Some Equations used in Extragalactic Astronomy

During this course, we will come across the following equations; you will need to know what they mean, and how to use them.

Connection between mass and orbital speed

If a small body is in orbit around a big one, then there's a connection between the radius of the orbit, the speed of the orbit, and the mass of the central body. For circular orbits,

                                              2
               ( G * M )                     v  * R
    v  =  sqrt ( ----- )      or      M  =  -------
               (   R   )                       G

where

        M       is the mass of the central object
        R       is the radius of the orbit
	v       is the orbital speed                            N * m^2
	G       is the Gravitational Constant:  6.67 x 10^(-11) -------
                                                                  kg^2

For astronomical situations involving objects orbiting around galaxies, we use units

        M       in solar masses
	R       in kiloparsecs
	v       in km/sec

in which case it works out to

	                                 2
        M   =  230,000 solar masses  *  v  * R

Parallax measurments

When one looks at a distant object from two different points separated by a baseline distance B, and one sees that the object appears to shift its position by an angle theta, then the distance Lto the object is

                   B/2                        B
         L  =  ------------         ~=    ---------
               tan(theta/2)               tan(theta)

Gravitational microlensing

Suppose that a body of mass M lies a exactly halfway between you and a distant object, at a distance D from you and from the distant object. Light rays from the distant object will be bent by an angle theta_E given by

                         (  4 * G * M  )
       theta_E  =   sqrt ( ------------)
                         (   D * c^2   )

where

          M         is the mass of the lensing body
	  G         is the gravitational constant
	  D         is the distance from you to the lensing body
	                  (and from the lensing body to the distant object)
	  c         is the speed of light

If the background source is lined up exactly with the lensing body, then you will see a perfect ring of light around the lens. If the background source is slightly offset, however, so that it appears an angular distance b away from the lensing body, then the source will appear as a distorted image (or set of images). This distorted image will be amplified in brightness by a factor

                           u^2  +  2                            b
     amp factor   =   -------------------      where    u  = -------
                       u * sqrt(u^2 + 4)                     theta_E

Stellar luminosity

A star (or any very dense spherical object) with radius R and surface temperature T will emit blackbody radiation according to the Stefan-Boltzmann formula:

                        2            4 
        L  =  4 * pi * R  * sigma * T

where

        L        is the luminosity (Watts)
	pi       is 3.14159...
        R        is the radius of the star (meters)
        T        is the temperature (Kelvin)                          W
	sigma    is the Stefan-Boltzmann constant: 5.67 x 10^(-8) ---------
                                                                  m^2 * K^4

The Hubble Law

When we look at other galaxies, we see that almost all of them are moving away from us, with velocities which increase with distance (they farther away, the faster they move away). The ratio between radial velocity and speed is called the Hubble Constant H:

                  v
         H   =   ----
	          D

where

         v        is the radial velocity (km/sec)
	 D        is the distance of the galaxy (Mpc)
	 H        is the Hubble Constant (km/sec/Mpc)

Redshifts

When an object is moving towards us or away from us, the wavelength lambda of radiation from it is shifted: this is called the Doppler Effect. For radial velocities much smaller than the speed of light (the usual situation), the size of the shift is

                                  v          positive if moving away us
    shift in lambda  =  lambda * ---              
                                  c          negative if moving towards us

where

         lambda           is the rest wavelength
	 v                is the radial velocity of the object
	 c                is the speed of light

Most galaxies move away from us, which causes a positive shift in wavelength; that means a shift towards the red end of the spectrum. Astronomers call this a redshift. We often measure the size of the redshift with this formula:

                shifted lambda
        z   =   --------------   -  1 
                    lambda

where

        z                 is the "redshift" of the object
	lambda            is the rest wavelength 
	shifted lambda    is the observed wavelength