MNRAS 000, 1–13 (2015)
Preprint 11 December 2015
Compiled using MNRAS LATEX style file v3.0
Numerical computation of gravitational field of infinitely-thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33 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 an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’ algorithm. The new method is of around 12 digit accuracy and sufficiently fast because requiring only one-dimensional integration. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density. Although the number of free parameters is a little larger, the latter model provides a significantly better fit. The Fortran 90 programs of the new method are electronically available. Key words: accretion, accretion discs — celestial mechanics — galaxies: individuals: M33 — galaxies: spiral — gravitation — method: numerical
1 INTRODUCTION The existence of the dark matter was first suggested by the unexplained rotation curve of spiral galaxies (Sofue & Rubin 2001, §1). The standard approach to estimate the dark matter distribution from the observed rotation curves is the deconvolution method: (i) to compute the squared circular velocity, V 2 , contributed by the gravitational field of stars and gas both of which are assumed to lie on the galactic plane, (ii) to calculate the other contributions, if necessary, such as the central black hole regarded as a masspoint or the bulge approximated as a spheroid, (iii) to subtract their sum from the observed values of V 2 , and (iv) to estimate the parameters of the dark matter distribution by fitting its contribution to the residual. Good examples of the deconvolution are Corbelli et al. (2014) for M33 and Sofue (2015) for the Milky Way where the spherical distribution of the dark matter was determined by using one or a few popular models (Burkert 1995; Navarro, Frenk, & White 1996). The deconvolution method demands an accurate evaluation of the gravitational field of a thin disc. Nevertheless, its computation is not so easy as that of a spherical mass distribution (Kellogg 1929). Especially, the external mass gravitationally affects the internal motion unless its distribution is (i) spherically symmetric, ⋆
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© 2015 The Authors
or (ii) spheroidally or ellipsoidally symmetric in the sense that the distribution happens to follow that of a confocal homoeoid (Binney & Tremaine 2008, §2.5.1). Also, the gravitation is not a local phenomenon. A point value of the gravitational field is determined by a global distribution of mass. Indeed, a high rotation velocity at some radius does not automatically mean a large gravitational potential nor a high mass density at that radius but does a strong gravitational acceleration there. It may be caused by (i) a rapid change of the local and small mass distribution, and/or (ii) a gradual variation of the global and massive one. As a result, the rotation curve computation becomes more complicated if the disc is finite and/or has a hole at the center as in the case of typical accretion discs (Pringle 1981). Of course, this issue is classic and has been extensively investigated: from the early days in electrostatics (Green 1835; Weber 1873), and later in astrophysics (Wyse & Mayall 1942; Schmidt 1956; Brandt 1960). In fact, known are a variety of exact solutions of the gravitational potential, the acceleration vector, and/or the rotation curve of the infinitely-thin axisymmetric disc (Binney & Tremaine 1987; Evans & de Zeeuw 1992; Binney & Tremaine 2008). Their representatives are (i) a finite uniform disc (Durand 1953; Lass & Blitzer 1983; Fukushima 2010), (ii) the Kuzmin-Toomre discs (Kuzmin 1956; Toomre 1963), (iii) the Mestel-Kalnajs discs (Hunter 1963; Mestel 1963;
2
T. Fukushima
M33 Surface Mass Density: Stars
M33 Surface Mass Density: Gas
1000
Σ (MSun pc−2)
Σ (MSun pc−2)
10 100 10 1
1
0.1
0.1 1
10
1
10
R (kpc)
R (kpc)
Figure 1. The log-log plot of the surface mass density of stars of M33 quoted from Corbelli et al. (2014, Table 1). The curves are (i) an exponentiallydamped power function fitted to the data for R < 10 kpc, and (ii) an exponential function fitted to the data for R > 10 kpc. Their detailed models are described in §5.1 of the main text.
Figure 3. The log-log plot of the surface mass density of gas of M33 quoted from Corbelli et al. (2014, Table 1). Same as Fig. 1 but for the gas component. The curves are (i) a double power-law function fitted to the data for R < 10 kpc, and (iii) a power function fitted to the data for R > 10 kpc. Their detailed models are described in §5.1 of the main text.
M33 Surface Mass Density: Stars
M33 Surface Mass Density: Gas
1000 100
Σ (MSun pc−2)
Σ (MSun pc−2)
10
10 1
1
0.1 0.1 0
5
10
15
20
25
0
5
10
15
20
25
R (kpc)
R (kpc)
Figure 2. The logarithmic plot of the surface mass density of stars of M33. Same as Fig. 1 but plotted in a single logarithmic manner.
Figure 4. The logarithmic plot of the surface mass density of gas of M33. Same as Fig. 3 but plotted in a single logarithmic manner.
Kalnajs 1972; Gonzalez & Reina 2006), (iv) the (infinite) Mestel disc (Mestel 1963) and its generalization termed power-law discs (Schmitz & Ebert 1987; Evans 1994), and (v) an infinite exponential disc (Freeman 1970). However, the reality is not so simple. See Figs 1 through 4 showing the surface mass density profiles of the stars and the gas in M33 (Corbelli et al. 2014, Table 1). The stars component can be approximated by a combination of two different curves, say an exponentially-damped power function and an exponential function switched at R ≈ 10 kpc. Meanwhile, that of the gas component seems to contain two parts: a double power-law function, and a single power function. The details of the fitted models will be de-
scribed later in §5. At any rate, this example is a trigger of our trial to develop a method to evaluate the gravitational field of an infinitelythin axisymmetric disc with an arbitrary profile of its surface mass density. There exist six approaches to do this task: (i) the direct numerical quadrature of line integrals (Kellogg 1929), (ii) the forward and inverse Hankel transforms (Weber 1873; Toomre 1963), (iii) an approximation by the field of a disc with a finite but sufficiently small thickness (Casertano 1983), (iv) a superposition of homoeoids (Cuddeford 1993, Eq. (34)), (v) a splitting of the density profile into the uniform component at the evaluation point and the residual (Pierens & Hure 2004; Hure & Pierens 2005), and (vi) the partial MNRAS 000, 1–13 (2015)
Gravitational field of infinitely-thin disc transformation of the potential expression in the first approach into the derivative of a super potential (Hure & Dieckmann 2012). The first approach is ideal but known to suffer from serious numerical problems caused by (i) the logarithmic singularity of the complete elliptic integral of the first kind in the case of potential computation, and/or (ii) the algebraic singularity of the integrand and a cancellation between the positive and negative parts in the case of acceleration computation (Binney & Tremaine 2008, §2.6). This is the reason why Cuddeford (1993) developed its alternative as the fourth option. In the second case, the oscillatory behavior of the Bessel functions (Olver et al. 2010, §10.3), which are the kernel of both Hankel transforms, makes their accurate numerical evaluation difficult. The third approach is more general in the sense it can treat thick discs as well. However, unchanged is the issue of the integrand singularity on the disc plane. On the other hand, the last two options are promising since they avoid the singularity issues by (i) splitting the density into the analytically-integrable part and the remainder, or (ii) careful rewriting of the integrand. The fifth option works well if the density profile is close to a constant (Hure & Pierens 2005). However, its precision soon degrades when eminent is the departure of the density profile from the constant. Notice that the last expression contains the complete elliptic integrals of all three kinds. Among them, that of the third kind is a complicated special function (Byrd & Friedman 1971). Its computation is not an easy task (Bulirsch 1969; Carlson 1979) although its fast and precise computation is now available (Fukushima 2013). Meanwhile, the second, third, and fourth methods demand two-dimensional numerical integrations, and therefore, are time-cosuming even if the convergence of their numerical quadrature is assured and sufficiently fast. In this article, we reexamine the first approach and present a simpler method requiring only one-dimensional numerical quadrature. Below, we describe the new method in §2, confirm its correctness numerically in §3, provide the rotation curves of some sample discs to show its performance in §4, and apply it to a couple of model fittings to the rotation curve of M33 in §5.
2 METHOD Consider an infinitely-thin axisymmetric annular disc of an inifite or finite size. Adopt the cylindrical polar coordinate system, (R, Z, ϕ). Denote the inner and outer radii of the disc by Rin (≥ 0) and Rout (≤ ∞), and the surface mass density of the disc by Σ(R), respectively. Then, the gravitational potential of the disc is expressed as a line integral (Kellogg 1929) as ∫ R out ( ) Φ(R, Z ) = Ψ R ′ ; R, Z dR ′, (1) R in
where the integrand is written as ( ) ( ( )) ( ) −4GΣ R ′ K m R ′ ; R, Z R ′ , Ψ R ′ ; R, Z ≡ P (R ′ ; R, Z )
(2)
while G is Newton’s gravitational constant, K (m) is the complete elliptic integral of the first kind (Byrd & Friedman 1971) with the pa( ) ( ) rameter m ≡ k 2 as its argument, and m R ′ ; R, Z and P R ′ ; R, Z are functions defined as ( ) m R ′ ; R, Z ≡ ( ) P R ′ ; R, Z ≡
4RR ′ (R + R ′ ) 2 + Z 2 √
,
(R + R ′ ) 2 + Z 2 .
MNRAS 000, 1–13 (2015)
(3)
(4)
3
Refer to Fukushima (2010) for the geometric meaning of these quantities. In order to compute the integral accurately and quickly, we evaluate it by the double exponential (DE) quadrature rule (Takahashi & Mori 1973), which is explained in Press et al. (2007, §4.5). If the disc consists of multiple parts and Σ(R) is described by a piecewise function as ( ) Σ(R) = Σ j (R), R j < R < R j+1 ; j = 1, 2, . . . , J (5) where R1 ≡ Rin and RJ +1 ≡ Rout , then Φ(R, Z ) is computed as the sum of the contributions of each piece as Φ(R, Z ) =
J ∑
Φ j (R, Z ),
(6)
j=1
where Φ j (R, Z ) =
∫ R out R in
( ) Ψ j R ′ ; R, Z dR ′,
(7)
while
( ) ( ( )) ( ′ ) −4GΣ j R ′ K m R ′ ; R, Z R ′ Ψ j R ; R, Z ≡ . P (R ′ ; R, Z )
(8)
The number of pieces, J, would be small. For example, it becomes two in the case of the known disc components of M33 galaxy. Refer to Figs 1 through 4 in the previous section. At any rate, since the DE rule assumes the analyticity of the integrand, it is wise to apply the rule to evaluate each Φ j (R, Z ) separately and to sum them up. When R j < R < R j+1 for a certain index j and | Z | is sufficiently small, the direct application of the quadrature rule to Φ j (R, Z ) encounters with a numerical difficulty caused by the logarithmic singularity of K (m) at m = 1. This becomes fatal when Z = 0. Even if Z , 0, we cannot escape from it completely if | Z | is relatively small, say when | Z |/R < 0.1. In that case, the integrand has a sharp peak at R ′ = R. This is due to the existence of a complex singularity at R ′ = R ± i Z. The singularity hinders a proper convergence of almost all of the common quadrature rules including the Newton-Cotes rules and the Gaussian-type quadratures. Even the powerful DE rule is no exception. This is because the singularity prohibits a proper convergence of the trapezoidal rule, which the DE rule utilizes after the variable transformation by a doublly-exponential function. In order to resolve this issue, following the treatment in computing a general integral of the Fermi-Dirac distribution (Fukushima 2014), we split the integration interval at R ′ = R as Φ j (R, Z ) = Φ j L (R, Z ) + Φ j R (R, Z ), where Φ j L (R, Z ) ≡
Φ j R (R, Z ) ≡
∫ R Rj
( ) Ψ j R ′ ; R, Z dR ′,
∫ R j+1 R
( ) Ψ j R ′ ; R, Z dR ′,
(9)
(10)
(11)
and apply the double exponential rule to Φ j L (R, Z ) and Φ j R (R, Z ) separately. This device is simple but works quite well since the DE rule can properly handle the integrable singularities if they are on the end-points of the integration interval. On the other hand, the cylindrical polar coordinate components of the acceleration vector are originally defined as the partial derivatives of the gravitational potential as ( ) ∂Φ(R, Z ) A R (R, Z ) ≡ − , (12) ∂R Z
4
T. Fukushima (
AZ (R, Z ) ≡ −
) ∂Φ(R, Z ) . ∂Z R
(13)
Rotation Curve: Finite Uniform Disc
If we exchange the order of the partial differentiation and the integration, we arrive at their integral representations as ∫ R out ( ( )) ∂Ψ R ′ ; R, Z A R (R, Z ) = − dR ′, (14) ∂R R in R ′, Z
Nevertheless, their integrands share a small divisor, √ 2 ′ 2 (R − R) + Z , which completely vanishes when R ′ = R if Z = 0. This is an algebraic singularity. Thus, the integral becomes improper, and therefore, its value must be understood in the sense of Cauchy’s principal value (Olver et al. 2010, §4.5). In other words, its numerical computation becomes quite difficult. Fortunately, there exists a primitive but effective device: the numerical differentiation. Namely, using the original definitions, Eqs (12) and (13), we numerically differentiate Φ(R, Z ) obtained by the numerical quadrature as described earlier. As the method of numerical differentiation, we adopt Ridder’s algorithm (Ridder 1982). The algorithm assumes the analyticity of Φ(R, Z ). Therefore, as in the same way in evaluating Φ(R, Z ), the application of the algorithm must be piecewise, namely must be applied to each Φ j (R, Z ) and to sum up the results. At any rate, if A R (R, 0) ≤ 0, the circular orbital velocity is obtained as √ V (R) = −RA R (R, 0). (16) Thus, the method is completed. As for the actual implementation of the DE rule, we recommend Ooura’s intde and intdei (Ooura 2006) for the finite and infinite intervals, respectively. Also, K (m) is precisely and quickly computed by ceik, its piecewise minimax rational approximation (Fukushima 2015a). Further, a good implementation of Ridder’s algorithm is dfridr provided by Press et al. (2007). We suggest setting h, the nondimensional initial step size for dfridr, to be 0.01–0.001 times the scaled radial distance of the evaluation point. Typically, the DE rule requires 50–100 evaluations of the integrand when the 15 digits accuracy is demanded. Meanwhile, the averaged CPU time of ceik, the evaluation of which is the most computational labor of the integrand evaluation, is as small as that of the exponential function (Fukushima 2015a). Thus, the averaged CPU time of the potential calculation is only a few µs at CPUs running at 2–4 GHz clock. On the other hand, dfridr usually calls the subprogram to compute the potential 4–10 times. Consequently, the total CPU time to compute a single point of the rotation curve amounts to 10 µs or so at recent PCs. If the final precision requirement is relaxed as 6 digits or so, this value beomes 2–3 times smaller.
3 VALIDATION We examine the accuracy of the new method by comparing its results with a couple of the exact solutions. Begin with a finite uniform disc of the unit radius, namely when Rin = 0 and Rout = 1. The surface mass density is expressed as Σ(R) = Σ0, (R < 1)
(17)
V(R)/V0
(15)
2
V(R)∝1/√R
V(R)∝R
1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 R
Figure 5. Rotation curve of a finite uniform disc where R in = 0 and R out = 1. The thick curve shows the rotation curve computed analytically. The blowup singularity at R = 1 is due to the edge effect caused by a sudden drop of the surface density there. Meanwhile, the two thin√curves are its asymptotes: (i) V (R) ∝ R when R ≪ 1. and (ii) V (R) ∝ 1/ R when R → ∞.
Rotation Curve Error: Finite Uniform Disc -10 log10|Vnew(R)/Vexact(R)−1|
∫ R out ( ( )) ∂Ψ R ′ ; R, Z AZ (R, Z ) = − dR ′ . ∂Z ′ R in R ,R
2.5
-11 -12 -13 -14 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 R
Figure 6. Relative error of the rotation curve of a finite uniform disc. The logarithmic blow-up singularity of the gravitational potential at the edge of the disc, R = 1, hinders the accurate evaluation of the partial derivative by the numerical differentiation.
where Σ0 denotes the central density. Using the analytical expression of A R (R, Z ) (Fukushima 2010), we explicitly write V (R) as √ V (R) = m(1; R, 0) GΣ0 P(1; R, 0)S(m(1; R, 0)), (18) where (i) m(R ′ ; R, Z ) and P(R ′ ; R, Z ) are already given in Eqs (3) and (4), respectively, (ii) S(m) is a special complete elliptic integral (Fukushima 2010) defined as (2 − m)K (m) − 2E(m) , (19) m2 while (iii) E(m) is the complete ellpitic integral of the second kind (Byrd & Friedman 1971). A precise procedure to compute S(m) ≡
MNRAS 000, 1–13 (2015)
Gravitational field of infinitely-thin disc
Rotation Curve Error: Exponential Disc
Rotation Curve: Power-Law Disc
-11
100
-12
Σ(R) ∝ R−α, Rin=10−8, Rout=104
10 V(R)/V(1)
log10|Vnew(R)/Vexact(R)−1|
5
-13 -14
1
α=3
0.1 α=10−4 α=0
-15 -16 0
1
2
3
4
5
6
7
8
9 10
0.1
R Figure 7. Relative error of the rotation curve of an exponential disc. Same as Fig. 6 but for an exponential disc where Σ(R) ∝ exp(−R).
S(m) is found in Fukushima (2010). Recently we developed ceis, a faster procedure to compute S(m) (Fukushima 2015b, Appendix). Fig. 5 illustrates the normalized rotation curve of the finite uniform disc computed analytically. The same rotation curve is numerically obtained by the new method. Its relative errors are shown in Fig. 6. Around 12 digit accuracy is maintained except on the edge of the disc, where the logarithmic singularity of S(m) hinders Ridder’s algorithm to converge properly. Move to the case of an infinite disc where Rin = 0 and Rout = ∞. We conducted a similar examination for an infinite exponential disc (Freeman 1970). Assume that the scale radius of the exponential decay is unity. Then, its surface mass density is expressed as Σ(R) = Σ0 exp(−R),
(20)
where Σ0 is again the central density. Its rotation curve is analytically obtained (Binney & Tremaine 2008, Eq. (2.165)). The feature of the rotation curve will be seen as a thick curve in Fig. 17 later. The comparison of the numerically-obtained result with the exact solution confimed the 12 digit accuracy of the new method as shown in Fig. 7. The accuracy of 11–12 digits achieved here is a typical limit of the numerical differentiation in the double precision environment. This is realized by the best balance between the truncation and round-off errors of the second-order central difference formula, which Ridder’s method is based on. Although the achieved accuracy is not the ultimate, we think it is far more enough for the purpose of astrophysical simulation and data analysis.
4 EXAMPLES 4.1
Truncated Power-law disc
As an illustration of the performance of the new method, we prepared Fig. 8 showing the rotation curve of (almost) infinite powerlaw discs. Their surface mass density profiles are expressed as Σ(R) ∝ R−α ,
(21)
where α is a general real number. If we strictly apply this functional form to an infinite disc covering the entire range, 0 ≤ R < ∞, it MNRAS 000, 1–13 (2015)
1
10
100
R Figure 8. Rotation curve of power-law discs: log-log plot. Shown are the loglog plots of the rotation curve of truncated power-law discs. Their surface mass density profile is of the form as Σ(R) ∝ R −α . In order to realize physically-meaningful discs, we restricted the inner and outer radii as R in = 10−8 and R out = 104 , respectively. Plotted are the curves for various powerlaw exponents as α = 0, 10−4 , 10−3 , 10−2 , 0.1, 0.2, . . . , 1, 1.1, . . . , 2, and 3. The thick solid ones are the cases for α = 0 and α = 1, corresponding to the uniform and Mestel discs, respectively. The thin sold ones are those for 0.1 < α < 3 while the thin broken ones are those for α < 0.1 The thick broken ones are those for α = 0.1 and α = 3, respectively. We omitted the curves when α > 3 since their deviations from that of α = 3 are not visible at this scale.
becomes unphysical. Indeed, (i) the total mass of the disc becomes infinite if α ≤ 1, while (ii) the disc has a blow-up singularity at R = 0 when α > 0 (Evans 1994). Therefore, in order to realize a physically-meaningful disc of the similar property, we set the inner radius as tiny as Rin = 10−8 and the outer one sufficiently large as Rout = 104 . These restrictions were necessary so as to integrate the gravitational potential correctly. The figure shows the rotation curves in a log-log manner for the range, 10−2 ≤ R < 100. This range is sufficeintly narrow in order to suppress the effects of the center pin hole and of the finiteness of the disc size. This is experimentally proven by the straightness of the case of α = 0 drawn by a thick slant curve in the figure, which corresponds to the case of uniform disc. The curves are plotted for several power-law indices as α = 0, 10−4 , 10−3 , 10−2 , 0.1, 0.2, . . . , 1, 1.1, . . . , 2, and 3. If 0.1 ≤ α, the curves are almost straight lines in a log-log graph. In that case, roughly speaking, the rotation curves follow power-laws as V (R) ∝ R µ .
(22)
Fig. 9 illustrates the relation of µ and α, the power-law indices of the rotation curve and of the surface mass density profile, respectively. Here, we approximated µ as √ µ ≈ V (10)/V (0.1), (23) which is easily evaluated from the computed rotation curve. Theoretically, µ is related to α (Schmitz & Ebert 1987, §3) as µ = (1 − α)/2, (1/2 < α < 2)
(24)
where the special case α = 1 corresponds to the infinite Mestel disc (Mestel 1963). The restriction on α comes from the formulas
T. Fukushima
Power-Law Exponent of Rotation Curve 0.5 0.4 Power-Law Disc 0.3 Σ(R) ∝ R−α 0.2 V(R) ∝ Rµ 0.1 Mestel disc 0 -0.1 -0.2 -0.3 -0.4 µ=−1/2 -0.5 -0.6 µ=(1−α)/2 -0.7 0 0.5 1 1.5 2 2.5 3
Hole Effect in Accretion Disc
.
V(R)/V0
µ
6
3.6 3.2 2.8 2.4 2 1.6 1.2 0.8 0.4 0
Σ(R) ∝ R−β√1−(Rin/R)
0.2 0
2
4
β=2.0 β=0.1
6
α
R
log10|V(R)/V0−1|
Figure 9. Power-law exponent of rotation curve of power-law discs. Plotted are µ, the power-law exponent of the rotation curve of a power-law disc, as functions of α, the power-law index of the surface mass density profile. The bullet indicates the case of (infinite) Mestel disc (Mestel 1963).
Size Dependence of Truncated Mestel Disc 1 0 Rout=102 -1 -2 103 -3 -4 104 -5 -6 105 -7 -8 106 -9 -10 -11 -12 0.1 1 10 100
Figure 11. Rotation curve of model accretion discs with a non-negligible center √ hole. The surface mass density profile is expressed as Σ(R) ∝ R −β 1 − (R in /R) while the inner and outer radii are R in = 1 and R out = 104 . Plotted are the rotation curves for several power-law indexs as β = 0.1, 0.2, . . . , 2.0. The thick solid curve is that for β = 1 corresponding to those observed in the simulation of a collapsing protoplanetary disc (Hueso & Guillot 2005, Fig. 1) while the thick broken one is for β = 1.5 corresponding to the minimal mass solar nebula (Hayashi 1981). The arrow indicates the location of the inner edge of the disc, R = R in .
Mestel discs where the outer radius is changed as Rout = 102 , 103 , . . . , 108 while the inner radius is fixed as Rin = 10−8 . Obviously, the observed departure is caused by the finiteness of the disc.
4.2 Effect of center hole
R Figure 10. Size dependence of rotation curve of truncated Mestel disc. Shown are the log-log plots of the departure from the theoretical prediction of V (R) of infinite Mestel disc for the truncated Mestel discs of various sizes as R out = 102 , 103 , . . . , 108 while the inner radius is fixed as R in = 10−8 . The thick solid curve is the case R out = 108 .
for the forward and inverse Hankel transforms (Olver et al. 2010, §10.22(i)) of general power functions used in the derivation of the relation. Apparently Fig. 9 seems to support an extension of this theoretical prediction as µ = (1 − α)/2. (0 < α < 2)
8 10 12 14 16 18 20
(25)
Nevertheless, a detailed examination reveals that there exists a small but non-negligible deviation from the predictions. For example, Fig. 10 shows the difference between the computed value of V (R) and its theoretical prediction for the case of the infinite Mestel disc, namely α = 1 and µ = 0. Plotted are the curves for some truncated
In the above case of truncated power-law discs, a tiny hole at the center is needed to avoid the singularity at the origin, R = 0. Then, a natural question comes out: what will happen if there is a non-negligible center hole? This is an important issue for the study of accretion discs, especially in discussing the dynamics of particle/fluid inside the center hole. Therefore, by setting Rin = 1 and Rout = 104 , we computed V (R) of a typical accretion disc the surface mass density of which is modelled (Pringle 1981) as √ Σ(R) ∝ R−β 1 − (Rin /R). (26) Fig. 11 plots the rotation curves for several power-law indices as β = 0.1, 0.2, . . . , 2.0. It is interesting that an almost flat rotation curve is realized by the power-law indices of 1.2–1.4, which result a finite value of the total mass even if Rout → ∞. Notice that [V (R)]2 becomes negative if R is sufficiently small but larger than Rin . This is no unphysical situation. In that case, the gravitational attraction by the external disc supersedes that of the internal one, and therefore, the total gravitational acceleration of the disc is in the outward direction. As a result, the centrifugal force cannot maintain the circular orbit. In conclusion, a not-so-small center hole has an effect to pull down the rotation curve near the center. Thus, this feature makes the resulting curves somewhat rising. See the case of β = 1 drawn as a thick solid curve. If the central hole does not exist, this should be flat since the curve reduces to that of the infinite Mestel disc. MNRAS 000, 1–13 (2015)
Gravitational field of infinitely-thin disc
Edge-Softening of Truncated Mestel Disc
Σ(R) ∝ 1/(R(exp((R−RE)/RF)+1)) RE=1
Σ(R)/Σ(1)
2.5 2
RF→0
1.5 1
RF=1
0.5
Rotation Curve: Double Power-Law Disc 0.5 0 log10|V(R)/V(100)|
3
c=1
-0.5
c=0
-1 -1.5 -2 -2.5
0
7
Σ(R)∝(R/RS)−c(1+(R/RS)1/a)((c−b)a) a=0.2, b=0.9, RS=2
-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-2 -1.5 -1 -0.5
0
0.5
1
1.5
2
R
log10R
Figure 12. Edge-softening of truncated Mestel disc. Plotted are normalized surface mass density profiles expressed as Σ(R) ∝ 1/(R[exp {(R − R E ) /R F } + 1]) where R E = 1. Thin curves are for several softening scales as R F = 1, 1/2, 2−2 , . . . , 2−5 , while the thick curve corresponds to the limit R F → 0, which is nothing but the (infinite) Mestel disc truncated at R = R E .
Figure 14. Rotation curve of double power-law discs in log-log plot: cdependence. The surface mass density profile takes the form as Σ(R) ∝ ] (c−b)a [ . Plotted are the curves when a = 0.2, (R/R S ) −c 1 + (R/R S ) 1/a b = 0.9, R S = 2, R in = 10−8 , and R out = 108 for various values of c as c = 0, 0.1, . . . , and 1. The thick solid curve is the case c = 0 while the thick broken one is the case c = 1
ening the edge by adding a multiplication factor as Σ(R) ∝
V(R)/V(0)
Edge-Softening Effect 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7
RF→0
RF=1 RE=1
Σ(R) ∝ 1/(R(exp((R−RE)/RF)+1)) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
R−α , exp [(R − RE ) /RF ] + 1
(27)
where RE is the location of the center of the softening, which is usually set to be the edge of unsoftened disc, while RF is a scale radius representing the softness. This functional form of the multiplication factor was hinted from the Fermi-Dirac distribution (Ashcroft & Mermin 1976). The limit case of RF → 0, which means the sharp drop of the surface mass density at R = RE , corresponds to the Sommerfeld approximation of the Fermi-Dirac distribution (Sommerfeld 1927). At any rate, Fig. 12 illustrates the softened profiles of the surface mass density of the truncated Mestel disc where α = 1 and RE = 1 for some values of RF as RF = 1, 1/2, 2−2 , . . . , 2−5 , and RF → 0. The corresponding rotation curves computed by the new method are plotted in Fig. 13. Clearly, the edge-softening also softens the rotation curve as anticipated.
R 4.4 Double power-law disc Figure 13. Effect of edge-softening of truncated Mestel disc. Same as Fig. 12 but for the rotation curves computed by selecting R in = 10−8 and R out = 108 . The thick curve corresponds to the limit R F → 0, which is the (infinite) Mestel disc truncated at R = R E .
4.3
Effect of edge-softening
At the same time, another question arises naturally: what will be the effect of truncating the disc at a certain finite radius. The result for the uniform disc is already displayed in Fig. 5: the logarithmic blow-up singularity at the edge of the disc. The same phenomenon is observed for other power-law discs. We omitted their figures because of the appearance of similar blow-up singularities at the edge. In order to avoid this unphysical situation, let us consider softMNRAS 000, 1–13 (2015)
As seen in the previous subsections, a single power-law of the surface mass density roughly results a single power-law of the rotation curve. Thus, we anticipate that a double power-law of the surface mass density may lead to a double power-law of the rotation curve. Here, by a double power-law, we mean a following functional form such as Σ(R) is expressed as ( ) [ ( ) ] (c−b)a , (28) Σ(R) ∝ R/RS −c 1 + R/RS ) 1/a where a, b, and c are power-law indices while RS is a certain scale radius. We derived this functional form by translating a general profile of the volume mass density proposed by Zhao (1996) into that of the surface mass density. In order to examine the above conjecture, we prepared Figs 14 through 16 illustrating the dependence of the computed rotation curves on three indices, c, b, and a, respectively, while the other
8
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Rotation Curve: Double Power-Law Disc
1
Σ(R)∝(R/RS)−c(1+(R/RS)1/a)((c−b)a) a=0.2, c=0.4, RS=2 b=1
V(R)/V(2)
log10|V(R)/V(0.01)|
1.5
Exponentially-Damped Power-Law Disc
b=0.1
0.5 0
b=0
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2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
γ=5/3
−5/3 −4/3 −1 −2/3
1 2/3
1/3
Σ(R) ∝ R−γexp(−R) 0
0.5
1
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2
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3
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4
log10R
R
Figure 15. Rotation curve of double power-law discs in log-log plot: bdependence. Same as Fig 14 but when a = 0.2, c = 0.4, and R S = 2 for various values of b as b = 0, 0.1, . . . , and 1. The thick solid curve is the case b = 0 while the thick broken one is the case b = 1
Figure 17. Rotation curve of exponentially-damped power-law discs. The surface mass density profile is assumed such that Σ(R) ∝ R −γ exp(−R). Plotted are the curves for various power-law indices as γ = −5/3, −4/3, . . . , 5/3. The thick curve is the case γ = 0, namely the standard exponential disc.
4.5 Exponentially-damped power-law disc
Rotation Curve: Double Power-Law Disc
As another kind of example, we depicted Fig. 17 showing the rotation curve of discs with exponentially-damped power-law profiles of the surface mass density as
log10|V(R)/V(0.01)|
1 0.9
a=0.1
0.8
Σ(R) ∝ R−γ exp(−R).
a=1
0.7 0.6 0.5 0.4 -0.5
(30)
Figs 1 and 2 have already shown examples of this kind of profile. Fig. 17 plots the curves for some power-law indices as γ = ±5/3, ±4/3, ±1, ±2/3, ±1/3, and 0. When γ < 0 and R is sufficiently small, V (R) has no real solution. This is again no unphysical situation.
Σ(R)∝(R/RS)−c(1+(R/RS)1/a)((c−b)a) b=0.9, c=0.4, RS=2 0
0.5
1
1.5
2
log10R
4.6 Sinusoidally-modulated exponential disc
Figure 16. Rotation curve of double power-law discs in log-log plot: adependence. Same as Fig 14 but when b = 0.9, c = 0.4, and R S = 2 for various values of a as a = 0.1, 0.2, . . . , and 1. The thick solid curve is the case a = 0.1 while the thick broken one is the case a = 1
parameters are fixed as RS = 2, Rin = 10−8 , and Rout = 108 . Except the middle range, say 1 < R/RS < 30, the rotation curve of the disc with a double power-law profile of the surface mass density seems to exhibit a power-law-like feature. The inner and outer power-law indices of the rotation curve, µ and ν, roughly satisfy the extended predictions on the relation with the corresponding indices of the surface mass density, c and b, respectively, as µ = (1 − c)/2, ν = (1 − b)/2,
(29)
This fact implies a possibility to construct a model profile of the surface mass density of a disc as a double power-law if the observed rotation curve exhibits a double power-law-like feature as will be seen later in §5.3.
So far, we have dealt with the monotonically-changing profile of the surface mass density. Then, what will happen if the profile is oscillating such as caused by arm-like features? In order to answer this question, we calculated the rotation curve of an exponential disc with a sinusoidal modulation as Σ(R) ∝ exp(−R) [1 + A sin (2πR/R M )] ,
(31)
where R M is the wavelength of the modulation. Fig 18 illustrates the rotation curve of the case when R M = 1 and A = 0.5. The wavy feature in the rotation curve has the same wavelength as that of Σ(R) but the −90 degree shift of the phase, which is well explained by the differential relation between Σ(R), Φ(R, 0), A R (R, 0), and V (R). Nonetheless, the amplitude relation is so complicated that its theoretical prediction seems to be diffiult. Notice that, since A is set to be sufficiently large in this case, [V (R)]2 becomes negative when R is sufficiently small, say when R < 0.06. This is caused by a strong attraction by the nearest mass concentration at R = 0.5. MNRAS 000, 1–13 (2015)
Gravitational field of infinitely-thin disc
Rotation Curve of M33 Disc
1.2
1.2
1
1 V(R)/Vstars(2)
V(R)/Vexp(2)
Sinusoidally-Modulated Exponential Disk
0.8 0.6 0.4
Σ(R) ∝ exp(−R)(1+Asin(2πR/RM)) RM=1, A=0.5
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9
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Figure 18. Rotation curve of a sinusoidally-modulated exponential disc. Same as Fig. 17 but for a disc with the surface mass density profile described as a sinusoidally-modulated exponential function, Σ(R) ∝ exp(−R) [1 + A sin(2π R/R M )] when R M = 1 and A = 0.5. This time, a thin curve is the case A = 0, namely the standard exponential disc.
Figure 19. Rotation curve of M33 disc components. Plotted are the normalized value of the circular velocity contributed by the disc of the stars as a thin sold curve, that of the gas as another thin sold curve, and their sum as the thick sold curve. The surface mass density of the stars and the gas were modelled as the functional forms described in §5.1 of the main text while their inner and outer radii are set as R in = 10−5 pc and R out = 100 Gpc. The thin broken vertical line indicates the place of the switching radius, R D = 10.18 kpc.
5 APPLICATION Let us apply the new method to a case study of the rotation curve of an actual galaxy, M33. The existing works on this spiral galaxy are well summarized in Corbelli et al. (2014). 5.1
RD = 10.18.
Contribution of stars and gas components
As an example for the piecewise profile of the surface mass density, we estimate the contribution of known disc components to the rotation curve of M33. Through fitting to the surface mass density data provided in Corbelli et al. (2014, Table 1), we find a piecewise model for those of the stars and of the gas as Σstars (R) = Σ A (R/R A ) −1/3 exp (−R/R A ) , (R < RD )
(32)
Σstars (R) = Σ B exp (−R/R B ) , (R > RD )
(33)
( ) ] (c−b)a ( ) [ , (R < RD ) Σgas (R) = ΣC R/RC −c 1 + R/RC 1/a (34) ( ) Σgas (R) = Σ D R/RC −3 . (R > RD )
(36)
(ii) the radius constants in the unit of kpc are R A = 2.2, RB = 6.3, RC = 7.2,
(37)
(iii) the power-law indices are a = 0.05, b = 5.5, c = 0.05, MNRAS 000, 1–13 (2015)
(39)
Figs 1 through 4 already depicted the manner of fitting of these model curves to the observed values of the surface mass densities. Based on these models, we computed the rotation curves of the stars component, the gas component, and the total disc component by the new method. The inner and outer radii of these discs are set as Rin = 10−5 pc ≈ 2 AU and Rout = 100 Gpc, respectively. They are sufficiently small and large so as to cover the whole observable universe except a negligiblly small pin hole of the radius of the order of the Sun-Earth distance at the center of the galaxy. The results are shown in Fig 19. See also Table 1 for their numerical values at the radii where the observed values of the rotation curve are available (Corbelli et al. 2014, Table 1). A plateau in the interval of 3–8 kpc is caused by a composite of the stars and gas components. Their peaks are located at different radii, say at 3 kpc and at 8 kpc, respectively. On the other hand, it is only vaguely seen that the decreasing manner of the curves change slightly at the switching point, R = RD .
(35)
Here the model parameters are determined by a multi-dimensional bisection method as follows: (i) the surface mass density constants in the unit of MSun pc−2 are Σ A = 169, Σ B = 5, ΣC = 6, Σ D = 2.5,
while (iv) the radius of the switching point determined by solving the continuity condition of the surface mass density profiles is expressed in the unit of kpc as
(38)
5.2 Deconvolution Using the contributions of the stars and the gas components obtained in the previous subsection, we conduct a deconvolution of the observed rotation curve by adding the contribution of the dark matter distributed spherically to those of the stars and gas disc components. As for the volume mass density distribution of the dark matter, consulting with the conclusion of Corbelli et al. (2014), we concentrate ourselves on the model of Navarro, Frenk, & White (1996): ρ0 (40) ρNFW (r) ≡ ( )[ ( )] , r/r S 1 + r/r S 2
10
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Rotation Curve of M33 160 140
Data: Corbelli et al. (2014)
120 100 80
V(R) (km/s)
V(R) (km s−1)
160 140
Approximation of M33 Rotation Curve
NFW
60 40 20
stars+gas
0
Data: Corbelli et al. (2014)
120 100 80 V(R)=VV (R/RV)−γ(1+(R/RV)1/α)(γ−β)α VV=103 km s−1, RV=3.2 kpc α=0.14, β=−0.1, γ=−0.4
60 40 20 0
0
5
10
15
20
25
0
5
10
15
20
25
R (kpc)
R (kpc)
Figure 20. Rotation curve of M33. The data points plotted with the error bars are quoted from Table 1 of Corbelli et al. (2014). The solid curve is the rotation curve of a model composite of the stars and gas disc components and the spherical dark matter component following the NFW density model (Navarro, Frenk, & White 1996). The broken and dotted curves are the contributions of the dark matter and the disc components, respectively.
Figure 22. Approximation of M33 rotation curve. The data points plotted with the error bars are quoted from Table 1 of Corbelli et al. (2014). A solid curve is a model rotation curve approximated by a double power-law function as described in §5.3 of the main text.
where VD (R) is the contribution of the stars and the gas disc components. We had a difficulty in obtaining the best-fit model by keeping the surface mass density values of the disc components as those given in Corbelli et al. (2014, Table 1). Thus, we add one free parameter, VR , a renormalization constant of the disc contribution of the rotation curve computed in Table 1. In other words, we assume that VD (R) is given as ( ) VD (R) = VR V (R)/Vstars (2kpc) (43)
Rotation Curve of M33
V(R) (km s−1)
Data: Corbelli et al. (2014) 100
NFW
stars+gas
1
10 R (kpc)
Figure 21. Rotation curve of M33: log-log plot. Same as Fig. 20 but plotetd in a log-log manner.
√ where r ≡ R2 + Z 2 is the distance from the galactic center while ρ0 and r S are the nominal volume density and the scale radius, respectively. Since the distribution is spherically symmetric, the corresponding rotation curve is analytically given (Binney & Tremaine 2008, §2.2.2(g)) as ( [ ) ( )] ln 1 + R/r S 1 VNFW (R) = VN − ( ) , (41) R/r S 1 + R/r S where VN is the corresponding nominal velocity. Thus, the total rotation curve is given as √ [ ] V (R) = VNFW (R) 2 + [VD (R)]2, (42)
where V (R)/Vstars (2kpc) are the values listed in column 2 of Table 1. Then, we optimized three parameters, namely VR , VN , and r S , to obtain a best-fit to the observed values of the rotation curve. By using the multi-dimensional bisection method again, we determined the numerical values of these three parameters in the unit of km s−1 for VR and VN and in the unit of kpc for r S as VR = 73, VN = 5.1 × 103, r S = 10.
(44)
The determined model rotation curve is shown in Figs 20 and 21 together with the observed data as well as the determined components of the disc of the stars and gas and of the spherical dark matter. Obviously, troublesome is a hump centered at R = 8 kpc, which was caused by the rapid drop in the surface mass density profile of the gas component in that region.
5.3 Direct fit The result of the deconvolution method presented in the previous subsection is not satisfactory. Recall that the new method enables us to compute the rotation curve of the axisymmetric disc of an arbitrary profile of its surface mass density. Therefore, changing the viewpoint, let us try to explain the observed rotation curve of M33 directly by a single disc mass component with an unknown profile of the surface mass density. In order to find a trigger of the functional form of the profile, we first approximate the observed rotation curve by a double power-law MNRAS 000, 1–13 (2015)
Gravitational field of infinitely-thin disc
Approximation of M33 Rotation Curve
Rotation Curve of M33 160 Data: Corbelli et al. (2014) 140 V(R) (km s−1)
V(R) (km s−1)
128
64
32
11
Data: Corbelli et al. (2014)
120 100 80
Σ(R)=ΣS(R/RS)−c(1+(R/RS)1/a)((c−b)a) ΣS=1480 MSun pc−2, RS=2 kpc a=0.2, b=0.9, c=0.4
60 40 20 0
1
10
0
5
10
15
20
25
R (kpc)
R (kpc)
Figure 23. Approximation of M33 rotation curve: log-log plot. Same as Fig. 22 but plotted in a log-log manner.
Figure 24. Rotation curve of M33. The data points plotted with the error bars are quoted from Table 1 of Corbelli et al. (2014). A solid curve is the rotation curve of a model disc mass, the surface mass density of which is of the form of a double power-law function as described in §5.3 of the main text.
function as
[ ] (γ−β)α V (R) = VV (R/RV ) −γ 1 + (R/RV ) 1/α .
(45)
By using a multi-dimensional bisection method once again, we experimentally determined 5 model parameters; (i) three power-law indices, α, β, and γ, as α = 0.14, β = −0.1, γ = −0.4,
(46)
Rotation Curve of M33
(ii) the scale radius, RV , as
and (iii) a normalization constant, VV , as VV = 103 kms−1 .
(48)
The approximated rotation curve is depicted together with the observational data in Figs 22 and 23. The goodness of the approximation of V (R) by a double powerlaw function encouraged us to approximate Σ(R) as a similar double power-law function as ) ] (c−b)a ) [ ( ( . (49) Σ(R) = ΣS R/RS −c 1 + R/RS 1/a By using a multi-dimensional bisection method once again, we experimentally determined 5 model parameters; (i) three power-law indices, a, b, and c, as a = 0.2, b = 0.9, c = 0.4,
(51)
and (iii) the nominal value of the surface mass density, ΣS , as ΣS = 1.48 × 103 MSun pc−2 .
(52)
The rotation curve obtained by the new method from the determined surface mass density profile is plotted together with the observational data in Figs 24 and 25. On the other hand, Figs 26 and 27 show that the surface mass density of the determined disc is much larger than those of the stars and gas components. Notice that the comparison of the total mass is meaningless since that of the double power-law disc determined strongly depends on the manner of MNRAS 000, 1–13 (2015)
64
32
Data: Corbelli et al. (2014) 1
10 R (kpc)
Figure 25. Rotation curve of M33: log-log plot. Same as Fig. 24 but plotted as a log-log graph.
(50)
(ii) the scale radius, RS , as RS = 2 kpc,
128
(47)
V(R) (km s−1)
RV = 3.2 kpc,
decrease at the region much far from the location of the stars and gas disc components of the galaxy. At any rate, the comparison of the surface mass density profiles supports the existence of the dark matter itself without doubt. Clearly, the graphs show a possibility to explain the observed rotation curve of M33 by a single disc mass component. If so, the smoothness of the determined surface density profile of the disc mass when compared with those of the stars and the gas components, both of which exhibit different manners of variation, may lead to another question why the behavior of these three disc components differ from each other. Since its discussion is out of the scope of the present article, nevertheless, we shall leave this issue for the future studies.
12
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Surface Mass Density: M33 10000
Data: Corbelli et al. (2014)
Σ (MSun pc−2)
1000
determined
100 stars+gas
stars
10
gas 1 0.1 0
5
10
15
20
R (kpc) Figure 26. Surface mass density profiles of M33. Plotted are the surface mass density profiles of disc components of M33 in a single logarithmic manner. The solid thick curve is that of a single disc mass with a double power-law functional form determined from the fitting to the observed rotation curve. Meanwhile, the data points are quoted from Table 1 of Corbelli et al. (2014): (i) open circles are for the stars disc compopnent, (i) filled squares are for the gas disc component, and (iii) filled circles are for their sum.
disc reveals that the new method is of around 12 digit accuracy if conducted in the double precision environment. Since the computation requires only one-dimensional integration and utilizes a fast procedure to compute the complete elliptic integral of the first kind, its CPU time is sufficiently small, say at most 10 µs to compute a single point of the rotation curve by recent PCs. Using the new method, we presented the rotation curves of various kind of discs: (i) truncated power-law discs, (ii) typical accretion discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we conducted a primitive investigation of the rotation curve of M33 by trying to find a single disc mass component to explain both the sharp increase near the galactic center and the gradual rise when the radius is large. The comparison of the obtained fit to that of the standard deconvolution method assuming the spherical mass distribution of the dark matter indicates a possibility to explain the unexpectedly high and slightly rising rotation curve of M33 by a disc-like distribution of the dark matter. 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/
ACKNOWLEDGEMENTS The author appreciates the referee’s valuable advices to improve the quality of the present article.
Surface Mass Density: M33 determined
Σ (MSun pc−2)
1000 100 stars
10
stars+gas
gas 1 0.1
Data: Corbelli et al. (2014) 1
10 R (kpc)
Figure 27. Surface mass density profiles of M33: log-log plot. Same as Fig. 26 but plotted in a log-log manner.
6 CONCLUSION We developed a new numerical method to compute the gravitational potential and the acceleration vector of an infinitely-thin axisymmetric disc with arbitrary density profile. The method is a combination of (i) the direct numerical integration of the potential by means of the split quadrature using the double exponential rule, and (ii) the numerical differentiation of the numerically-integrated potential by Ridder’s algorithm. The new method is applicable to both the internal and external gravitational fields. The comparison with the exact solutions of the rotation curves of a finite uniform disc and of an infinite exponential
REFERENCES Ashcroft, N.W., Mermin, N.D. 1976, Solid State Physics, Holt, Rinehalt, and Winston, Dumfries Binney, J., Tremaine, S. 1987, Galactic Dynamics, Princeton Univ. Press, Princeton Binney, J., Tremaine, S. 2008, Galactic Dynamics (2nd ed.), Princeton Univ. Press, Princeton Brandt, J.C. 1960, ApJ, 131, 211 Bulirsch, R. 1969, Numer. Math., 13, 305 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 Carlson, B.C. 1979, Numer. Math., 33, 1 Casertano, S. 1983, MNRAS, 203, 735 Corbelli, E., Thilker, D., Zibetti, S., Giovanardi, C., Salucci, P. 2014, A&A, 572, A23 Cuddeford, P. 1993, MNRAS, 262, 1076 Durand, E. 1953, Electrostatique et Magnetostatique, Masson et Cie, Paris Evans, N.W. 1994, MNRAS, 267, 333 Evans, N.W., de Zeeuw, P.T. 1992, MNRAS, 257, 152 Freeman, K.C. 1970, ApJ, 160, 811 Fukushima, T. 2010, CMDA, 108, 339 Fukushima, T. 2013, J. Comp. Appl. Math., 253, 142 Fukushima, T. 2014, Appl. Math. Comp., 238, 485 Fukushima, T. 2015a, J. Comp. Appl. Math., 282, 71 Fukushima, T. 2015b, AJ, revised Gonzalez, G.A., Reina, J.I. 2006, MNRAS, 371, 1873 Green, G. 1835, Trans. Cambr. Phil. Soc., 5, 395 Hayashi, C. 1981, Progr. Theor. Phys. Suppl., 70, 35 Hueso, R., Guillot, T. 2005, A&A, 442, 703 Hunter, C. 1963, MNRAS, 126, 299 Hure, J.-M., Pierens, A. 2005, ApJ, 624, 289 Hure, J.-M., Dieckmann, A. 2012, A&A, 541, A130 MNRAS 000, 1–13 (2015)
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MNRAS 000, 1–13 (2015)
13
Table 1. Normalized circular velocity of M33 disk components. Shown are the normalized value of the circular velocity of the model disk components of M33 plotted in Fig. 19. The normalization constant is chosen such that Vstars = 1 when R = 2 kpc. R (kpc) 0.24 0.28 0.46 0.64 0.73 0.82 1.08 1.22 1.45 1.71 1.87 2.20 2.28 2.69 2.70 3.12 3.18 3.53 3.66 4.15 4.64 5.13 5.62 6.11 6.60 7.09 7.57 8.06 8.55 9.04 9.53 10.02 10.51 10.99 11.48 11.97 12.46 12.95 13.44 13.93 14.41 14.90 15.39 15.88 16.37 16.86 17.35 17.84 18.32 18.81 19.30 19.79 20.28 20.77 21.26 21.74 22.23 22.72
Vstars+gas
Vstars
Vgas
0.48101 0.51092 0.62063 0.70451 0.73995 0.77197 0.84913 0.88308 0.92980 0.97160 0.99256 1.02644 1.03302 1.05867 1.05915 1.07389 1.07526 1.08024 1.08095 1.07944 1.07322 1.06455 1.05560 1.04882 1.04681 1.04865 1.04533 1.03053 1.00746 0.98049 0.95186 0.92174 0.88951 0.86605 0.84563 0.82750 0.81104 0.79590 0.78185 0.76871 0.75660 0.74491 0.73382 0.72327 0.71319 0.70355 0.69430 0.68541 0.67702 0.66876 0.66078 0.65306 0.64559 0.63836 0.63133 0.62466 0.61803 0.61159
0.48023 0.51004 0.61930 0.70270 0.73788 0.76962 0.84592 0.87937 0.92517 0.96580 0.98596 1.01800 1.02410 1.04696 1.04736 1.05868 1.05951 1.06104 1.06034 1.05274 1.03900 1.02091 0.99985 0.97683 0.95263 0.92783 0.90338 0.87853 0.85399 0.82982 0.80590 0.78151 0.75600 0.73692 0.72013 0.70516 0.69155 0.67903 0.66740 0.65653 0.64650 0.63682 0.62763 0.61888 0.61052 0.60250 0.59480 0.58739 0.58038 0.57347 0.56678 0.56030 0.55402 0.54792 0.54199 0.53634 0.53072 0.52525
0.02742 0.02998 0.04064 0.05056 0.05538 0.06014 0.07367 0.08089 0.09269 0.10603 0.11426 0.13131 0.13548 0.15702 0.15755 0.18010 0.18337 0.20274 0.21008 0.23859 0.26885 0.30168 0.33853 0.38188 0.43395 0.48866 0.52596 0.53868 0.53448 0.52226 0.50651 0.48872 0.46871 0.45496 0.44329 0.43301 0.42373 0.41519 0.40727 0.39987 0.39304 0.38646 0.38023 0.37431 0.36867 0.36329 0.35814 0.35322 0.34859 0.34405 0.33969 0.33549 0.33145 0.32754 0.32378 0.32021 0.31669 0.31329
