vcdisk.vcbulge_sersic

vcdisk.vcbulge_sersic(rad, mtot, re, n, q=0.99, inc=0.0)

Circular velocity of a flattened Sersic bulge.

This is the same as vcdisk.vcbulge(), but for an analytic [Sersic68] surface density profile. The implementation follows Eq.s (10)-(14) in [Noordermeer08].

Parameters

  • rad (list or numpy.ndarray) – array of radii in \(\rm kpc\).

  • mtot (float) – total mass in \(\rm M_\odot\) of the Sersic bulge.

  • re (float) – effective radius in \(\rm kpc\) of the Sersic bulge.

  • n (float) – Sersic index, \(\rm 0 < n \leq 8\).

  • q (float, optional) – intrinsic axis ratio of the spheroid. This is related to the ellipticity of the observed isophotal contours \(\epsilon\) and the inclination angle \(i\) (i.e. the inc parameter) by \((1-\epsilon)^2 = q^2+(1-q^2)\cos^2 i\). This parameter is \(0<q<1\) for oblate bulges. The spherical case \(q=1\) is singular in this formulation and will fallback to vcdisk.vcbulge_sph().

  • inc (float, optional) – inclination in degrees of the line-of-sight with respect to the symmetry axis of the spheroid. inc=0 is edge-on, inc=90 is face-on.

Returns

array of \(V_{\rm bulge}\) velocities in \(\rm km/s\).

Return type

numpy.ndarray

See also

vcdisk.vcbulge(), vcdisk.vcbulge_sph(), vcdisk.sersic

Notes

This function calculates \(V_{\rm bulge}\) for a spheroidal bulge, whose observed surface density can be approximated with a Sersic profile

\[I(R) = I_e \exp\left\{ -b_n\left[\left(\frac{R}{R_e}\right)^\frac{1}{n}-1\right] \right\},\]

where \(R_e\) is the effective radius, i.e. the radius containing half the total mass of the spheroid, \(I_e\) is the surface density at the effective radius, and \(n\) is the Sersic index, which determines the concentration of the density profile. The derivative of \(I(R)\) is also analytic:

\[\frac{{\rm d}I(R)}{{\rm d}R} = -\frac{I_e\,b_n}{n\,R_e} \exp\left\{ -b_n\left[\left(\frac{R}{R_e}\right)^\frac{1}{n}-1\right] \right\} \left(\frac{R}{R_e}\right)^{\frac{1}{n}-1}.\]

With this expression, recalling that \(e=\sqrt{1-q^2}\) is the intrinsic ellipticity, the circular velocity profile of the bulge becomes (e.g. Eq. 2.132 in [BT2008]):

\[V^2_{\rm bulge}(r) = \mathcal{C} \int_0^r \left[ \int_m^\infty \frac{ \exp\left\{ -b_n\left[\left(R/R_e\right)^{1/n}-1\right] \right\} \left(R/R_e\right)^{1/n-1} }{\sqrt{R^2-m^2}} {\rm d}R \right]\frac{m^2}{\sqrt{r^2-e^2m^2}}{\rm d}m,\]
\[\mathcal{C} = \frac{4\,G q I_e\,b_n}{nR_e} \sqrt{\sin^2i+\frac{1}{q}\cos^2i}.\]

As in vcdisk.vcbulge(), it is convenient to change integration variables since both integrals present singularities: \(u={\rm arccosh}{(R/m)}\) is used for the inner integral in \({\rm d}R\), while \(t=\arcsin{me/r}\) is used for the outer integral in \({\rm d}m\).

References

Sersic68

Sersic, 1968, Argentina: Observatorio Astronomico. Atlas de Galaxias Australes. https://ui.adsabs.harvard.edu/abs/1968adga.book…..S/