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

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

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^2For astronomical situations involving objects orbiting around galaxies, we use units

M in solar masses R in kiloparsecs v in km/secin which case it works out to

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

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

( 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

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 * Twhere

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

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 = ---- Dwhere

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

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 uswhere

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 lambdawhere

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

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.