Hernquist profile, mass and scale radius?

[Richard Massey]

How do I specify the mass and scale radius of a Hernquist profile (16) in Lenstool?
The input .par file still requires variables "v_disp" and "core_radius" - how are they related?

Some information from the source code is below, but I'm not sure about the normalisations. Any authoritative answers could go here: https://projets.lam.fr/projects/lenstool/wiki/HernquistProfile

Thanks,
Richard




In e_grad.c,
g.x = dpl_rad * (1. - ilens->emass) * Q.x / u; // corrections to elliptical potential
g.y = dpl_rad * (1. + ilens->emass) * Q.y / u;
where
dpl_rad = hern_dpl(u, ilens->rc, ilens->b0)


I'm not sure what DPL stands for, but hern_dpl() is in e_hernquist.c, where the variable ilens->b0 is locally called kappas, and
dpl = kappas * rs * x * (1. - hern_f(x)) / (x * x - 1.);
so I'm guessing b0 is proportional to mass.


Moreover, b0 is read in set_lens.c, as
ilens->b0 = 6 * pia_c2 * ilens->sigma * ilens->sigma
but I have no idea what pia_c2 is. Any ideas?




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PS: I now suspect that v_disp is never optimised, but b0 is. That means if you think you're getting a flat prior on v_disp, you're getting a flat prior on mass instead. Is that right?

[Johan Richard]

Hi Richard

I do not know about the Hernquist profile specifically, as I did not code this part, but the information below is generic for most of

the potentials in lenstool.

dpl_rad is the radial displacement for an axisymetric (e=0)profile, then converted into g.x and g.y by applying the ellipticity correction and projection (Q vector).

pia_c2 is simply pi over c^2 with pi in arcsec, it is used to convert velocity dispersion into arcsec displacement for an isothermal profile and the same factorisation is used here. Definition of constants is done in include / constant.h

b0 is the normalisation factor for all sort of potentials, and typically is the parameter optimised. You are correct that the flat prior on v_disp is converted into a flat prior in mass.

Maybe Eric has more details about the Hernquist profile.

Best

Johan

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[Richard Massey]

Thank you Johan for unpacking the acronyms - and Peter Taylor, who replied off-thread.

D'oh! pia_c2=pi/c^2 explains everything.
I was wracking my brain for French words beginning pia. The simplest one hadn't even occurred to me. :)

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[Eric Jullo]

Hi 

Note that you can also parametrize the Hernquist model in terms of its Virial mass in Msun. In this case, the relation to rc and b0 is (taken from set_lens.c:420)

                ilens->rckpc = halo_rvir(ilens->masse, ilens->z)*0.03*0.551; // Giocoli+12 

                ilens->rc = ilens->rckpc / (d0 / C.h * distcosmo1(ilens->z));

                ilens->b0 = ilens->masse/ilens->rckpc/ilens->rckpc /2./M_PI/(1E12*cH0_4piG*C.h/distcosmo1(ilens->z));

The halo_rvir() function is defined in cosmoratio.c, and uses the critical matter density at a given redshift to derive the Virial radius of a halo from its mass. The multiplicative factors 0.03*0.551 to convert from Virial to scale radius come from Giocoli et al. 2012. Note that this has been little tested so to use with caution.

Cheers

Eric