Astronomy: Mass of Elliptical Galaxy from Homework

[theown1]

Homework Statement

The measured velocities in elliptical galaxies pertain to the spread in random velocities of
the stars; in spirals, to the rotational velocities of either the stars or the gas about the
galaxy's center.
When a spectrum is taken of an elliptical galaxy, we measure not the light from any single
star, but the light from a signi cant part of the entire galaxy of stars. The superposition of
billions of stars moving at dierent random velocities along the line of sight will, because
of the Doppler effect, broaden the spectral lines. A comparison of the broadened galaxy
spectrum with the intrinsic spectrum of an appropriately chosen star allows an estimate
for the velocity dispersion of the stars along the whole line of sight of the galaxy V (which
is related to the 3-D (deprojected) velocity at a point by v =$$\sqrt{3}$$V

(a) The intrinsic absorption line-width of a K giant star might be 0.5 $$A$$ at a central
wavelength of $$\lambda$$ = 5000 $$A$$. Suppose the spectrum of an elliptical galaxy is observed that
has a line width of
 = 3 $$A$$. Assume that the increase is wholly due to broadening
by random velocities of the constituent stars, and calculate V from the Doppler formula

$$\Delta$$$$\lambda$$₀=V/c.
(b) Additionally, you are given that the elliptical galaxy is 500 million lyr away from us
and 1 arcmin in angular size. Find the mass of the elliptical galaxy. You may approximate
the elliptical galaxy as a sphere. Express your answer in M$$\odot$$ .(solar mass)

Homework Equations

given in the problem

The Attempt at a Solution

the first part is fairly straight forward, all i did to solve the problem was (3/5000)c=V
and I got V to equal 179,880 m/s I think my units are right?

but I'm not sure how to solve for the mass of the galaxy only knowing the velocity dispersion of the stars along the whole line of sight of the galaxy, and the distance, and how much space it takes up in the sky, can someone help?

 

[nate808]

To solve for the mass of the galaxy, we can use the virial theorem, which states that the total kinetic energy of a system is equal to half of its total potential energy. This can be written as:

KE = 1/2 * PE

Where KE is the kinetic energy and PE is the potential energy.

In the case of a galaxy, the kinetic energy can be represented by the random velocities of the stars, while the potential energy can be represented by the gravitational potential energy of the galaxy. This can be written as:

KE = (1/2) * M * V^2

Where M is the total mass of the galaxy and V is the velocity dispersion of the stars.

The potential energy can be represented by:

PE = -G * M^2 / R

Where G is the gravitational constant, M is the total mass of the galaxy and R is the distance from the center of the galaxy.

Since we know the velocity dispersion and the distance of the galaxy, we can rearrange the equations to solve for the mass:

M = (V^2 * R) / (G * 2 * 3 * 10^8)

Where R is the distance in meters and G is the gravitational constant in m^3 / kg * s^2.

Plugging in the values, we get:

M = (179,880^2 * 500 * 9.46 * 10^15) / (6.67 * 10^-11 * 2 * 3 * 10^8)

M = 4.52 * 10^41 kg

To convert this to solar masses, we can divide by the mass of the sun, which is approximately 1.99 * 10^30 kg:

M = 2.27 * 10^11 M_sun

Therefore, the mass of the elliptical galaxy is approximately 2.27 * 10^11 solar masses.