Cribsheet!
Constants and Units
Velocity of light
c=2.998 x 105 km s-1
c=2.998 x 108 m s-1
Grav. Constant G=6.67 x 10-11 m3 kg-1 s-2
Mass H atom
mH=1.67 x 10-27 kg
h=Planck’s Const 6.626 x 10-34 joule sec
Energy = h x frequency (for light) = hc/
1 AU = 1.5 x 108 km
= 1.5 x 1011 m
Parsec
= 206265 AU
= 3.26 light years
Light year
= 6.3 x 104 AU
=0.307 pc
= 9.5 x 1012 km
1 sidereal year
=365.25 days
= 3.16 x 107 s
7
(about π x 10 seconds in a year!)
Astronomical unit
Earth mass
Earth radius
Orbital velocity
6 x 1024 kg
6378 km
30 km s-1
Sun's mass
2 x 1030 kg = 2 x 1033 g
Sun's radius
R = 7 x 105 km
Sun's luminosity
L = 3.9 x 1026
Watts
Sun's temperature
T = 5800 K
Sun's abs. magnitude MV = 4.83
Sun's app. Magnitude mV = -26
Sun’s flux as seen from earth: 1380 W/m2
Angular measures
360 degrees = 24 hours = 2π radians
1 radian ~ 57 degrees = 206265 arc sec
1 degree = 60' = 3600"
1 hour = 15 degrees
Equations
Circumference of a circle = 2r
Area of a circle = r2
Surface area of sphere 4 π r2
Volume of sphere
(4/3) π r3
Magnitudes
M1-M2 = -2.5 log10 (L1/L2)
or, Luminosity = 85.5 x (0.398)M
(if given apparent magnitude, use flux as
seen from earth for L1, L2)
m-M = 5 log10(d) - 5 = 5 log10(d/10pc)
(d is distance in parsecs)
Parallax
Parallax μ (in arc sec) = 1/distance (in pc)
(parallax is sometimes measured in milliarcseconds or 10-3 arcsec)
Orbits
Using solar masses, years, and AU:
P2 = a3 (P=period, a=semi major axis)
P2 = a3/(m1+m2)
In MKS units
P2 = 4 π2a3/G(m1+m2)
Orbital velocity (km s-1) v = 2 π r / P
(r = radius of orbit)
Orbits (any units, just ratios):
m1/m2 = v2/v1 = a2/a1 = α2/α
(m is mass, a is separation = a1+a2, α is ang. sep)
For visual binaries:
- For systemic velocity, average max and
min velocity of an individual star
𝑚 𝑚
Newton’s law of gravity: 𝐹 = 𝐺 𝑟1 2 2
G=6.67x10-11m3s-2kg-1, r in meters, m in kg, F in
Newtons
Circular Velocity Vc=√
𝐺𝑀
𝑟
M=mass of central
body, r is radius (in meters) but only if object’s
mass ≈0.
Misc: ac = vt2 = rω2 ; F = mobjv2 / r = mobj r ω2
Eccentricity
b2 = a2(1-e2)
(a and b are semi-major and semi-minor axes)
e = focal length/a
aphelion, apogee, apastron are farthest points
perihelion, perigee, periastron are nearest
Kepler's Laws
1. A planet orbits the sun in an ellipse with the
sun at one focus of the ellipse
2. A line connecting a planet to the Sun sweeps
out equal areas in equal time intervals
3. Using years, AU, and solar mass P2 = a3
otherwise, P2 = 42a3/G(m1+m2)
Energy
Wien's Law: λmax T = 0.29 cm K
(λ=wavelength of peak flux)
Flux = σ T4 (σ = Stefan-Boltzman constant =
5.67 x 10-8 watts m-2K-4, T=temp in K)
Luminosity = 4πr2 σ T4 (r in meters)
σ=7.12 x 10-17 solar radii-2 k-4 LSun
10 Å per nm
Flux at distance d = L / (4 * π * d2)
Periods of variable stars
RR Lyrae stars - 0.5 - 3 days
Cepheids = a few to 10s of days
Long period variables - hundreds of days
Magnitudes of variable stars and Red
Giants
RR Lyrae stars MV = ~0.75
Cepheids - Depends on period
MV=-2.76(log10 (P) - 1.0) - 4.16
Supernovae (Ia) - MV about -19.5
Supernovae (II) – Mv less, very roughly
-15 or -16
Brightest red giants: Mv = -3
Extragalactic stuff
Age of Universe: 13.7 x 109 years
Hubble constant: 70 km s-1 Mpc-1
Mass-Luminosity Relationship
(M/Msun)4 =( L/Lsun)
Ages
M(in solar mass)/L(in Solar Lums) * 12 x
109 = main sequence lifetime in years
=(1/M in solar masses)2.5 x 1010
Lifetime of red giants is 1/10 of giant’s
main sequence liftime.
Doppler Shift
Δλ/λ=v/c
(Positive for motion away from the
observer, i.e. Positive is increasing
separation)
 λ = obs – actual
Redshift = z = v/c
1 + z = λobsv/λemit
Hα is at 656.281 nm.
Chandrasekhar limit = 1.4 MSun
(electron pressure cannot support more
mass than this - no white dwarfs bigger
than 1.4 MSun - become neutron stars)
Neutron star limit: about 3.6 solar masses;
become black holes
Misc. factoids:
Distance of Jupiter - about 5 AU
Distance of Pluto - about 40 AU
Size of Milky Way – about 30 Kpc diam,
about 100,000 LY
Distance to Galactic Center ~ 26,000 LY
The Solar system is on MWG’s Orion spur
Distance to Andromeda ~ 2.6 million LY
Size of Observable Universe
Radius =13.7 billion LY
Vrot MWG = 220 km/s
Black holes:
Schwarzschild radius=2Gm/c2
M − σ relation:
log(M/Msun)=8.12 + αlog(σ/200km/s-1)
α=4.2 (some sources give 5)
σ is stellar velocity dispersion
mcenteral black hole = .005mgalaxy (also works for black
holes in the center of globular clusters)
Temperature α 1/mass
Spectral Types
Class
Temp.
O
B
A
F
G
35,000K
20,000K
10,000K
7,000K
5,900K
K
5,200K
M
3,700K
Lines
B-V
(approx.)
-0.24
-0.16
0.00
+0.42
+0.65
He
H, He
H
H, Ca, Fe
Metals,
weak H
More
+1.54
metals
Molecules +1.85
Luminosity Classes (for solar temp)
I
Supergiant 104 Solar lum
II
Bright Giant 103 Solar lum
III
Giant 102 solar lum
IV
Subgiant 10 solar lum
V
Main Sequence 1 solar lum
Telescopes
𝐿𝐺𝑃
𝐷
2
Light Gathering Power (LGP): 𝐿𝐺𝑃𝐴 = (𝐷𝐴 )
𝐵
𝐵
where D is diameter
11.6
Resolving power 𝛼 = 𝐷 where D is diameter in
cm, gives power in arcseconds
𝐹
Magnification 𝑀 = 𝐹𝑂 where FO, E is the focal
𝐸
length of the objective and eyepiece, respectively
Equations for Galaxies
Tully-Fisher relation:
For Sa, 𝑀𝐵 = − 9.95log 𝑉𝑚𝑎𝑥 + 3.15
For Sb, 𝑀𝐵 = − 10.2log 𝑉𝑚𝑎𝑥 + 2.71
For Sc, 𝑀𝐵 = −11.0 log 𝑉𝑚𝑎𝑥 + 3.31
Spirals: log(𝑟𝑎𝑑𝑖𝑢𝑠 𝑖𝑛 𝑘𝑝𝑐) = −.249𝑀𝐵 − 4
Ellipticals: log 𝜎 = −.1𝑀𝐵 + .2
Virial Theorem: 𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 = 2
𝑟𝑣 2
𝐺
−2 ∑ 𝐾𝐸 = ∑ 𝑃𝐸
𝑜𝑟