016)

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

( ) ( ) Ψ3 R ′ ; R, z = Ψ4 R ′ ; R, z ≡ ∫ ∞

4

Figure A3. Potential contour map of uniform prolate spheroid. Same as Fig. A2 but when the semi-axes are a = 1 and c = 3. Obviously, this figure is different from that of Fig. A2 after the 90 degree rotation. In fact, clearly visible is the non-analyticity of the potential on the spheroid surface.

V/V0

Φ3 (R, z) ≡

3

R

Figure A2. Potential contour map of uniform oblate spheroid. Plotted are the contours of the gravitational potential of a uniform spheroid with the semi-axes, a = 3 and c = 1. The contours are drawn at every 1 per cent level of the bottom value realized at the coordinate center. The potential is not analytic on the surface of the spheroid where R 2 /a 2 + z 2 /c 2 = 1 although the non-analyticity is not obvious at this scale.

∫ R

23

Figure A4. Rotation curve: uniform spheroid. Shown are the rotation curves of uniform spheroids of various values of c as c = 0.003, 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, and 30.0 while a is fixed as a = 1. The normalization constant V0 is adjusted such that the total mass of the object becomes the same. As a result, the limit curve of R → ∞ becomes the same, namely that of a point mass. At this scale, invisible are the departure of the curves when c ≤ 0.03 from the limit of c = 0. All the curves have a kink at the surface of the spheroid, namely when R = a.

24

T. Fukushima

modification of intermediate results during the random calling of the same child subprogram by multiple parent subprograms. A sample fortran code of the resulting parallel evaluation of the double integral is real*8 function phiint(r,z) ... common /param1/r1,z1,rp1 common /param2/r2,z2,rp2 common /param3/r3,z3,rp3 common /param4/r4,z4,rp4 ... r1=r; z1=z; r2=r; z2=z; r3=r; z3=z; r4=r; z4=z ... !$omp parallel !$omp sections !$omp section call intder1(psiint1,0.d0,r,eps,phi1,err1) !$omp section call intdeir1(psiint2,r,eps,phi2,err2) !$omp section call intder2(psiint3,0.d0,r,eps,phi3,err3) !$omp section call intdeir2(psiint4,r,eps,phi4,err4) !$omp end sections !$omp end parallel phiint=phi1+phi2+phi3+phi4 return; end where the statements including "$omp" are OpenMP directives for fortran. The line integral computation is conducted by differently named but essentially the same subprograms as real*8 function psiint1(rp) ... common /param1/r1,z1,rp1 rp1=rp call intdeiz1(fint1,-z1,eps1,psiint1,err1) return; end real*8 function psiint2(rp) ... common /param2/r2,z2,rp2 rp2=rp call intdeiz2(fint2,-z2,eps2,psiint2,err2) return; end real*8 function psiint3(rp) ... common /param3/r3,z3,rp3 rp3=rp call intdeiz3(fint3,z3,eps3,psiint3,err3) return; end real*8 function psiint4(rp) ... common /param4/r4,z4,rp4 rp4=rp call intdeiz4(fint4,z4,eps4,psiint4,err4) return; end where intdeiz1, intdeiz2, intdeiz3, and intdeiz4 are the same subprogram as intdei but named differently. Also, the integrand is computed by almost the same functions,

fint1, fint2, fint3, and fint4. They are, after setting G = 1 by adopting an appropriate unit system, defined as real*8 function fint1(zp) real*8 zp,r1,z1,rp1,rho1,green1 common /param1/r1,z1,rp1 fint1=-rho1(rp1,-zp)*green1(rp1,-zp,r1,z1) return; end real*8 function fint2(zp) real*8 zp,r2,z2,rp2,rho2,green2 common /param2/r2,z2,rp2 fint2=-rho2(rp2,-zp)*green2(rp2,-zp,r2,z2) return; end real*8 function fint3(zp) real*8 zp,r3,z3,rp3,rho3,green3 common /param3/r3,z3,rp3 fint3=-rho3(rp3,zp)*green3(rp3,zp,r3,z3) return; end real*8 function fint4(zp) real*8 zp,r4,z4,rp4,rho4,green4 common /param4/r4,z4,rp4 fint4=-rho4(rp4,zp)*green4(rp4,zp,r4,z4) return; end where (i) rho1 through rho4 are differently named but essentially the same functions to calculate the volume mass density profile, ( ) ρ R ′, z ′ , while (ii) green1 through green4 are differently named but actually the same functions to return the azimuthally integrated ( ) Green’s function, g R ′, z ′ ; R, z , as specified in equation (17). We warn the readers that fint1 and fint2 are the same functions but numbered differently, both of which are to compute the integrand for −z, while fint3 are fint4 are a similar pair for +z. This difference in the argument sign was necessary because Ooura’s intdei is designed to evaluate an integral over a positively-open semi-infinite interval such as t 0 ≤ t < +∞. In this example, the named common blocks are carefully distributed to the subprograms so as to avoid the interference among them during the parallel execution of subprograms. This is another important point to guarantee the correctness of the parallel computation.

APPENDIX C: GRAVITATIONAL FIELD OF MIYAMOTO-NAGAI MODEL The Miyamoto-Nagai model (Miyamoto & Nagai 1975) is a popular potential-density pair of an infinitely extended axisymmetric object. Let us introduce a pair of convenient quantities, pM and qM , defined as √ pM ≡ b2M + z 2, (C1) √ qM ≡

( ) R2 + aM + pM 2,

(C2)

where aM and bM are the parameters of the Miyamoto-Nagai model. Then, the volume mass density of the object is described as a twoparameter function as ( )( ) Mb2M aM R2 + aM + 3pM aM + pM 2 + ρMN (R, z) ≡ * , (C3) 3 q5 pM , 4π M MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of Uniform Cylinder, a=c=1

where M is the total mass of the object. By means of pM and qM , the gravitational potential and the associated acceleration vector of the object are compactly expressed as (C4)

3 FMN, R (R, z) = −GM R/qM ,

(C5)

) ( ) ( 3 FMN, z (R, z) = −GM z aM + pM / pM qM .

(C6)

APPENDIX D: GRAVITATIONAL FIELD OF FINITE UNIFORM OBJECTS Here, we present the contour map of the gravitational potential and/or the rotation curve of a series of finite uniform axisymmetric objects. Although such objects are rarely found in nature, the information on their gravitational fields will give us some hints on those of inhomogeneous objects with similar shapes. Considered are the following finite uniform objects: (i) a circular cylinder, the volume of which is described as 0 ≤ ξ ≤ 1, 0 ≤ |η| ≤ 1,

(D1)

where ξ ≡ R/a and η ≡ z/c are normalized cylindrical coordinates, (ii) a rhombic spindle; 0 ≤ ξ + |η| ≤ 1,

(D2)

(iii) a linearly flared circular cylinder; 0 ≤ |η| ≤ ξ ≤ 1,

4 3 z

ΦMN (R, z) = −GM/qM,

(D3)

25

2 1 0 0

1

2

3

4

R Figure D1. Potential contour map of finite uniform cylinder. Same as Fig. 25 but for a finite uniform cylinder of the radius a = 1 and the half height c = 1. Namely, the object occupies the volume defined as 0 ≤ R ≤ a and 0 ≤ |z | ≤ c. The contours are drawn at every 2 per cent level of the bottom value. Easily visible is the non analyticity of the potential on the surface of the cylinder, R = a and/or |z | = c. The map is not symmetric with the 90 degree rotation.

(iv) a bipolar circular cone; 0 ≤ ξ ≤ |η| ≤ 1,

(D4)

(v) a vertically separated pair of circular cylinders; 0 ≤ ξ ≤ 1, 0 ≤ |η ± ∆η| ≤ 1,

(D5)

where ∆η is a positive constant offset satisfying the condition, ∆η > 1, (vi) a toroid with a square cross-section; 0 ≤ |ξ − ξ0 | ≤ 1, 0 ≤ |η| ≤ 1,

(D6)

where ξ0 is another constant offset, (vii) a toroid with a circular cross-section; ( ) 0 ≤ ξ − ξ0 2 + η 2 ≤ 1, (D7) (viii) an upper hemisphere; 0 ≤ ξ 2 + η 2 ≤ 1, 0 ≤ η ≥ 0,

(D8)

where c = a, and (ix) a single cone; 0 ≤ ξ ≤ η ≤ 1.

(D9)

The last two objects are not plane symmetric. Figs D1–D11 illustrate the contour map of the gravitational potential of these objects computed by the new method when the parameters are chosen as a = 1 or 3, c = 1, 2, or 3, and ξ0 = η 0 = 2. The shape of the potential contours are significantly different from that of the density contours. In general, the potential contours approach to circles when the distance from the object increases. This is a natural consequence that, independently on its shape, the point mass potential becomes a good approximation of the finite √ 2 2 object for the evaluation points when r ≡ R + z → ∞. Also, MNRAS 000, 1–43 (2016)

the cross-section of the object surfaces are sometimes visible as the dense concentration of contours since the potential is not analytic there. Meanwhile, Fig. D12 compares the rotation curves of the nonflared cylinders for various values of c as c = 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10, and 30 while a is fixed as a = 1. In drawing Fig. D12, we normalized the computed circular speed such that their asymptotic forms are the same as that of a point of the same total mass. Similarly, Figs D13–D18 indicate the rotation curves of other uniform objects with the plane symmetry. These rotation curves exhibit kinks when crossing the surface of the object if it lies on the z-plane. In some cases, the rotation curve blows up infinitely at the kink point. This happens for infinitely thin cylinders with and/or without flaring. It is noteworthy that the imaginary circular velocity is realized in an inner region for toroidal objects and some flared cylinders. Refer to Figs D14, D17, and D18. This is not unphysical situation. It simply means that the gravitational attraction from the external mass is stronger than that from the internal one.

T. Fukushima

Φ of Uniform Cylinder, a=3, c=1 4

Φ of Uniform Rhombic Spindle 4

3

3

2

2

z

z

26

1

1

0

0 0

1

2

3

4

0

1

R

2

3

4

R

Figure D2. Potential contour map of finite uniform cylinder, a = 3 and c = 1. Same as Fig. D1 but when a = 3 and c = 1.

Figure D4. Potential contour map of finite uniform rhombic spindle. Same as Fig. D1 but for a finite uniform rhombic spindle of the radial and vertical semi-axes, a = 3 and c = 1, respectively. Namely, the object occupies the volume defined as 0 ≤ R/a + |z |/c ≤ 1.

Φ of Uniform Cylinder, a=1, c=3 4

Φ of Uniform Flared Cylinder 4

3 2 z

z

3 2

1 1 0 0

1

2 R

3

4

Figure D3. Potential contour map of finite uniform cylinder, a = 1 and c = 3. Same as Fig. D1 but when a = 1 and c = 3. Notice that this figure is not the same as Fig. D2 even after the 90◦ rotation.

0 0

1

2 R

3

4

Figure D5. Potential contour map of finite uniform flared cylinder. Same as Fig. D1 but for a finite uniform flared cylinder of the radial and vertical semi-axes, a = 3 and c = 1, respectively. Namely, the object occupies the volume defined as 0 ≤ R/a ≤ |z |/c ≤ 1.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of Uniform Square Toroid

4

4

3

3

2

2

z

z

Φ of Uniform Bipolar Cone

1

1

0

0 0

1

2

3

4

R

Φ of Uniform Double Cylinder 4

z

3 2 1 0 1

2 R

3

4

Figure D7. Potential contour map of pair of finite uniform cylinders. Same as Fig. D1 but for a pair of cylinders with the radius and the half height as a = c = 1 and separated by ∆z = 2. Namely, the objects occupy the volume defined as 0 ≤ R ≤ a and 0 ≤ |z ± ∆z | ≤ c.

MNRAS 000, 1–43 (2016)

0

1

2

3

4

R

Figure D6. Potential contour map of finite uniform bipolar cone. Same as Fig. D1 but for a finite uniform bipolar cone of the radial and vertical semiaxes, a = 1 and c = 3, respectively. Namely, the object occupies the volume defined as 0 ≤ |z |/c ≤ R/a ≤ 1.

0

27

Figure D8. Potential contour map of finite uniform square toroid. Same as Fig. D1 but for a finite uniform square toroid of the inner and outer radii, R in = 1 and R out = 3. Namely, the object occupies the volume defined as R in ≤ R ≤ R out and 0 ≤ |z | ≤ (R out − R in ) /2.

28

T. Fukushima

Φ of Uniform Circular Toroid

Φ of Uniform Cone

4 3 3 2

z

z

2 1 1 0 0 0

1

2

3

4

0

1

2

R

4

R

Figure D9. Potential contour map of finite circular uniform toroid. Same as Fig. D1 but for a finite uniform toroid of the inner and outer radii, R in = 1 and R out = 3, respectively. Namely, the object occupies the volume defined as 0 ≤ [R − (R in + R out ) /2]2 + z 2 ≤ [(R out − R in ) /2]2 . Thus, the figure center of the cross section of the toroid is at R = 2 and z = 0. Meanwhile, the location of the potential maximum is at R ≈ 1.45 and z = 0.

Figure D11. Potential contour map of finite uniform cone. Same as Fig. D1 but for a finite uniform cone of the radius a = 1 and the height c = 2. Namely, the object occupies the volume defined as 0 ≤ z/c ≤ R/a ≤ 1.

Circular Speed: Uniform Cylinder

Φ of Uniform Hemisphere 1.5 V/V0

1 z

3

0.5

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

0.3 c=1 c=3 c=10 c=30 0

0

δ=10-8, a=1

c=0

0.5

1

1.5

2

2.5

3

3.5

4

R

-0.5 0

0.5

1 R

1.5

2

Figure D10. Potential contour map of finite uniform hemisphere. Same as Fig. D1 but for a finite uniform hemisphere of the radius a = 1. Namely, the object occupies the volume defined as 0 ≤ R 2 + z 2 ≤ a 2 and 0 ≤ z.

Figure D12. Rotation curve: finite uniform cylinder. Shown are the rotation curves of finite uniform cylinders of the radius a and the half height c. Depicted are the results for various values of c as c = 0.003, 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, and 30.0 while a is fixed as a = 1. The normalization constant V0 is adjusted such that the total mass of the object is the same. As a result, the limit curve of R → ∞ becomes the same, namely that of a point mass. At this scale, invisible are the departure of the curves when c ≤ 0.03 from the limit of c = 0, namely that of the infinitely thin uniform disc (Fukushima 2016, Fig. 5). The curves have a kink at the surface of the cylinder, namely when R = a. The kink approaches to a logarithmic blow-up singularity when c → 0.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

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

Circular Speed: Uniform Bipolar Cone

δ=10-8, a=1

c=0 0.3 c=1

V/V0

V/V0

Circular Speed: Uniform Rhombic Spindle

c=3 c=10 c=30 0

0.5

1

1.5

2

2.5

3

3.5

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

4

δ=10-8, a=1

c=0 0.1 c=0.3 c=1 c=3 c=10 0

0.5

1

1.5

R

V/V0

V/V0 0.3 c=1 c=3 c=10 c=30 1

1.5

2

2.5

3

3.5

4

R Figure D14. Rotation curve: finite uniform flared cylinder. Same as Fig. D12 but for finite uniform flared cylinders.

MNRAS 000, 1–43 (2016)

2.5

3

3.5

4

Circular Speed: Uniform Double Cylinder

δ=10-8, a=1

c=0

0.5

2

Figure D15. Rotation curve: finite uniform bipolar cone. Same as Fig. D12 but for finite uniform bipolar cones.

Circular Speed: Uniform Flared Cylinder

0

c=30 R

Figure D13. Rotation curve: finite uniform rhombic spindle. Same as Fig. D12 but for finite uniform rhombic spindles.

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

29

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

δ=10-8, a=c=1 ∆z=0

∆z=3

0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure D16. Rotation curve: finite uniform double cylinder. Same as Fig. D7 but for the rotation curve for several values of ∆z as ∆z = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.2, 2.4, 2.6, 2.8, and 3.0 while the radius and the half height are fixed as a = c = 1.

30

T. Fukushima

Circular Speed: Uniform Square Toroid 4

Density Profiles of Some Spheroids

δ=10-8, Rout=2

3.5 3 Rin=0.1

1.5

1.7 Rin=1.8

1

Rin=1.9

2

ρ/ρ0

V/V0

2.5

0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

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

4

Modified Exponential G Burkert

0

Exp.

0.2 0.4 0.6 0.8

R

1

1.2 1.4 1.6 1.8

2

s

Figure D17. Rotation curve: finite uniform circular toroid. Same as Fig. 38 but for a uniform circular toroid for several values of R in as R in = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, and 1.9, and the outer radius is fixed as R out = 2. Meanwhile, the half height is arranged as c ≡ (R out + R in ) /2 such that the cross section is square-shaped. All the curves have a kink at R = R out .

Figure E1. Density profile of some spheroids. Plotted are the volume mass density (i) the Gaussian spheroid, ρ G (s) ≡ ( ) of some spheroids: √ ρ 0 exp −s 2 , where s ≡ (R/a) 2 + (z/c) 2 , (ii) the exponential spheroid, ρ X (s) [≡ ρ 0/exp(−s), ( √ (iii))]the modified exponential spheroid, ρ M (s) ≡ ρ 0 exp −s 2 1 + 1 + s 2 , and (iv) the Burkert spheroid (Burkert 1995), ( ) −1 ρ B (s) ≡ ρ 0 (1 + s) −1 1 + s 2 . The thick curve shows the Gaussian spheroid. The exponential and Burkert spheroids have a cusp at the coordinate center.

Circular Speed: Uniform Circular Toroid 3.5

Density Profiles of Plummer Spheroids

-8

δ=10 , Rout=2

3 Rin=0.1

2 1.5

ρ/ρ0

V/V0

2.5

1 0.5 1.9 0 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure D18. Rotation curve: finite uniform circular toroid. Same as Fig. 38 but for a uniform circular toroid for several values of R in as R in = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, and 1.9, while the outer radius is fixed as R out = 2. All the curves have a kink at R = R out .

APPENDIX E: GRAVITATIOANL FIELD OF INFINITELY EXTENDED SPHEROIDS

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

j=2 j=3 4 j=5 0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

s Figure E2. Density profile of indexed Plummer spheroids. Same as Fig. E1 ( ) − j /2 but for some indexed Plummer spheroids, ρ P j (s) ≡ ρ 0 1 + s 2 , where √ s ≡ (R/a) 2 + (z/c) 2 . Here (i) the case j = 2 is the modified isothermal model (King 1962), (ii) the case j = 3 is the modified Hubble model (Rood et al. 1972), (iii) the case j = 4 is the perfect spheroid (de Zeeuw 1985a,b), and (iv) the case j = 5 is the original Plummer model (Plummer 1911).

Although the gravitational fields for general spheroids are computable by another approach (Binney & Tremaine 2008, section 2.5.2), we present the contour map of the gravitational potential and the rotation curve of some spheroids below. This is because most of the information is rarely accessible in the literature. MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

8 6 Jaffe 4 H 2 0 -2 -4 NFW -6 -8 -10 -12 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 log10 s

Figure E3. Density profile of double power-law spheroids. Same as Fig. E2 but plotted are double logarithmic graphs for some spheroids with the double power-law density profile: √ (i) the Jaffe spheroid (Jaffe 1983), ρ J (s) ≡ ρ 0 s −2 (1 + s) −2 where s ≡ (R/a) 2 + (z/c) 2 , (ii) the Hernquist spheroid (Hernquist 1990), ρ H (s) ≡ ρ 0 s −1 (1 + s) −3 , and (iii) the Navarro-FrenkWhite (NFW) spheroid (Navarro et al. 1997), ρ N (s) ≡ ρ 0 s −1 (1+ s) −2 . The result of the Jaffe spheroid is depicted in a broken curve while those of the NFW and Hernquist spheroids are shown by a thick and a thin solid curve.

6 4

i=0.5 i=1

2

i=2 0 i=4 -2 i=7 -4 -6 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 log10 s

Figure E5. Density profile of Einasto spheroids. Same as Fig. E2 but plotted are modified logarithmic graphs for the Einasto spheroids. Their ( ) volume mass density profile is expressed as ρ Ti (s) ≡ ρ 0 exp −d i s 1/i where s ≡ √ (R/a) 2 + (z/c) 2 while the coefficient d i are approximated (Merritt et al. 2006, equation (24)) as d i = 3i − 1/3 + 0.0079/i. Compared are the curves of some indices as n = 1/2, 1, 2, 4, and 7. Except the scaling constants, the curves of n = 1/2 and n = 1 are the same as those of the Gaussian and exponential spheroids, respectively.

Figs E1–E5 show the density profiles of a group of spheroids: (i) the Gaussian spheroid; ( ) ρG (s) ≡ ρ0 exp −s2 , (E1) √ where s = (R/a) 2 + (z/c) 2 , (ii) the exponential spheroid;

Density Profiles of Sersic Spheroids 2 1.5 1 log10 |ρ/ρ0|

Density Profiles of Einasto Spheroids

log10 |1/[log10 (ρ0/ρ)]|

log10 |ρ/ρ1|

Density Profiles of Double Power-Law Spheroids

31

ρX (s) ≡ ρ0 exp(−s),

0.5

(E2)

(iii) the modified exponential spheroid (Robin et al. 2003, table 3); [ ( )] √ ρM (s) ≡ ρ0 exp −2s2 / 1 + 1 + 4s2 , (E3)

0 -0.5 -1

n=4

-1.5

n=2

-2

n=1

-2.5 0

0.5

1

1.5

2

2.5

3

3.5

4

s Figure E4. Density profile of Sérsic spheroids. Same as Fig. E2 but plotted are the single logarithmic graphs for the spheroids approximately resulting the Sérsic law of the surface intensity profile (Sérsic 1963, 1968). ( Here) the volume mass density is written as ρ Sn (s) ≡ ρ 0 s −p n exp −s 1/n √ where s ≡ (R/a) 2 + (z/c) 2 while the power index p n are approximated (Lima Neto et al. 1999) as p n = 1 − 0.6097n + 0.05463n 2 . Compared are the curves of (i) n = 1 drawn by a thick line, (ii) n = 2 by a dotted line, and (iii) n = 4 by a thin line. The case of the index n = 4 corresponds to the famous de Vaucouleurs’ law (de Vaucouleurs 1948).

(iv) the Burkert spheroid (Burkert 1995); ( ) −1 ρB (s) ≡ ρ0 (1 + s) −1 1 + s2 ,

(E4)

(v) the Plummer spheroids of index j; ( ) − j/2 , ρP j (s) ≡ ρ0 1 + s2

(E5)

where (v-a) j = 2 corresponding to the modified isothermal model (King 1962), (v-b) j = 3 corresponding to the modified Hubble model (Rood et al. 1972), (v-c) j = 4 corresponding to the perfect spheroid (de Zeeuw 1985a,b), and (v-d) j = 5 corresponding to the original Plummer model (Plummer 1911), (vi) the Jaffe spheroid (Jaffe 1983); ρJ (s) ≡ ρ0 s−2 (1 + s) −2,

(E6)

(vii) the Hernquist spheroid (Hernquist 1990); ρH (s) ≡ ρ0 s−1 (1 + s) −3,

(E7)

(viii) the Navarro-Frenk-White spheroid (Navarro et al. 1997); ρN (s) ≡ ρ0 s−1 (1 + s) −2, MNRAS 000, 1–43 (2016)

(E8)

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(ix) the Sérsic spheroids of index n (Sérsic 1963, 1968); ( ) ρSn (s) ≡ ρ0 s−p n exp −s1/n ,

(E9)

and (x) the Einasto spheroids of index i (Einasto 1965); ( ) ρEi (s) ≡ ρ0 exp −d i s1/i .

(E10)

In the above, (i) n is a real-valued index termed the Sérsic index and the power index pn is approximated (Lima Neto et al. 1999) as pn ≡ 1 − 0.6097n + 0.05463n2,

(E11)

and (ii) i is a real-valued index and the coefficient d i is approximated (Merritt et al. 2006, equation (24)) as d i ≡ 3i − 1/3 + 0.0079/i.

(E12)

In the Besançon galactic model (Robin et al. 2003, table 3), the Gaussian and the modified exponential spheroids are used in representing the thin disc of young and old stars. The Sérsic spheroid with the index 4 is nothing but the de Vaucouleurs spheroid (de Vaucouleurs 1948, 1953). Also, the Einasto spheroids with the index 1/2 and 1 are, except the radial scaling factor, the same as the Gaussian and exponential spheroids, respectively. The adopted formulas of pn and d i are of approximate nature. However, through independent numerical deprojection of the surface intensity curves for pn and numerical solutions of a nonlinear equation involving the incomplete gamma functions for d i , we confirm that they are sufficiently precise for the purpose to draw the contour maps and the rotation curves. Even if the volume mass density profiles are spheroidally symmetric, the resulting gravitational potentials are not so. Therefore, the two-dimensional distribution of the gravitational potential is indispensable in order to understand its global behavior. Figs E6– E22 show the contour maps of the gravitational potentials of these spheroids while the semi-axes are fixed the same as a = 1 and c = 0.3. Although the contours in these maps are not genuine ellipses, their departure from ellipses is difficult to notice. More visible is the difference in the spatial distribution of the contour levels, namely the radial and/or vertical distribution of the variation of the potential such as the circular speed on the symmetric plane as shown in Figs E23–E39. The rotation curves of these spheroids are depicted there for several values of c as c = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5 while a is fixed as a = 1. Obviously, the functional form of the rotation curve is significantly different spheroid by spheroid. For example, almost flat rotation curves are produced by the modified isothermal spheroids. This situation is mostly the same as the Mestel disc in the case of infinitely thin axisymmetric discs (Mestel 1963). Also, the Jaffe spheroids result the finite value of the circular speed at the object center although it is unphysical. Except these two cases, all the other rotation curves (i) initially rise as a linear function, (ii) reach the maximum value when 0.1 < R/a < 6, and (iii) decays monotonically. The differences are in (i) the inclination of the initial rise, (ii) the location of the peak, and (iii) the decreasing manner when R → ∞. Refer to Fig. E40 showing the rotation curves after their peak values and the location of peaks become the same by appropriate scaling. See also Table E1 for a quantitative comparison. It is interesting that, for each kind of spheroid, almost flat rotation curves are realized when c/a is large. This is a natural consequence of the facts that (i) very prolate spheroids resemble thin rods, (ii) their gravitational potential are similar to that of an infinitely thin rod, namely the logarithmic potential

Table E1. Rotation curve indices of spheroids. Summarized are the numerical values of two indices of the rotation curve, W1/2 and W2 , for a group of infinitely extended spheroids when c/a = 0.3. The rotation curve index for a real-valued factor, k, is defined as Wk ≡ V (k R max ) /Vmax where (i) Vmax ≡ max R V (R) is the peak value of the circular speed, and (ii) R max is the location of the peak such that Vmax = V (R max ). The spheroids are listed in the increasing order of W1/2 . Although this shows a special case when c/a = 0.3, we experimentally confirmed that the order is hardly sensitive to the aspect ratio. spheroid

W1/2

W2

Gaussian Plummer modified exponential exponential perfect Sérsic (n = 1) modified Hubble Sérsic (n = 2) Burkert Hernquist Einasto (i = 2) Navarro-Frenk-White Einasto (i = 4) de Vaucouleurs Einasto (i = 7)

0.773 0.817 0.834 0.853 0.860 0.884 0.890 0.910 0.911 0.930 0.936 0.952 0.954 0.968 0.968

0.726 0.864 0.836 0.830 0.878 0.866 0.930 0.910 0.936 0.930 0.938 0.958 0.947 0.972 0.976

(MacMillan 1930) as Φ(R, z) ∝ − ln R, then (iii) the radial acceleration, FR (R, z) ≡ (∂Φ/∂ R), is in √ proportion to 1/R, and therefore, (iv) the rotation curve, V (R) ≡ −RFR (R, 0), becomes asymptotically flat. Qualitatively speaking, we notice that the overall feature of the rotation curves depend on the aspect ratio c/a rather weakly. Therefore, one may quote the spherically symmetric case, a = c, which is analytically and easily computed thanks to Newton’s theorems, as the representative rotation curve. If a better approximation is required for oblate spheroids where 0 ≤ c/a ≤ 1, we suggest a linear interpolation between two major cases: (i) the spherically symmetric case where c/a = 1, and (ii) the infinitely thin case where c/a = 0, the result of which is quickly computed (Fukushima 2016).

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of Gaussian Spheroid

Φ of Modified Exponential Spheroid 4

3

3

2

2

z

z

4

1

1

0

0 0

1

2

3

4

0

1

R

2

3

4

R

Figure E6. Potential contour map of Gaussian spheroid. Shown is the contour map of the gravitational potential ( ) √ of the Gaussian spheroid defined as ρ G (s) ≡ ρ 0 exp −s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3. The contours are drawn at every 5 per cent level of the bottom value realized at the coordinate origin. Despite their elliptical outlook, the contours are not genuine ellipses.

Figure E8. Potential contour map of modifed exponential spheroid Same as Fig. E6 but for the modified exponential spheroid (Robin et al. 2003) defined [ /√ ] √ √ as ρ M (s) ≡ ρ 0 exp −s 2 1 + 1 + s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

Φ of Burkert Spheroid

Φ of Exponential Spheroid

4

4

3 z

3 z

33

2

2 1

1

0

0 0

1

2 R

3

4

Figure E7. Potential contour map of exponential spheroid Same as Fig. E6 but for √ the exponential spheroid defined as ρ X (s) ≡ ρ 0 exp (−s) while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

MNRAS 000, 1–43 (2016)

0

1

2 R

3

4

Figure E9. Potential contour map of Burkert spheroid. Same as Fig. E6 but for the Burkert spheroid (Burkert 1995) defined as ρ B (s) ≡ ρ 0 (1 + ( ) −1 √ s) −1 1 + s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

T. Fukushima

Φ of Modified Isothremal Spheroid 4

Φ of Perfect Spheroid 4

3

3

2

2

z

z

34

1

1

0

0 0

1

2

3

4

0

1

R

Φ of Plummer Spheroid

z

4

3

3

2

2

z

4

1

1

0

0 2 R

4

Figure E12. Potential contour map of perfect spheroid. Same as Fig. E6 but for the perfect spheroid (de Zeeuw 1985a,b) defined as ρ P4 (s) ≡ ( ) −2 √ ρ0 1 + s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

Φ of Modified Hubble Spheroid

1

3

R

Figure E10. Potential contour map of modified isothermal spheroid. Same as Fig. E6 but for a modified isothermal spheroid (King 1962) defined as ( ) −1 √ ρ P2 (s) ≡ ρ 0 1 + s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

0

2

3

4

Figure E11. Potential contour map of modified Hubble spheroid. Same as Fig. E6 but for the modified Hubble spheroid (Binney & Tremaine 2008) ( ) −3/2 √ defined as ρ P3 (s) ≡ ρ 0 1 + s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

0

1

2 R

3

4

Figure E13. Potential contour map of Plummer spheroid. Same as Fig. E6 but for the Plummer spheroid (Plummer 1911) defined as ρ P5 (s) ≡ ( ) −5/2 √ ρ0 1 + s 2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Φ of NFW Spheroid

4

4

3

3

2

2

z

z

Φ of Jaffe Spheroid

1

1

0

0 0

1

2

3

4

0

1

R

2

3

4

R

Figure E14. Potential contour map of Jaffe spheroid. Same as Fig. E6 but for a√Jaffet spheroid (Jaffe 1983) defined as ρ J (s) ≡ ρ 0 s −2 (1 + s) −2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

Figure E16. Potential contour map of NFW spheroid. Same as Fig. E6 but for a Navarro-Frenk-White (NFW) spheroid (Navarro et al. 1997) defined as √ ρ N (s) ≡ ρ 0 s −1 (1 + s) −2 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

Φ of Hernquist Spheroid

Φ of Sersic Spheroid, n=1

4

4

3

3

2

z

z

35

1

2 1

0 0

1

2 R

3

4

Figure E15. Potential contour map of Hernquist spheroid. Same as Fig. E6 −1 but for a Hernquist √ spheroid (Hernquist 1990) defined as ρ H (s) ≡ ρ 0 s (1+ s) −3 while s ≡ (R/a) 2 + (z/c) 2 when a = 1 and c = 0.3.

MNRAS 000, 1–43 (2016)

0 0

1

2 R

3

4

Figure E17. Potential contour map of Sérsic spheroid, n = 1. Same as Fig. E6 but for a Sérsic spheroid (Sérsic 1963, √ 1968) of index n = 1 defined as ρ S1 (s) ≡ ρ 0 s −p 1 exp (−s) while s ≡ (R/a) 2 + (z/c) 2 and p 1 ≈ 0.44493000 when a = 1 and c = 0.3.

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Φ of Einasto Spheroid, i=2

4

4

3

3

2

2

z

z

Φ of Sersic Spheroid, n=2

1

1

0

0 0

1

2

3

4

0

1

R

Φ of Einasto Spheroid, i=4 4

3

3

2

2

z

z

4

1

1

0

0 2 R

4

Figure E20. Potential contour map of Einasto spheroid, i = 2. Same as Fig. E6 but for an Einasto spheroid (Einasto√1965) of index 2 de( √ ) fined as ρ E2 (s) ≡ ρ 0 exp −d 2 s while s ≡ (R/a) 2 + (z/c) 2 and d 2 ≈ 5.7061667 when a = 1 and c = 0.3.

Φ of de Vaucouleurs Spheroid

1

3

R

Figure E18. Potential contour map of Sérsic spheroid, n = 2. Same as Fig. E6 but for a Sérsic spheroid (Sérsic 1963, 1968) of index n = 2 ( √ ) √ defined as ρ S2 (s) ≡ ρ 0 s −p 2 exp − s while s ≡ (R/a) 2 + (z/c) 2 and p 2 ≈ 0.70880750 when a = 1 and c = 0.3.

0

2

3

4

Figure E19. Potential contour map of de Vaucouleurs spheroid. Same as Fig. E6 but for a de Vaucouleurs spheroid (de Vaucouleurs 1948),(namely) Sérsic spheroid of index n = 4 defined as ρ S4 (s) ≡ ρ 0 s −p 4 exp −s 1/4 √ while s ≡ (R/a) 2 + (z/c) 2 and p 4 ≈ 0.85098937 when a = 1 and c = 0.3.

0

1

2 R

3

4

Figure E21. Potential contour map of Einasto spheroid, i = 4. Same spheroid (Einasto√1965) of index 4 deas Fig. E6 but for an Einasto ( ) fined as ρ E4 (s) ≡ ρ 0 exp −d 4 s 1/4 while s ≡ (R/a) 2 + (z/c) 2 and d 4 ≈ 11.686417 when a = 1 and c = 0.3.

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

37

Φ of Einasto Spheroid, i=7 Circular Speed: Exponential Spheroid

4

3.5 3 2.5 V/V0

z

3

δ=10-8, a=1

c=5

2

1

2 1.5 1 c=0.1

0.5

1

0 0

2

4

6

8

10 12 14 16 18 20 R

0 0

1

2

3

4

R

Figure E24. Rotation curve of modified exponential spheroid. Same as Fig. E23 but for a spheroid with √ the volume mass density expressed as ρ X (s) ≡ ρ 0 exp(−s) where s ≡ (R/a) 2 + (z/c) 2 .

Figure E22. Potential contour map of Einasto spheroid, i = 7. Same as Fig. E6 but for an Einasto spheroid (Einasto√1965) of index 7 de( ) fined as ρ E7 (s) ≡ ρ 0 exp −d 7 s 1/7 while s ≡ (R/a) 2 + (z/c) 2 and d 7 ≈ 20.677952 when a = 1 and c = 0.3.

Circular Speed: Modified Exponential Spheroid 4

-8

δ=10 , a=1

c=5

3.5 3

V/V0

Circular Speed: Gaussian Spheroid δ=10-8, a=1

c=5

V/V0

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

1

2.5 2 1.5 1

c=0.1

0.5

1

0 0

2

4

6

8

10 12 14 16 18 20 R

Figure E25. Rotation curve of modified exponential spheroid. Same as Fig. E23 but for [a spheroid mass √ density expressed as )] ( with √ the volume ρ M (s) ≡ ρ 0 exp −2s 2 / 1 + 1 + 4s 2 where s ≡ (R/a) 2 + (z/c) 2 .

c=0.1 0

2

4

6

8

10 12 14 16 18 20 R

Figure E23. Rotation curve of Gaussian spheroid. Shown are the rotation curves of a group of with the ( spheroids ) √ volume mass density expressed as ρ G (s) ≡ ρ 0 exp −s 2 where s ≡ (R/a) 2 + (z/c) 2 . Plotted are the curves for various values of c as c = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5 while a is set as a = 1 and the nominal density value, ρ 0 , is fixed. If the total mass is fixed instead, V0 is adjusted such that the asymptotic forms of all the curves become the same. The thick curve represents the spherically symmetric case, c = a, which is analytically computable by means of the error function.

MNRAS 000, 1–43 (2016)

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T. Fukushima

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

c=5

Circular Speed: Modified Hubble Spheroid 3

-8

δ=10 , a=1

-8

c=5

δ=10 , a=1

2.5 1

2 V/V0

V/V0

Circular Speed: Burkert Spheroid

1

1.5 1

c=0.1

c=0.1

0.5 0

0

2

4

6

8

10 12 14 16 18 20

0

2

4

6

8

R

R

Figure E26. Rotation curve of Burkert spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ B (s) ≡ ρ 0 (1 + ( ) −1 √ s) −1 1 + s 2 where s ≡ (R/a) 2 + (z/c) 2 (Burkert 1995).

Figure E28. Rotation curve of modified Hubble spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ P3 (s) ≡ ( ) −3/2 √ ρ0 1 + s 2 where s ≡ (R/a) 2 + (z/c) 2 (Binney & Tremaine 2008).

δ=10-8, a=1

Circular Speed: Perfect Spheroid

c=5

c=1 V/V0

V/V0

Circular Speed: Modified Isothermal Spheroid 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

c=0.1

0

2

4

6

8

10 12 14 16 18 20

10 12 14 16 18 20 R

Figure E27. Rotation curve of modified isothermal spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ( ) −1 √ ρ P2 (s) ≡ ρ 0 1 + s 2 where s ≡ (R/a) 2 + (z/c) 2 (King 1962).

2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

δ=10-8, a=1

c=5

1

c=0.1

0

2

4

6

8

10 12 14 16 18 20 R

Figure E29. Rotation curve of perfect spheroid. Same as Fig. E23 but for a ( ) −2 spheroid with the volume mass density expressed as ρ P4 (s) ≡ ρ 0 1 + s 2 √ where s ≡ (R/a) 2 + (z/c) 2 (de Zeeuw 1985a,b).

MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

Circular Speed: Plummer Spheroid 1.8

Circular Speed: Hernquist Spheroid

-8

δ=10 , a=1

1.6

c=5

1.4 1

V/V0

V/V0

1.2 1 0.8 0.6 0.4

c=0.1

0.2 0 0

2

4

6

8

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

-8

δ=10 , a=1 c=5

1

c=0.1

10 12 14 16 18 20

2

4

6

8

R

Figure E32. Rotation curve of Hernquist spheroid. Same as Fig. E23 but for −1 a spheroid with the √ volume mass density expressed as ρ H (s) ≡ ρ 0 s (1 + s) −3 where s ≡ (R/a) 2 + (z/c) 2 (Hernquist 1990).

Circular Speed: NFW Spheroid

δ=10-8, a=1

V/V0

V/V0

Circular Speed: Jaffe Spheroid

c=5 1 c=0.1 2

4

6

8

10 12 14 16 18 20 R

Figure E31. Rotation curve of Jaffe spheroid. Same as Fig. E23 but for a −2 −2 spheroid with √ the volume mass density expressed as ρ J (s) ≡ ρ 0 s (1+s) where s ≡ (R/a) 2 + (z/c) 2 (Jaffe 1983). Notice that the circular speed approaches to a finite value when R → 0. This is unphysical.

MNRAS 000, 1–43 (2016)

10 12 14 16 18 20 R

Figure E30. Rotation curve of Plummer spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ P5 (s) ≡ ( ) −5/2 √ ρ0 1 + s 2 where s ≡ (R/a) 2 + (z/c) 2 (Plummer 1911).

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

39

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

δ=10-8, a=1

c=5

1

c=0.1

0

2

4

6

8

10 12 14 16 18 20 R

Figure E33. Rotation curve of Navarro-Frenk-White (NFW) spheroid. Same as Fig. E23 but for a spheroid with the volume√ mass density expressed as ρ NFW (R, z) ≡ ρ 0 s −1 (1 + s) −2 where s ≡ (R/a) 2 + (z/c) 2 (Navarro et al. 1997).

40

T. Fukushima

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Circular Speed: de Vaucouleurs Spheroid 1.8

δ=10-8, a=1

c=5

-8

1.6

δ=10 , a=1

c=5

1.4 1.2

1 V/V0

V/V0

Circular Speed: Sersic Spheroid, n=1

1

1 0.8 0.6 0.4

c=0.1

c=0.1

0.2 0 0

0

2

4

6

8

2

4

6

8

10 12 14 16 18 20

10 12 14 16 18 20 10-3R

R Figure E34. Rotation curve of Sérsic spheroid, n = 1. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ S1 (s) ≡ √ ρ 0 s −p 1 exp(−s) where p 1 ≈ 0.44493000 and s ≡ (R/a) 2 + (z/c) 2 . This density profile approximately reproduces the surface brightness profile following the Sérsic law of index n = 1 (Sérsic 1963, 1968).

Figure E36. Rotation curve of de Vaucouleurs spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ S4 (s) ≡ ( ) √ ρ 0 s −p 4 exp −s 1/4 where p 4 ≈ 0.85098937 and s ≡ (R/a) 2 + (z/c) 2 . This time, further extended is the range of radial distance so as to cover the feature of rise and fall. If the object is spherically symmetric such that a = c, the volume density profile approximately reproduces the surface brightness profile following the de Veaucouleurs’ law (de Vaucouleurs 1948).

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

δ=10-8, a=1

Circular Speed: Einasto Spheroid, i=2

c=5

1 V/V0

V/V0

Circular Speed: Sersic Spheroid, n=2

c=0.1

0

2

4

6

8

10 12 14 16 18 20 -1

10 R

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

δ=10-8, a=1 c=5

1

c=0.1 0

Figure E35. Rotation curve of Sérsic spheroid, n = 2. Same as Fig. E23 but for a spheroid as ρ S2 (s) ≡ ( √ ) with the volume mass density expressed √ ρ 0 s −p 2 exp − s where p 2 ≈ 0.70880750 and s ≡ (R/a) 2 + (z/c) 2 . Enlarged is the range of radial distance so as to cover the feature of rise and fall.

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2

4

6

8

10 12 14 16 18 20 R

Figure E37. Rotation curve of Einasto spheroid, i = 2. Same as Fig. E23 but for a spheroid with the volume mass density √ expressed as ρ E2 (s) ≡ ρ 0 exp (−d 2 s) where d 2 ≈ 5.7061667 and s ≡ (R/a) 2 + (z/c) 2 .

Brooks, D. R. 1997, Problem Solving with Fortran 90, Springer, Berlin Burkert, A. 1995, ApJ, 447, L25 Byrd, P. F., Friedman, M. D. 1971, Handbook on Elliptic Integrals for Engineers and Physicists, 2nd ed., Springer-Verlag, Berlin Casertano, S. 1983, MNRAS, 203, 735 Chandrasekhar, S. 1969, Ellipsoidal Figures of Equilibrium, Yale Univ. Press, Boston (reprinted 1987, Dover Publ.) Chandrasekhar, S. 1995, Newton’s Principia for the Common Reader, Oxford Univ. Press, Oxford MNRAS 000, 1–43 (2016)

Gravitational field of axisymmetric object

2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Spheroidal Law Dependence of Rotation Curve

δ=10-8, a=1

c=5

c=1

V/Vmax

V/V0

Circular Speed: Einasto Spheroid, i=4

c=0.1

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

Einasto, i=7

Gaussian

δ=10-8, a=1, c=0.3 0

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2.5

3

R Figure E38. Rotation curve of Einasto spheroid, i = 4. Same as Fig. E23 but for (a spheroid as ρ E4 (s) ≡ ) with the volume mass density expressed √ ρ 0 exp −d 4 s 1/4 where d 4 ≈ 11.686417 and s ≡ (R/a) 2 + (z/c) 2 (Einasto 1965). This time, the radial argument range is diminished so as to emphasize the initial rising feature.

Figure E40. Spheroidal law dependence of rotation curve. Same as Fig. 47 but compared are the rotation curves of most of the spheroids when a = 1 and c = 0.3. The exceptions are (i) the Jaffe spheroid where the curve reaches the maximum at the coordinate origin, and (ii) the modified isothermal spheroid where the curve increases monotonically and no maximum value is attained. Among the curves shown here, that of the Einasto spheroid of the index i = 7 increases most quickly and decreases most gradually. Meanwhile, that of the Gaussian spheroid grows most slowly and decays most rapidly. See Table E1 for the detailed comparison.

V/V0

Circular Speed: Einasto Spheroid, i=7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

δ=10-8, a=1

c=5

c=1

c=0.1

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

R Figure E39. Rotation curve of Einasto spheroid, i = 7. Same as Fig. E23 but for (a spheroid as ρ E7 (s) ≡ ) with the volume mass density expressed √ ρ 0 exp −d 7 s 1/7 where d 7 ≈ 20.677952 and s ≡ (R/a) 2 + (z/c) 2 (Einasto 1965). Again, the radial argument range is diminished so as to emphasize the initial rising feature.

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