# Springel Model - Initial conditions of galaxy for simulation

1. May 1, 2013

### fab13

Hello,

I'am studying a code (called "starscream") which allows to make initial conditions (positions and velocities) for NBody simulation. They are based on Springel and White (1999) model. I have several problems about the underlying equations used in this code.

Firstly, See attachment "fig1.png", for computing the disk scale length, into the "disk_scale_length_func" routine, the author seems to have forgotten to multiply the line 128 by the line 129, i.e the
factor "sqrt(1.0/f_c_func(gal->halo_concentration))".

Indeed, the definition of the disk scale length is :

$$R_{d}=\dfrac{1}{\sqrt{2}}\,\bigg(\dfrac{j_{d}}{m_{d}}\bigg)\,\lambda\,r_{200}\,f_{c}^{-1/2}\,f_{R}(\lambda,c,m_{d},j_{d})$$

where $$f_{R}(\lambda,c,m_{d},j_{d})=2\bigg[\int_{0}^{\infty}\,e^{-u}\,u^{2}\,\dfrac{v_{c}(R_{d}\,u)}{v_{200}}\,du\bigg]^{-1}$$

and $$f_{c}=\dfrac{c}{2}\,\dfrac{1-1/(+c)^{2}-2\,(ln(1+c))/(1+c)}{[c/(1+c)-ln(1+c)]^{2}}$$

Here is the original part of this code :

Code (Text):

// This function determines the scale length of the disk to within 15% or so using
// the fitting formula provided in MMW97.
double disk_scale_length_func(galaxy *gal) {

int i;
double base, power, c, f_conc, f, scale, old_scale, v_c;

c = gal->halo_concentration;
f_conc = 2.0/3.0+ pow((c/21.5),0.7);
base = (gal->j_d*gal->lambda)/(0.1*gal->m_d);
power = (-0.06+2.71*gal->m_d+0.0047*gal->m_d/(gal->j_d*gal->lambda));
f = pow(base,power)*(1.0-3.0*gal->m_d+5.2*gal->m_d*gal->m_d) *
(1.0-0.019*c+0.00025*c*c+0.52/c);
gal->disk_scale_length = (1.0/sqrt(2.0))*(gal->j_d/gal->m_d)*gal->lambda*
gal->r200*(f/sqrt(f_conc));

// Now try to iterate for the correct value.
for (i = 0; i < 50; ++i) {
scale = sqrt(1.0/2.0)*(gal->j_d/gal->m_d)*gal->lambda*gal->r200*
[B]2.0*(gal->v200/j_d_func(gal))             // product missing "*" with the next line
sqrt(1.0/f_c_func(gal->halo_concentration));[/B]
gal->disk_scale_length = scale;
printf("scale = %f\n",scale);
}

return scale;
}

Is anyone can confirm this error ?

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Secondly, See attachment "fig2.png", into the "dj_d_func" routine, I don't understand the lines 199 to 201. Actually, from this equation in the doc :

$$v_{c}^{2}(R)=R\,\dfrac{\partial\Phi}{\partial R}=4\pi\,G\,\Sigma_{0}\,\dfrac{R^{2}}{4\,R_{d}}\,\bigg[I_{0}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)\,K_{0}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)+I_{1}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)\,K_{1}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)\bigg]$$ (eq 1)

this should be :

$$v_{c}^{2}=\dfrac{GM_{d}(r)}{r} \rightarrow M_{d}(r)=\dfrac{v_{c}^{2}\,r}{G}$$

with $$v_{c}^{2}$$ computed from the (eq 1) equation.

The mass density of the disk is :

$$\Sigma(R)=\Sigma_{0}\,e^{-R/R_{d}}=\dfrac{M_{d}}{2\pi\,R_{d}^{2}}\,e^{-R/R_{d}}$$ (eq 2)

With the expression of $$\Sigma_{0}$$ on (eq 2), we can have :

$$M_{d}(r)=\dfrac{v_{c}^{2}\,r}{G}=M_{d,total}\,\dfrac{R^{3}}{2\,R_{d}^{3}}\,\bigg[I_{0}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)\,K_{0}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)+I_{1}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)\,K_{1}\,\bigg(\dfrac{R}{2\,R_{d}}\bigg)\bigg]$$

So into the code, on line 199, with $$y=R/(2\,R_{d})$$ and we should have :

disk_mass=4*total_mass_disk*y*y*y*(I_{0}*K_{0}+I_{1}*K_{1})

and not only one product "y*y".

Here is the original part of this code (with in bold my correction) :

Code (Text):

double dj_d_func(double radius, void *params) {

int status;
galaxy *gal = (galaxy *) params;
double I_0, K_0, I_1, K_1, halo_mass, disk_mass, y, a;
double total_disk_mass, v_c, scale;
gsl_sf_result result;

scale = gal->disk_scale_length;
total_disk_mass = gal->disk_mass;
a = gal->halo_a_value;
halo_mass = gal->halo_mass;

status = gsl_sf_bessel_I0_e(y,&result);
I_0 = result.val;
status = gsl_sf_bessel_K0_e(y,&result);
K_0 = result.val;
status = gsl_sf_bessel_I1_e(y,&result);
I_1 = result.val;
status = gsl_sf_bessel_K1_e(y,&result);
K_1 = result.val;

[B]  // ORIGINAL VERSION
disk_mass = 2.0*total_disk_mass*y*y*(I_0*K_0 - I_1*K_1);
// END ORIGINAL VERSION[/B]

[B] //THIS SHOULD BE
//disk_mass = 4.0*total_disk_mass*y*y*y*(I_0*K_0 + I_1*K_1);
// END CORRECTION[/B]

}

Is my modification correct ?

I make you notice too that there is a minus sign on the original code at line 199 : (I_{0}*K_{0} - I_{1}*K_{1}) : is this good ?

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