D 
Small angle equation:
d
206,265
where D is the linear size of the object,  is the angular size of the object in arcseconds, and d is the distance
to the object.
Time:
Sidereal Time = HA() = HA(*) + RA(*) = HA() + RA()
Solar Time = HA() + 12h
Orbits:
Inferior planets:
1
1 1


P
E S
1
1 1


Superior planets:
P
E S
where P is the planet’s sidereal (orbital) period,
S is its synodic period, and E = 365.2564 days.
Kepler’s Third Law: (M1 + M2) (in M) = a3/P2 , for a in A.U. and P in years.
Light Energy: E  h 
hc
, where  is the frequency,  is the wavelength, c is the speed of light, and h =

6.6261 × 10–34 J s = Planck’s constant.
0.0029
, wavelength of maximum light flux.
T
Wien’s Law:
max (in meters ) 
Black Body output:
F (flux )   T 4 , T is temperature,  the Stefan-Boltzmann constant.
Doppler shift:
vR
c
Magnification:
Magnificat ion
Angular resolution:
 (in arcseconds) = 2.5 × 105 /D , where D is the telescope diameter.

Telescope image scale:
  0
,  is the measured wavelength and 0 the rest wavelength.
0

focal length of objective lens
focal length of eyepiece lens
206265 arcseconds/f (mm), where f is the focal length.
2 1
 29.8 km s 1   
 r a
–1
where 29.8 km s is the Earth’s orbital velocity, and r and a are the distance of the object from the Sun and
the semi-major axis of its orbit, respectively, in Astronomical Units (A.U.).
Vis-viva Equation:
Roche limit: rR  2
Orbital Velocity
1
3
  planet 
Rplanet 

  satellite 
1
3
 1.44 Rplanet Planet temperature (A = albedo): Tbb 
Titius-Bode Relation: Semi  major axis , a A.U. 
Distance:
d (in parsecs, pc) 
Distance modulus:
mM
1  A0.25 279 K
0.5
r A.U.
0, 3, 6,12, 24, 48...  4
10
1
,  = parallax angle.
 arcseconds 
 5 log d  5 , m = apparent magnitude, M = absolute magnitude.
1 parsec (pc) = 206265 A.U. = 3.086 × 1013 km = 3.26 light years.
Tangential velocity:
vt (in km/s )  4.74  d ,  = proper motion (arcsec. yr–1), d = distance (pc).
Space velocity:
vspace  vt2  vR2 . In a moving cluster, vt = vspace sin , vR = vspace cos .
Inverse square law of light:
Stellar luminosity:
L
L
LSun
b 
 mc2 .
 4R 2Te4ff , R = stellar radius, Teff = effective temperature. Teff (Sun) = 5779 K.
 R 

 
 RSun 
Magnitude relationship:
m1  m2
2
 T

 TSun



4
b 
  2.5 log  1  , magnitudes m1 and m2, brightnesses b1 and b2.
 b2 
L
LSun
Mass-Luminosity relationship:
Spectral Sequence:
L
. Energy generation: E
4 d 2
 M
 
 M Sun
4

 , M = mass, for main-sequence stars types O–K.

O B A F G K M L T (hottest to coolest, LT types are brown dwarfs).
Stellar Radii: Giants ~ 10–50× Dwarfs, Supergiants ~ 100–1000× Dwarfs, Degenerates ~ 0.01× Dwarfs.
Stellar Masses: 32 M (O5), 14 M (B0), 2 M (A0), 1.5 M (F0), 1.0 M (G2), 0.8 M (K0), 0.4 M (M0).
t lifetime 
Stellar Lifetimes:
Energy Available
xM
x

 3 , x = some fraction of initial mass.
Rate of Consumptio n
L
M
NUV
Lifetime of Sun for H burning: ~1010 years. Strömgren Spheres (H II Regions), R  4
.
N e N H  2
3 
Virial Theorem:
Energy available through gravitational contraction ~ ½ gravitational energy = GM2/2R.
Distance to a star in a moving cluster:
d  pc  
Mass of the Milky Way Galaxy: M Galaxy 
Redshift, z 


vR km s -1 tan 
,  = angle from convergent point.
4.74  arcsec. yr -1


3
aSun
 1.31011 solar masses .
2
PSun
v obs  rest

(non-relativistic), where v = velocity of the object and c = speed of light.
c
rest
 1  z 2  1
i.e. v  z c (non-relativistic). For velocities approaching c, v  
 c (relativistic).
2
 1  z   1
v
Hubble relation (for distances to galaxies): d 
, where H0 is the Hubble “constant”  70 km/s/Mpc.
H0
1
 14 10 9 years .
Hubble time (approximate age of the universe): t H 
H0
Gh
Gh
 4.0510 35 meter . Planck time: t P 
 1.3510 43 second , prior to which
3
5
c
c
2GM
the universe was equivalent to a black hole. Schwarzschild (black hole) Radius: Rs  2 .
c
6
3c
1
 2 (the larger the mass, the lower the density of a black hole).
Black hole density:  S 
3
2
32 G M
M
Planck length: P 

3 H 02
Critical density of the universe:  c 
. Density parameter:  0  measured .
c
8 G