29:62 GENERAL ASTRONOMY
Mathematical Formulae
29:62 General Astronomy
Winter Semester 1999
Note: All units MKS unless otherwise indicated.
Wien's Law: Wavelength at which a blackbody radiator is brightest
where 
(Planck's Constant),
(speed of light), 
.
Kinetic Energy of Thermal Motion:
where m is the mass of the particle (electron, atom, ion, molecule) and
v is its speed.
Proper Motion:
Units are kilometers per second. 
is the proper motion in arcseconds per year.
is the distance to the star in parsecs.
Doppler Effect:
where 
is the measured wavelength, 
is the
rest wavelength, v is the velocity toward or away from the observer
(velocity along the line of sight) and c is the wave speed.
Relation between Flux and Magnitude:
is the flux (Watts/m 
) from object 1, 
is that from
object 2, 
is the magnitude of object 1, and 
is the magnitude
of object 2. Then,
Distance Modulus - Distance:
where m is the apparent magnitude, M is the absolute magnitude, and
d is the distance in parsecs.
Energy of a Photon: 
where 
is the wave frequency (units Hertz). Photon energy is often
conveniently expressed in electron volts, 1 eV = 
.
Energy Levels of the Hydrogen Atom with respect to zero energy at the
unbound continuum.
where the energy is in electron volts, and n is the principal quantum
number, or more simply, the number of the energy level.
Energy - Mass Equivalence:
where m is the mass.
Fermi Energy:
where m is the mass of the particle and n is the density in
particles 
Mass-Luminosity Relation:
where L and M are luminosity and mass of the star, and Sun, respectively.
Stefan-Boltzmann Law:
Units of P are Watts/m . T is the temperature of the blackbody, and
in MKS units.
Potential Energy of Self-Gravitating Sphere:
where 
is the gravitational constant,
M is the mass of the sphere, and 
is its radius
Frequency- Wavelength Relation for Electromagnetic Waves:
where is the frequency and 
is the wavelength of observation.
Beamwidth of a Radio Telescope:
where 
is the width of the radio telescope beam (full width
at half maximum) is the wavelength of observation, and D is the
diameter of the antenna. With this formula, is in radians.
Definition of the Jansky: A Jansky is a unit of flux density, and
is defined as 
Watts-meters 
Hertz 
.
Kinetic Energy of a Rotating Object:
where I is the moment of inertia of the object, and 
is the
angular velocity (radians per second). The moment of inertia I depends on
the mass of the object and the distribution of mass within the object; for
a sphere of uniform density it is 
where M is the
mass of the object and R is its radius. The angular velocity if given
by 
where P is the rotational period.
Schwarzschild Radius:
where is the gravitational constant and c is the speed of light.
Attenuation of Radiation by an Absorbing and Scattering Medium:
where 
is the initial flux and 
is the absorption and
scattering per unit length.
Rotational Energy of a Quantum Rotor:
where J is the rotational angular momentum quantum number, I is the moment
of inertia, and 
. The moment of inertia for a
diatomic molecule is 
.
Expression for Galactic Distribution of Stars:
where N is the density of stars (number per cubic parsec).
Perfect Gas Law:
where p is the gas pressure (Newtons/m ), n is the gas number density,
and T is the gas temperature (K).
Circular Orbit Equation:
where M(r) is the mass interior to r.
Geometry for Galactic Structure:
Velocity of Object in Inner Galaxy:
where 
is line of sight velocity with respect to the Sun and
is orbital velocity of Sun in the Milky Way.
The Law of Cosines for Oblique, Plane Triangles
Given sides b and c and the enclosed angle A, the remaining side
a is given by
Hubble's Law:
where v is the speed of recession of a galaxy (km/sec, not meters/sec)
d is the distance in Megaparsecs , and 
km/sec/Mpc. Note that this
value is (arguably) the current best estimate, not the definitive value as in a
fundamental constant.
Characteristic Frequency of Synchrotron Radiation:
The mean frequency of radiation 
emitted by energetic electrons of
energy E spiraling in a magnetic field of strength B is given by
where e is the electric charge of the particle, and m is its mass. In terms of more convenient units,
where is in Hertz, B is in Tesla (the unit of magnetic field strength in
the SI system of units, and E is in Joules.
Relativistic Doppler Effect:
Definitions same as for regular Doppler effect, equation (6).
Critical Density in Friedmann Universe:
where 
is the Hubble constant in natural units, i.e. inverse time.