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3DRAFT VERSIONJULY 13, 2018Preprint typeset using LATEX style emulateapj v. 5/2/11

AN EMPIRICAL FORMULA FOR THE DISTRIBUTION FUNCTION OF A THINEXPONENTIAL DISC

SANJIB SHARMA & JOSSBLAND -HAWTHORNSydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia

Draft version July 13, 2018

ABSTRACTAn empirical formula for a Shu distribution function that reproduces a thin disc with exponential surface

density to good accuracy is presented. The formula has two free parameters that specify the functional formof the velocity dispersion. Conventionally, this requiresthe use of an iterative algorithm to produce the correctsolution, which is computationally taxing for applications like Markov Chain Monte Carlo (MCMC) modelfitting. The formula has been shown to work for flat, rising andfalling rotation curves. Application of thismethodology to one of the Dehnen distribution functions is also shown. Finally, an extension of this formulato reproduce velocity dispersion profiles that are an exponential function of radius is also presented. Ourempirical formula should greatly aid the efficient comparison of disc models with large stellar surveys or N-body simulations.Subject headings:galaxies: kinematics and dynamics — galaxy: disk — galaxy: structure — methods: analyt-

ical — methods: numerical

1. INTRODUCTION

With the advent of large photometric, spectroscopic andproper motion surveys of stars, it has now become possibleto study in detail the properties of the Milky Way disc. Sub-stantial improvements in the richness and quality offered bythese new surveys demands increasing sophistication in themethods used to analyze them (McMillan & Binney 2012;Binney & McMillan 2011). Simultaneously, the vast increasein thesizeof the new data sets requires that these same meth-ods have greatly enhanced computational efficiency comparedto existing algorithms. There is a pressing need for wholesaleimprovements to theoretical modelling in order to cope withthe data deluge. In the context of modelling the kinematicsof the disc, for simplicity, 3D Gaussian functions have beenused. This form is barely adequate for radial and vertical mo-tions and entirely inappropriate for azimuthal motions. Theazimuthal distribution of stellar motions, the subject of thispaper, is very skewed because the surface density and radialvelocity dispersions are declining functions of radiusR. Atany given radius, we observe more stars with azimuthal ve-locity vφ less than the local circular velocityvc than stars withvφ > vc, a phenomenon known as ‘asymmetric drift’ (e.g.,Sandage 1969). The inadequacy of the Gaussian functionsin modelling disc kinematics has been discussed by Binney(2010) and Schonrich & Binney (2012).

A better way to model disc kinematics is to use a properdistribution functionf . For axisymmetric systems with poten-tial Φ = Φ(R), the distribution function from Jeans’ theoremis a function of energyE and angular momentumL alone,f = f(E,L). However, there is no unique solution for agivenΣ(R) (Kalnajs 1976). SpecifyingσR(R) can restrictthe solution space but does not remove the general degener-acy, the reason being that a finite set of functions of one vari-able cannot determine a function of two variables (Dehnen1999). One approach to arrive at a suitablef(E,L) as givenby Shu (1969) is to consider moderately heated (‘warmed up’)versions of cold discs, i.e.

f(E,L) =F (L)

σ2R(L)

exp

[

−E − Ec(L)

σ2R(L)

]

(1)

whereEc(L) is the energy of a circular orbit with angularmomentumL. HereF (L) is chosen such that in the limitσ(L) → 0, the function gives a surface density ofΣ(R). Re-cently Schonrich & Binney (2012) used this to derive impor-tant formulas for disc kinematics.

Forms of warm discs other than the Shu family can be foundin Dehnen (1999). Further examples include the distribu-tion functions for a quasi-isothermal disc described in Binney(2010, 2012). One problem with these warmed up functionsis that they reproduce the target density only very approxi-mately− the larger the value ofσR, the larger the discrep-ancy. One way to improve this is to iteratively solve forF (L)from the integral equation connectingΣ(R) andf(E,L) assuggested by Dehnen (1999). For applications like MCMCmodel fitting, where in each iteration the model parametersare changed and one has to recompute the solution (Sharmaet al 2013, in preparation), existing algorithms are simplytooinefficient. Hence, in this paper we attempt to find an empiri-cal formula forF (L) that reproduces a disc with exponentialsurface density. The form ofF (L) in general depends on thechoice ofσR(L), so to proceed we assumeσR(L) to be anexponential function with two free parametersa0 andq andthen find an empirical formula forF (L|a0, q). The formula isshown to work for rotation curves whose dependence with ra-dius is flat, rising and falling. Additionally we also show thata similar formula can also be derived for one of the distribu-tion function proposed by Dehnen (1999). Finally, we presentan extension of the formula for the case of velocity dispersionprofiles that are exponential in radius.

The paper is organized as follows. In Section 2, we reviewthe basic theory and the method to numerically compute thedistribution function. Using this in Section 3, we determinean empirical formula for the distribution function. In Section4, we analyze the accuracy of our results and compare it withalternate solutions. Finally, in Section 5, we conclude andsummarize our findings.

2. THEORY

2

In this paper, we consider two types of distribution func-tions for warm discs such that

fShu=F (L)

σ2R(L)

exp

(

−E − Ec(L)

σ2R(L)

)

and (2)

fDehnen,a=F (E)

σ2R(E)

exp

(

−E − Ec(L)

σ2R(E)

)

. (3)

HereEc(L) is the energy of a circular orbit with angular mo-mentumL andE − Ec(L) is the energy in excess of thatrequired for a circular orbit. LetRg andRE be the radiusof a circular orbit with angular momentumL and energyErespectively, which in general we denote byRc. Our aimis to computeF (Rc) that produces a disc with an exponen-tial surface density profile,Σ(R). For this, eitherσR(Rc)has to be specified orσR(R). If σR(R) is specified then wealso need to solve forσR(Rc) in addition toF (Rc). We re-duce the velocity dispersion to dimensionless form by defin-ing a = σR/vcirc, wherevcirc is the circular velocity. For theremainder of the paper, we assumea to be an exponentiallydecreasing function ofRc(orR) and specify it as follows

a=a0e−qRc/Rd (4)

The choice of the functional form is motivated by the desireto produce discs in which the scale height is independent ofradius. For example, under the epicyclic approximation, ifσz/σR is assumed to be constant, then the scale height is inde-pendent of radius forq = 0.5 (van der Kruit & Searle 1982).

Instead of solving forF (Rc) directly, we proceed bysolving for Σ(Rc), which is the surface density inRc

space and satisfies∫

Σ(Rc) 2πRc dRc = 1. The dis-tribution function is assumed to be normalized such that∫ ∫ ∫

f(E,L) 2πR dR dvφ dvR = 1. For a given dis-tribution function letP (R,Rc), be the probability distribu-tion in (R,Rc) space. Using the relation2πRcΣ(Rc) =∫

P (R,Rc)dR one can then writeF (Rc) in terms ofΣ(Rc).IntegratingP (R,Rc) over Rc one gets the integral equa-tion connectingΣ(R) to Σ(Rc) and a(Rc). This is themain equation that needs to be solved to determineΣ(Rc).The integral equation forfShu has already been discussed inSchonrich & Binney (2009, 2012); below we provide an al-ternate derivation, both for the sake of completeness and theneed to generalize it for arbitrary rotation curves. A detailedderivation forfDehnen,a is given in the appendix.

2.1. The Shu distribution function

The energyE of an orbit in potentialΦ(R) with angularmomentumL is given by

E =1

2v2R +

L2

2R2+ Φ(R) (5)

If Rg(L) is the guiding radius which is defined as the radiusof a circular orbit with angular momentumL, then the energyof a circular orbit can be expressed as

Ec(L) =L2

2R2g

+Φ(Rg) (6)

Defining

Φeff(R,L)=L2

2R2+Φ(R) (7)

we can write

E − Ec(L)=1

2v2R +Φeff(R,L)− Φeff(Rg, L) (8)

=1

2v2R +∆Φeff(R,L) (9)

We now show how to computeΣ(R) for a givenΣ(Rg). ForfShu, one can write the probability distribution in(R,Rg, vR)space as

P (R,Rg, vR)dR dvR dRg =2πf(E,L)dL dR dvR (10)

P (R,Rg, vR)=2πf(E,L)dL

dRgdvR (11)

Using E − Ec(L) = 12v

2R + ∆Φeff(R,Rg), dL/dRg =

2vcirc(Rg)/γ2(Rg), γ2(R) = 2/(1 + d ln vcirc(R)/d ln dR)

and integrating overvR one gets

P (R,Rg)= (2π)3/2F (Rg)

σR(Rg)

2vcirc(Rg)

γ2(Rg)×

exp

(

−∆Φeff(R,Rg)

σ2R(Rg)

)

(12)

If limr→∞ φ(R) = 0 then there is also an additional factor of

erf(√

−φeff (R,Rg)

σ2

R(Rg)

)

. For the cases considered in this paper

limr→∞ φ(R) = ∞ and this factor is unity. We now definea = σR(Rg)/vcirc(Rg) and

K(R,Rg) = exp

(

−∆Φeff(R,Rg)

σ2R(Rg)

)

(13)

then we get

P (R,Rg)= (2π)3/22F (Rg)

γ2(Rg)a(Rg)K(R,Rg) (14)

Using∫

P (R,Rg)dR = 2πRgΣ(Rg) one can write

F (Rg) =γ2(Rg)a(Rg)Σ(Rg)√

2πgK(a,Rg)(15)

and

P (R,Rg)=2πΣ(Rg)

gK(a,Rg)K(R,Rg) (16)

where

gK(a,Rg) =1

Rg

∫

K(R,Rg)dR (17)

Finally, the integral equation connectingΣ(R) andΣ(Rg) isgiven by

Σ(R) =1

R

∫

Σ(Rg)

gK(a,Rg)K(R,Rg)dRg (18)

For vcirc(R) = vc(R/R0)β , gK(a,Rg) is independent of

Rg. For flat rotation curve, the integral has an analytic solu-tion and has been given by Schonrich & Binney (2012) as

gK(a)=ecΓ(c− 1/2)

2cc−1/2with c =

1

2a2

≈√

π

2(c− 0.913)for large c.

3

For non-flat rotation curves, this has to be evaluated numeri-cally.

2.2. The Dehnen distribution function

Three variants of the Shu distribution function were pro-posed by Dehnen (1999) out of which we consider the fol-lowing case

fDehnen,a =F (RE)

σ2R(RE)

exp

(

E − Ec(L)

σ2R(RE)

)

. (19)

The integral equation connectingΣ(R) andΣ(RE) for thecase of flat rotation curve is given by (see Appendix forderivation)

Σ(R) =1

R

∫

Σ(RE)

gK(a,RE)K(R,RE)dRE (20)

with a = σR(RE)/vcirc(RE) and

K(R,RE) = (R/RE)(1+ 1

a2)(1− 2 ln(R/RE))

1

2a2 (21)

and

gK(a)=1

RE

∫

K(R,RE)dR

=

(

e

1 + c

)1+cΓ(1 + c)

2with c =

1

2a2

≈√

π

2(c+ 1)for large c.

2.3. The iterative method for solving the integral equation ofsurface density

In general if we have

P (R,Rc)=2πΣ(Rc)

gK(a,Rc)K(R,Rc). (22)

Then, as shown in previous section, the integral equation thatconnectsΣ(R) toΣ(Rc) is

Σ(R)=

∫

P (R,Rc)dRc

2πR

=1

R

∫

Σ(Rc)

gK(a,Rc)K(R,Rc)dRc. (23)

Also, the equation connectingσR(R) to σR(Rc) is given by

σR(R)=

∫

P (R,Rc)σ2R(Rc)dRc

∫

P (R,Rc)dRc

=1

RΣ(R)

∫

Σ(Rc)σ2R(Rc)

gK(a,Rc)K(R,Rc)dRc. (24)

In this paper, we are concerned with the following two cases.

• For a givenΣ(R) and σR(Rc) solve forΣ(Rc) andσR(R).

• For a givenΣ(R) and σR(R) solve for Σ(Rc) andσR(Rc).

An iterative Richardson-Lucy type algorithm to solve for theabove mentioned cases was given by Dehnen (1999) and is asfollows.

1. Start withΣ(Rc)=Σ(R). If the velocity dispersion alsoneeds to be constrained, setσR(Rc)=σR(R).

2. Solve for the current surface densityΣ′(R) and velocitydispersionσ′

R(R) profiles.

3. SetΣ(Rc) = Σ(Rc)Σ(R)Σ′(R) . If the velocity disper-

sion also needs to be constrained, setσR(Rc) =

σR(Rc)σR(R)σ′

R(R) .

4. If the answer is within some predefined tolerance thenexit else go to step 2.

The above algorithm is easy to setup but non trivial to tunefor efficiency and accuracy. We now discuss the technicaldetails of our implementation. We represent the profiles bylinearly spaced arrays with spacing∆R and rangeRmin =∆R to Rmax. We now determine the optimum step size tocompute the integrals. At a givenR, the typical width of thekernelK(R,Rc) is given byRa0 exp(−qR/Rd). Assumingat leastN points within this width, the criteria for step sizebecomes

∆Rint<Rmina0 exp(−qRmin/Rd)/N (25)∆Rint<Rmaxa0 exp(−qRmax/Rd)/N, (26)

The choice ofN in general should be greater than 1. Fromthe above equation, it can be seen that smalla0, large q,largeRmax and smallRmin, make the step size small andthe task of obtaining the solution computationally expen-sive. At a givenR, the contribution to the integral fromRc > R(1+5a0 exp(−qRmin/Rd)) is very small. So the up-per range of the integral is comfortably known. But the lowerrange is difficult to determine asΣ(Rc) increases with de-crease inRc. At largeR, we find that there can be a non negli-gible contribution fromRc < R(1− 5a0 exp(−qRmin/Rd)).To overcome this, in general we integrate on a linearly spacedarray with spacing∆Rint and range∆Rint to 2Rmax. Achoice ofN = 8 was found to be satisfactory. Additionally,we find that to avoid numerical ringing artifacts, the integralin step 2 should be done with step size∆Rint = ∆R/2α ,whereα ≥ 0.

3. DETERMINING THE EMPIRICAL FORMULA

In the previous section, we have shown how to computeΣ(Rc) given aΣ(R) andσR(Rc). Let us assume that thetarget surface density is given by

Σ(R) =1

2πR2d

e−R/Rd (27)

and the velocity dispersion profile satisfiesσR(Rc) =vcirc(Rc)a0 exp(−qRc/Rd), wherea0 andq are two free pa-rameters. Now starting with a density

Σ(Rc) =1

2πR2d

e−Rc/Rd (28)

and a givena0 andq, we can solve the integral equation itera-tively using the formalism of Dehnen (1999), and compute therequiredΣ(Rc). Let us denote the difference of the obtainedsolution from the initial guess by

∆Σ(Rc, Rd, a0, q)=e−Rc/Rd

2πR2d

− Σ(Rc, Rd, a0, q). (29)

4

TABLE 1EMPIRICALLY DERIVED PARAMETERS FOR

Rmax(q) = c1Rd/(1 + q/c2) AND ∆Σmax(a0) = c3ac40 /R2

d

f(E,L) PotentialΦ(R) β c1 c2 c3 c4fShu v2c ln(R/Rd) 3.740 0.523 0.00976 2.29

fShu v2c(R/Rd)

2β

2β0.2 3.822 0.524 0.00567 2.13

fShu v2c ln(R/Rd)+ -0.5 3.498 0.454 0.01270 2.12

v2c(R/Rd)

2β

2β

fDehnen,a v2c ln(R/Rd) 4.876 0.661 0.00062 1.62

0.1<q<0.7, 0.0<a0<1.0, a0<(0.5+q)

0 1 2 3 4Rg/Rg

max

-1.0

-0.5

0.0

0.5

1.0

∆ Σ(

Rg)

/∆ Σ

(Rgm

ax)

0.1<q<0.7, 0.0<a0<1.0, a0<(0.5+q)

0.0 0.2 0.4 0.6 0.8 1.0Rg/Rg

max

-30

-25

-20

-15

-10

-5

0

∆ Σ(

Rg)

/∆ Σ

(Rgm

ax)

FIG. 1.— The correction factor∆Σ that is required to reproduce an ex-ponential disc as a function ofRg . Rmax

g is the radius at which∆Σ ismaximum. Different curves in the figure correspond to different values ofa0andq. The magenta dashed lines correspond to an analytic fit. The bottompanel shows the same set of curves as the top panel but shows the region0 < Rg/Rmax

g < 1 in greater detail.

So, the iterative solution gives us the correction factor∆Σ butit will be different for different values ofa0 andq. But sincewe cannot write the correction factor as an analytic functionof Rc, Rd, a0 andq, we must try to derive an empirical func-tion. To this end, we first compute∆Σ for a range of val-ues ofa0 andq and then analyze them. We first study thedistribution functionfShu(E,L) for the case of the flat rota-tion curve and then generalize our results for non-flat rota-tion curves. Next, we apply our methodology to a distributionfunction other thanfShu(E,L), namelyfDehnen,a(E,L), butstudy only the case for flat rotation. Finally, we show how theformula can be extended for the case of velocity dispersionprofiles that are exponential in radius.

3.1. Case offShu with a flat rotation curve

0.0 0.2 0.4 0.6 0.8 1.0q

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Rgm

ax/R

d

0.0 0.2 0.4 0.6 0.8 1.0a0

0.000

0.002

0.004

0.006

0.008

0.010

∆ Σm

ax R

d2

0.0 0.2 0.4 0.6 0.8 1.0a0

0.6

0.8

1.0

1.2

1.4

Rgm

ax/R

gmax

(q,R

d)

0.0 0.2 0.4 0.6 0.8 1.0q

0.6

0.8

1.0

1.2

1.4

∆ Σm

ax/∆

Σm

ax(a

0,R

d)

FIG. 2.— The dependence ofRmaxg and∆Σmax on q anda0. The ma-

genta lines are the fitted functional forms forRmaxg and∆Σmax. The bottom

panels demonstrate thatRmaxg and∆Σmax are nearly independent ofa0 and

q respectively.

The correction factor∆Σ as a function ofRg for variousdifferent values ofa0 and q is shown in Figure 1. In gen-eral, we find that∆Σ is negative for smallRg; it then risesto a maximum and then for largeRg it asymptotically ap-proaches zero (black lines in Figure 1). If there exists aunique functional form, then we must first reduce∆Σ to ascale invariant form. To this end, we computes(Rg/R

maxg ) =

∆Σ(Rg)/∆Σ(Rmaxg ),Rmax

g being the value ofRg where∆Σis maximum. In Figure 1, we plot∆Σ(Rg)/∆Σ(Rmax

g ) forvarious different values ofa0 andq as derived by the itera-tive solution. It can be seen that they almost follow a uniquefunctional form, except forRg/R

maxg < 0.2, where slight dif-

ferences can be seen (see lower panel Figure 1). Next we fitthis by a function of the following form

s(x)=ke−x/b((x/a)2 − 1). (30)

Imposing the condition thatxmax = 1,∫

∞

−∞xf(x) = 0 and

s(xmax) = 1, one can solve fora, b andk. The final functionis then given by

s(x)=31.53e−x/0.2743(x/0.6719)2 − 1) (31)

and this is plotted as magenta dashed lines in Figure 1. It canbe seen that the proposed functional form provides a good fit.For greater accuracy one should replace this with a numericalfunction.

We now empirically determine the dependence ofRmaxg and

∆Σ(Rmaxg ) onRd, q anda0 and this is given below.

Rmaxg =

c11 + q/c2

Rd (32)

∆Σ(Rmaxg )=

1

R2d

c3ac40 . (33)

with c1 = 3.74, c2 = 0.523, c3 = 0.00976, c4 = 2.29,It was found that the position of maximumRmax

g is mainlydetermined byq, while the amplitude∆Σ(Rmax

g ) is mainlygoverned bya0. In Figure 2, we plotRmax

g and∆Σ(Rmaxg )

as functions ofq anda0 respectively. The fitted functionalforms Equation (32) and Equation (33) are shown as magentalines. The bottom panel shows the dependence ofRmax

g and

5

0.0<q<0.7, 0.1<a0<1.0, a0<(0.5+q)

0 1 2 3 4Rg/Rg

max

-1.0

-0.5

0.0

0.5

1.0∆

Σ(R

g)/∆

Σ(R

gmax

)

0.0<q<0.7, 0.1<a0<1.0, a0<(0.5+q)

0.0 0.2 0.4 0.6 0.8 1.0Rg/Rg

max

-30

-25

-20

-15

-10

-5

0

∆ Σ(

Rg)

/∆ Σ

(Rgm

ax)

FIG. 3.— The correction factor∆Σ for distribution functionfShu(E,L)with vcirc(R) = vc(R/Rd)

0.2 . For description see Figure1.

∆Σ(Rmaxg ) as a function ofa0 andq respectively after divid-

ing by the derived functional forms. It is clear that there isvery little dependence ofRmax

g ona0 or of∆Σ(Rmaxg ) on q.

The final functional form forΣ(Rg) can now be written as

Σcorr(Rg)=e−Rg/Rd

2πR2d

−0.00976a2.290

R2d

s

(

Rg

3.74Rd(1 + q/0.523)

)

(34)

3.2. Case offShu with a non-flat rotation curve

We now repeat the same exercise for cases where the rota-tion curve is not flat. Specifically, we investigate the followingtwo cases.

• Rising rotation curve: The circular velocity is assumedto be a power law with positive slope.

vcirc(R)= vc(R/Rd)β with β = 0.2 (35)

φ(R)= v2c (R/Rd)2β/(2β) with β = 0.2 (36)

• Falling rotation curve: In this case we assume the po-tential to be a superposition of a point mass and a flatrotation curve. The circular velocity decreases mono-tonically with radius but at largeR asymptotes to a con-stant value.

vcirc(R)= v2c (1 +√

Rd/R) (37)

φ(R)= v2c (log(R/Rd)−Rd/R) (38)

0.0 0.2 0.4 0.6 0.8 1.0q

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Rgm

ax/R

d

0.0 0.2 0.4 0.6 0.8 1.0a0

0.000

0.002

0.004

0.006

0.008

0.010

∆ Σm

ax R

d2

0.0 0.2 0.4 0.6 0.8 1.0a0

0.6

0.8

1.0

1.2

1.4

Rgm

ax/R

gmax

(q,R

d)

0.0 0.2 0.4 0.6 0.8 1.0q

0.6

0.8

1.0

1.2

1.4

∆ Σm

ax/∆

Σm

ax(a

0,R

d)

FIG. 4.— The dependence ofRmaxg and∆Σmax onq anda0 for distribu-

tion functionfShu(E,L) with vcirc(R) = vc(R/Rd)0.2. For description

see Figure2.

0.0<q<0.7, 0.0<a0<0.3

0 1 2 3 4Rg/Rg

max

-1.0

-0.5

0.0

0.5

1.0

∆ Σ(

Rg)

/∆ Σ

(Rgm

ax)

0.0<q<0.7, 0.0<a0<0.3

0.0 0.2 0.4 0.6 0.8 1.0Rg/Rg

max

-30

-25

-20

-15

-10

-5

0

∆ Σ(

Rg)

/∆ Σ

(Rgm

ax)

FIG. 5.— The correction factor∆Σ for distribution functionfShu(E,L)

with vcirc(R) = vc√

1 + Rd/R. For description see Figure1.

It should be noted that we do not study the case of a ro-tation curve falling as a power law, as the integral of theShu distribution function over all space or more specif-ically the factorgK(a,Rg) becomes infinite.

The results for the above two rotation curves are shown inFigure 3, Figure 4, Figure 5 and Figure 6 and the fit param-etersc1, c2, c3 andc4 are summarized in Table 1. It can beseen from Figure 3 and Figure 5 that the∆Σ curves followan almost universal form. The functional form differs only

6

0.0 0.2 0.4 0.6 0.8 1.0q

1.0

1.5

2.0

2.5

3.0

3.5

4.0R

gmax

/Rd

0.0 0.2 0.4 0.6 0.8 1.0a0

0.000

0.002

0.004

0.006

0.008

0.010

∆ Σm

ax R

d2

0.0 0.2 0.4 0.6 0.8 1.0a0

0.6

0.8

1.0

1.2

1.4

Rgm

ax/R

gmax

(q,R

d)

0.0 0.2 0.4 0.6 0.8 1.0q

0.6

0.8

1.0

1.2

1.4

∆ Σm

ax/∆

Σm

ax(a

0,R

d)

FIG. 6.— The dependence ofRmaxg and∆Σmax onq anda0 for distribu-

tion functionfShu(E,L) with vcirc(R) = vc√

1 + Rd/R. For descriptionsee Figure2.

slightly from the case for a flat rotation curve. From Figure 4and Figure 6, it can be seen that the functional form given byEquation (32) and Equation (33) provides a good fit for thedependence ofRmax

g and∆Σmax on q anda0 respectively. Aslight residual dependence onq of∆Σmax is visible for fallingrotation curve (lower right panel of Figure 6). If we ignore thedependence of parametersc1, c2, c3, c4 on the rotation curve,i.e.,β, the resulting surface density profiles will be slightly in-accurate. For(a0, q) = (0.5, 0.33), comparingΣcorr(Rg) ofβ = 0 with that ofβ = 0.2, we find that maximum deviationoccurs at aroundR ∼ Rmax

g and is about 10%.

3.3. Case offDehnen,a with a flat rotation curve

Finally, we study the case of a distribution function differ-ent from Shu. We use the functionfDehnen,a from Dehnen(1999). For this, we only study the case of a flat rotationcurve. It can be seen from Figure 7 that the∆Σ curves againfollow an almost universal form. However, the form is differ-ent from that offShu. The main difference being that asRE

approaches zero the curve moves upwards to large positivevalues. The functional form ofRmax

E (q) and∆Σmax(a0)) isthe same as infShu but with different values for constants,and the relationships are slightly less accurate than forfShu(see Figure 8). Interestingly,∆Σmax is much smaller than thecase for Shu distribution function.

To summarize, we find that for a given distribution func-tion and rotation curve, the correction factorΣ(Rc) requiredto reproduce exponential discs for different values ofa0 andq follows a universal functional form to high accuracy. Thescale lengthRmax

c and the amplitude of this function∆Σmax

has a simple dependence ona0 andq which can be parame-terized in terms of four constantc1, c2, c3 andc4. The factthat an empirical formula based on the above methodologycan be determined for different rotation curves and even foradistribution function other thanfShu is very promising. Thereason that the methodology works for different cases is re-lated somewhat to the following three facts

• The functional form of the required target density issame for all cases.

• We have parameterized the velocity dispersion in a waysuch that its dependence onRc is same for all cases.

0.0<q<0.7, 0.1<a0<1.0

0 1 2 3 4RE/RE

max

-1.0

-0.5

0.0

0.5

1.0

∆ Σ(

RE)/

∆ Σ(

REm

ax)

0.0<q<0.7, 0.1<a0<1.0

0.0 0.2 0.4 0.6 0.8 1.0RE/RE

max

0

20

40

60

80

100

∆ Σ(

RE)/

∆ Σ(

REm

ax)

FIG. 7.— The correction factor∆Σ for distribution functionfDehnen,a(E,L) with vcirc(R) = vc. For description see Figure1.

0.0 0.2 0.4 0.6 0.8 1.0q

1.0

1.5

2.0

2.5

3.0

3.5

4.0

REm

ax/R

d

0.0 0.2 0.4 0.6 0.8 1.0a0

0.000

0.002

0.004

0.006

0.008

0.010

∆ Σm

ax R

d2

0.0 0.2 0.4 0.6 0.8 1.0a0

0.6

0.8

1.0

1.2

1.4

REm

ax/R

Emax

(q,R

d)

0.0 0.2 0.4 0.6 0.8 1.0q

0.6

0.8

1.0

1.2

1.4

∆ Σm

ax/∆

Σm

ax(a

0,R

d)

FIG. 8.— The dependence ofRmaxg and∆Σmax onq anda0 for distribu-

tion functionfShu(E,L) with vcirc(R) = vc. For description see Figure2.

• The integral equation governing the computation of∆Σis the same for all cases except for the Kernel functionand its normalization functiongK .

Given an integral equation of the type in Equation (23), thedetermination ofΣ(R) from Σ(Rc) can be thought of as aconvolution operation with a Kernel functionK(R,Rc). Tofirst order, if the kernels are similar and of the same width, theeffect of convolution should be insensitive to the shape of thekernel. For the cases studied here the kernel in general peaks

7

a0=0.5, q=1/3

-0.2

-0.1

0.0

0.1

0.2

ln σ

R-ln

σR

,exp

fShu-Dehnen

fShu-Corr

fShu

-0.10.0

0.1

0.2

0.3

0.4

0.5

ln σ

R-ln

σR

,exp

fShu-Corr, 0.1<a0<0.8, q=1/3

0 2 4 6 8 10R/Rd

-0.020.00

0.02

0.04

0.06

0.08

0.10

ln σ

R-ln

σR

,exp

fShu-Corr, a0=1/3, 0.1<q<0.8

FIG. 9.— The logarithmic difference of the radial velocity disper-sion profile σR(R) with respect to the target velocity dispersion profileσR,exp(R) = vca0 exp(−qR/Rd). In the top panel the three cases shownare for the Shu distribution function with flat rotation curve, a) without theempirical formula (fShu) b) with the empirical formula (fShu−Corr) c) withDehnen’s ansatz (fShu−Dehnen). The bottom two panels are forfShu−Corronly. Here the different lines correspond to different values ofa0 andq andthe thickness of the line is proportional to the value ofa0 andq. The valuesof a0 andq increase in steps of 0.1.

at R = Rc and then falls of sharply for bothR < Rc andR > Rc. The exact shape is governed primarily by the formof the distribution function. For a constant rotation curve, thewidth of the kernel is an exponentially decreasing functionof Rc. The effect of a non-flat rotation curve is to simplymodulate this functional dependence of width as a function ofRc.

3.4. The case of velocity dispersion profiles that areexponential in radius

Till now, we had assumed theσR(Rc) profile to be fixedand exponential inRc. However, this does not result inσR(R)profiles that are strictly exponential in radiusR. ForfShu withflat rotation curve, the deviation ofσR(R) from that of an ex-ponential formσR(R) = vca0 exp(−qR/Rd) is shown in toppanel of Figure 9. The deviation is very similar to the devia-tion noticed for surface densities, so the effect can be thoughtof as an increase in scale length of the exponential functiongoverning the radial dependence of the dispersion profile. Thecase offShu both with and without the use of empirical for-mula result in very similarσR(R) profiles. Additionally, aform of fShu with Dehnen’s ansatz (see Equation (40)) alsogives a similarσR(R) profiles. This suggests that modifyingtheF (Rc) part of the distribution function only has a minoreffect on the dispersion profiles. The primary quantity thatdetermines the dispersion profile is the functionσR(Rc). It

Σ(Rg)=Σcorr(Rg)

0 1 2 3 4 5 6R/Rd

-1.0

-0.5

0.0

0.5

1.0

Σ(R

)/Σ e

xp(R

)-1

0<a0<0.65, 0.1<q<0.65

Σ(Rg)=Exp(-Rg / Rd)/(2 π Rd2 )

0 1 2 3 4 5 6R/Rd

-1.0

-0.5

0.0

0.5

1.0

Σ(R

)/Σ e

xp(R

)-1

0<a0<0.65, q=0.25

FIG. 10.— The fractional difference of the surface densityΣ(R) fromthe target surface density for different choices ofΣ(Rg). The target sur-face densityΣexp = exp(−R/Rd)/(2πR

2d) is that of an exponential disc.

Σcorr(Rg) is the formula presented in this paper which approximately re-produces an exponential disc. Different lines correspond to different choicesof a0 andq. In the bottom panelq is fixed and the thickness of the lines isproportional to the value ofa0.

can be seen from Figure 9 that the difference peaks at aroundR/Rd = 4 then decreases and finally for largeR it starts torise again. The bottom panel of Figure 9 shows that the lo-cation of the peak primarily depends onq (for small q, thepeak is at largerR) while the height depends on botha0 andq. The larger thea0 and smaller theq, the higher the peak.The rise at largeR is mainly dependent ona0 and is strongerfor largea0. This is because the distribution of guiding cen-ter Rg in a given annulus at largeR is bimodal. The mainpeak being from stars withRg ∼ R but there is also a sec-ondary peak from stars with very smallRg. The secondarypeak has a non-negligible contribution to the velocity disper-sion as the correspondingσR(Rg) is very large due to the ex-ponential functional dependence. Unlike us, this rise at largeR is not visible in Figure-1a of Dehnen (1999) and we believethat this is because in Dehnen (1999) the convolution integralis not done over the full domain ofRg but instead only aroundRg ∼ R.

If one desiresσR(R) = vcirc(R)a0 exp(−qR/Rd) thenone has to use the Dehnen (1999) iterative solution to solvefor bothσR(Rc) andΣ(Rc) simultaneously. We do this forthe case offShu with flat rotation curve. Doing so, we findthat theΣR(Rg) profile is almost unchanged from the casewhereσR(R) was not constrained. ForσR(Rg) we find thatan empirical formula as given below can reproduce to a good

8

0.0

0.2

0.4

0.6

0.8

1.0q

0.0

0.2

0.4

0.6

0.8

1.0q

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Empiricallog (|∆Σ|/Σ)Fixed Σ(R), σR(Rc)

(a)

Empiricallog (|∆Σ|/Σ)Fixed Σ(R), σR(R)

(b)

Empiricallog (|∆σR|/σR)Fixed Σ(R), σR(R)

(c)

0.0 0.2 0.4 0.6 0.8 1.0a0

0.0

0.2

0.4

0.6

0.8

1.0

q

0.0 0.2 0.4 0.6 0.8 1.0a0

0.0

0.2

0.4

0.6

0.8

1.0

q

Iterativelog (|∆Σ|/Σ)Fixed Σ(R), σR(Rc)

(d)

0.0 0.2 0.4 0.6 0.8 1.0a0

0.0 0.2 0.4 0.6 0.8 1.0a0

Iterativelog (|∆Σ|/Σ)Fixed Σ(R), σR(R)

(e)

0.0 0.2 0.4 0.6 0.8 1.0a0

0.0 0.2 0.4 0.6 0.8 1.0a0

Iterativelog (|∆σR|/σR)Fixed Σ(R), σR(R)

(f)

FIG. 11.— The mean absolute fractional difference of the final surface densityΣ(R) and velocity dispersionσR(R) from the target surface densityexp(−R/Rd)/(2πR

2d) and target velocity dispersionvca0 exp(−qR/Rd). Shown is the case offShu with flat rotation curve. The top panels are results

from using the empirical formula proposed in this paper. Thebottom panels are results from using the iterative algorithm. The panels a and d correspond to thecase whereΣ(R) andσR(Rc) are specified to follow exponential functional forms. The other panels are for the case whereΣ(R) andσR(R) are specified tofollow exponential functional forms. The mean difference was computed over equi-spaced bins in0 < R/Rd < 5, and was not weighted for mass variation elsethe contribution to the mean from large values ofR would be very small.

accuracy the desired velocity dispersion profiles.

σcorrR (Rg)= vca0 exp(−qRg/Rd)×

(

1− 0.25a2.040

q0.49fpoly(Rgq/Rd)

)

, (39)

Here fpoly is an 11 degree polynomial having coefficients-0.028476, −1.4518, 12.492, −21.842, 19.130, −10.175, 3.5214, −0.81052, 0.12311, −0.011851, 0.00065476,−1.5809× 10−5.

4. ACCURACY IN REPRODUCING THE TARGET DENSITY

In the top panel of Figure 10, we plot the fractional differ-ence of the final surface densityΣ(R), as compared to that ofan exponential disc, using our empirical formula as given byEquation (34) for the case offShu with flat rotation curve.Results for a range of values ofa0 and q are shown. ForR/Rd < 5, the difference is less than10%. The fit starts todeteriorate only forR/Rd > 5. In general, asa0 is increased,the fit progressively deteriorates. For comparison, the bottompanel shows the fractional difference ofΣ(R) when a sim-ple exponential form forΣ(Rg) is adopted. The differenceincreases asa0 is increased and is negligible only for verysmall values ofa0. The improvement offered by the empiri-cal formula is clearly evident here.

The accuracy of the formula as a function ofa0 andq canbe better gauged in panels a and d of Figure 11 . Here we

plot the mean absolute fractional difference of surface densityΣ(R) with respect to the target surface densityΣexp(R) =exp(−R/Rd)/(2πR

2d) as a function ofa0 andq. It can be

seen that the proposed solution works quite well over most ofthe(a0, q) space except for very high values ofa0. It shouldbe noted that in the regiona0 > q + 0.5, lower right corner,where the empirical formula fails the iterative solution also it-self fails (see panel d). So the actual inaccuracy of the formulais mainly confined to the region(q > 0.4, a0 > 0.75).

We now study the accuracy of our analytic formula givenby Equation (34) and Equation (39) for the case where boththe target surface density and the velocity dispersion are spec-ified to follow exponential forms. The mean absolute frac-tional difference ofΣ(R) andσR(R) with respect toΣ(R) =exp(−R/Rd)/(2πR

2d) and σR = vca0 exp(−qR/Rd) is

shown in panels b and c of Figure 11. For comparison,the panels e and f show the results of the iterative algo-rithm. It can be seen that the formula works well in the range0 < a0 < 0.5, where the mean error is less than 1%. How-ever fora0 > 0.5, the formula and the iterative solution bothdo not work well. It should be noted that the sharp transitionat a0 ∼ 0.5 is due to the fact that we measure the differenceover the range0 < R/Rd < 5. Extending the range wouldshift the transition value ofa0 to lower values and vice versa.In general, for a givena0 the solution provided by the iterativemethod and the empirical formula are both accurate in repro-ducing target profiles for small values ofR, they gradually

9

0 2 4 6 8 10R/Rgd

-1.0

-0.5

0.0

0.5

1.0

Σ(R

)/Σ e

xp(R

,Rd)

-1

a0=0.5, q=1/3

Rd/Rgd=1.00Rd/Rgd=1.05Rd/Rgd=1.10Rd/Rgd=1.15Rd/Rgd=1.20Rd/Rgd=1.25

FIG. 12.— The fractional difference of the surface densityΣ(R) fromthe target surface density,Σexp = exp(−R/Rd)/(2πR

2d), for Σ(Rg) =

exp(−Rg/Rgd)/(2πR2gd). The thickness of the line is proportional to the

value ofRd used forΣexp. The dashed line is the result of using our empir-ical formula withRgd = Rd.

become inaccurate at larger values ofR. It can also be seenfrom the figure that fora0 > 0.5, the empirical formula per-forms slightly better than the iterative solution at reproducingthe target surface densities. However, this is only at the ex-pense of not matching the target velocity dispersion profilesas well.

We now discuss the possible causes for the failure of theiterative solution for largea0. The iterative algorithm willonly work correctly if the contribution to the integral forΣ(R)or σR(R) (see Equation 23 and 24), is confined to a localregion aroundRc ∼ R. In general, at largeR this conditionis violated. The radius at which this violation occurs dependsupon the choice ofa0, and is lower for larger values ofa0.This violation of the locality condition is stronger forσR(R)profiles, asσR(Rc) increases exponentially with decrease inRc. The rise in∆σR/σR at largeR as discussed in Section 3.4and Figure 9 is also a manifestation of this effect. This isthe main reason that when the constraint of exponential radialdispersion profiles is applied the algorithm fails fora0 > 0.5.

4.1. Comparison with other approximate solutions

An alternative way to generate an exponential disc using aShu type distribution function was proposed in Binney (2010).They note that the effect of warming up the distribution func-tion is to expand the disc, and this can be thought of as anincrease in scale length of the disc. So, one way to take thisinto account is to start with a slightly smaller scale lengthwhen specifyingΣ(Rg). There are two problems with thisapproach. First, the factor by which the scale length will haveto be reduced will depend upon the values ofa0 andq. Sothe solution is no simpler than what we propose. Secondly,we find that even if one has the correct factor, the solution isless than optimal. To show this, we plot in Figure 12 the frac-tional difference of the surface densityΣ(R) obtained usingΣ(Rg) = exp(−Rg/Rgd)/(2π(Rgd)

2) from an exponential

a0=0.5, q=1/3

0 2 4 6 8 10R/Rgd

-0.6

-0.4

-0.2

-0.0

0.2

ln Σ

(R)-

ln Σ

exp(

R,R

d)

fCorr Rd/Rgd=1.0fShu-Dehnen-Norm Rd/Rgd=1.1fShu-Dehnen-Norm Rd/Rgd=1.0

fShu-Dehnen Rd/Rgd=1.0

FIG. 13.— The logarithmic difference of the surface densityΣ(R) fromthe target surface density,Σexp = exp(−R/Rd)/(2πR

2d), for Σ(Rg) =

exp(−Rg/Rgd)/(2πR2gd). Shown are the results for the Dehnen’s ansatz

with different scale length. The results of our empirical formula are alsoshown alongside.

surface density with different values ofRd. We seta0 = 0.5and q = 0.33 which is typical of old thin disc stars. Thecase considered is that offShu with flat rotation curve. It canbe seen that increasingRd improves the solution slightly forR/Rgd < 5, but at the expense of deteriorating it for largeR.Overall, the quality of the fit is quite poor. For comparison,the dashed line is the result using our empirical formula withRgd = Rd, which clearly performs better.

One might argue that the ansatz of Dehnen (1999)

fShu−Dehnen(E,L) =γ(Rg)Σ(Rg)

2πσ2R(Rg)

exp

(

Ec(Rg)− E

σ2R(Rg)

)

(40)

actually produces discs that are more close to an exponentialform, and so applying the Binney (2010) scale length reduc-tion to it might produce even better agreement. We now in-vestigate this issue. Firstly, the above distribution functionis not normalized. IfσR(Rg) is parameterized in termsa0andq as before then it has a normalization that depends uponthe choice ofa0 andq, so the full distribution function is notpurely analytical anymore. On top of that as said earlier, thefactor by which the scale length will have to be reduced willalso depend upon the values ofa0 andq. We assume it to be10% for the time being. In Figure 13, we show the fractionaldifference of surface density from the target density for thecase offShu−Dehnen with flat rotation curve. Results with nor-malization (denoted byfShu−Dehnen−Norm) and with Binney(2010) scale length reduction are also shown alongside. It canbe seen that after normalizing the Dehnen’s ansatz does pro-duce discs that match the target density better, i.e., the rangeof deviation is smaller than the thin red line correspondingtoRg/Rd = 1 in Figure 12. After applying the Binney (2010)scale length reduction the agreement gets even better, but onlyin regions withR/Rd < 4. For largeR, there is still a widediscrepancy. In comparison, it can be seen that the formulaproposed in this paper (dashed blue line) is still superior whilebeing simpler and fully analytic.

5. DISCUSSION

In this paper, we have presented an empirical formula forShu-type distribution functions which can reproduce withgood accuracy a disc with an exponential surface densityprofile. This should be useful in constructing equilibriumN-body models of disc galaxies (Kuijken & Dubinski 1995;

10

McMillan & Dehnen 2007; Widrow & Dubinski 2005) wheresuch functions are employed. It can also be employedin synthetic Milky Way modelling codes like TRILEGAL(Girardi et al. 2005), Galaxia (Sharma et al. 2011) and Be-sancon (Robin et al. 2003). Finally, the most importantuse of our new formalism is for MCMC fitting of theoreti-cal models to large data sets, e.g., the Geneva-Copenhagen(Nordstrom et al. 2004) , RAVE (Steinmetz et al. 2006) andSEGUE stellar surveys (Yanny et al. 2009). Here, separableanalytic approximations of this kind are required to make theproblem tractable (Sharma et al 2013, in preparation).

One of our main finding is that the part of the distributionfunction that determines the target surface density can be writ-ten as the target density plus a correction term which approx-imately follows a unique functional form. So, the problemreduces to determining the scale and the amplitude of this cor-rection function and their dependence ona0 (ratio of centralvelocity dispersion to circular velocity) andq (a parametercontrolling the gradient of the dispersion as a function of ra-dius). The problem is further simplified as we find that thescale is primarily a function of onlyq and the amplitude is

a primarily a function of onlya0. These findings are validfor not only flat rotation curves but also for cases with risingand falling rotation curves. Additionally, it was also found tobe valid for one of the Dehnen (1999) distribution functions.An extension of the formula to reproduce velocity dispersionprofiles that are exponential in radius was also found to workwell. This suggests that the methodology presented can, inprinciple, be also applied to other disc distribution functionsproposed by Dehnen (1999) and Binney (2010, 2012) whichare conceptually similar.

ACKNOWLEDGMENTS

We are thankful to the anonymous referee who among otherthings motivated us to generalize our results. We are alsothankful to James Binney for his comments that led to asubstantial improvement in the manuscript. JBH is fundedthrough a Federation Fellowship from the Australian Re-search Council (ARC). SS is funded through ARC DP grant120104562 which supports the HERMES project.

APPENDIX

INTEGRAL EQUATION FOR ONE OF THE DEHNEN DISTRIBUTION FUNCTIONS

We consider here the distribution function

fDehnen,a=F (RE)

σ2R(RE)

exp

(

−E − Ec(L)

σ2R(RE)

)

(A1)

from Dehnen (1999). ForfDehnen,a one can write the probability distribution in(R,RE , Rg) space as

P (R,RE , Rg)dRdREdRg =2πf(E,L)dLdR2d|vR| (A2)

P (R,RE , Rg)=2πf(E,L)dL

dRg

dE

dRE

2√

2(E − Φ(R))− L2/R2(A3)

To proceed further we assume a flat rotation curve. UsingE = Φeff(RE , RE), Ec(Rg) = Φeff(Rg, Rg), dL/dRg =2vcirc(Rg)γ2(Rg)

,

dE/dRE =2v2

circ(RE)

γ2(RE)RE, γ2(Rg) = γ2(RE) = 2 we get

P (R,RE, Rg)=4πF (RE)

a2(RE)

R−(1+ 1

a2)

E R1

a2

g√

1− 2 ln(R/RE)−R2g/R

2(A4)

Substituting Rg

R√

1−2 ln(R/RE)= x and

K(R,RE) = (R/RE)(1+ 1

a2)(1− 2 ln(R/RE))

1

2a2 (A5)

this simplifies to

P (R,RE , x)=4πF (RE)

a2(RE)K(R,RE)

x1

a2

√1− x2

(A6)

Integrating overx from 0 to 1 we get

P (R,RE)=4πF (RE)

a2(RE)K(R,RE)h(a) (A7)

where

h(a) =

∫ 1

0

x1/a2

√1− x2

=

√π

2

Γ( 12a2 + 1

2 )

Γ( 12a2 + 1)

=

√π

2

Γ(c+ 12 )

Γ(c+ 1)with c =

1

2a2(A8)

Using∫

P (R,RE)dR = 2πREΣ(RE) one can writeF (RE) in terms ofΣ(RE) to get

F (RE)=Σ(RE)a

2(RE)

2gK(a)h(a)(A9)

11

and then

P (R,RE)=2πΣ(RE)

gK(a)K(R,RE) (A10)

Here

gK(a)=1

RE

∫

K(R,RE)dR =

(

e

1 + c

)1+cΓ(1 + c)

2with c =

1

2a2

≈√

π

2(c+ 1)for large c.

REFERENCES

Binney, J. 2010, MNRAS, 401, 2318—. 2012, MNRAS, 426, 1328Binney, J., & McMillan, P. 2011, MNRAS, 413, 1889Dehnen, W. 1999, AJ, 118, 1201Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & daCosta, L.

2005, A&A, 436, 895Kalnajs, A. J. 1976, ApJ, 205, 751Kuijken, K., & Dubinski, J. 1995, MNRAS, 277, 1341McMillan, P. J., & Binney, J. 2012, MNRAS, 419, 2251McMillan, P. J., & Dehnen, W. 2007, MNRAS, 378, 541Nordstrom, B., et al. 2004, A&A, 418, 989Robin, A. C., Reyle, C., Derriere, S., & Picaud, S. 2003, A&A, 409, 523

Sandage, A. 1969, ApJ, 158, 1115Schonrich, R., & Binney, J. 2009, MNRAS, 396, 203—. 2012, MNRAS, 419, 1546Sharma, S., Bland-Hawthorn, J., Johnston, K. V., & Binney, J. 2011, ApJ,

730, 3Shu, F. H. 1969, ApJ, 158, 505Steinmetz, M., et al. 2006, AJ, 132, 1645van der Kruit, P. C., & Searle, L. 1982, A&A, 110, 61Widrow, L. M., & Dubinski, J. 2005, ApJ, 631, 838Yanny, B., et al. 2009, AJ, 137, 4377