Question

The H-alpha emission line from gas within a elliptical galaxy has a half-width of 6.5 Angstroms, representing a spread in velocity from opposite sides of the galaxy due to rotation. If the gas is effectively at a radius of 10 kpc from the center of the galaxy, what is the mass of the galaxy? Also, what is the expected mass of the supermassive black hole at the center of this galaxy?

Answer 1

given
spectral halfwidth = d(lambda) = 6.5 A
hence, this means that the wavelength from far end of the galaxy is shifted by this amount from the central wavelength
hence let its linear speed relative to center of the galaxy be v
distance from the center of the galaxy = r
then
v/c = d(lambda)/lambdao
lambdao = 656.28 nm ( H-alpha line)
hence
v = 6.5*10^-10/656.28*10^-9 *c
v = 9.9043*10^-4 c = 297129.2740 m/s = 297.12927409032 km/s

now, r = 10 kpc = 3.086*10^20 m
hence
mass of galaxy = M
GM/r = v^2
M = v^2*r/G = 4.0847*10^35 kg
considering that 1% mass of the universe is in blackholes
hence mass of blackhole = 4.0847*10^33 kg