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/l

Astronomical unit 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

(about π x 107 seconds in a year!)


Earth mass 6 x 1024 kg

Earth radius 6378 km

Orbital velocity 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 = 2pr

Area of a circle = pr2

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 milli-arcseconds 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

Newton’s law of gravity: F=Gm1m2r2

G=6.67x10-11m3s-2kg-1, r in meters, m in kg, F in Newtons

Circular Velocity Vc=GMr 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 = 4p2a3/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)

D 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. / Lines / B-V (approx.)
O / 35,000K / He / -0.24
B / 20,000K / H, He / -0.16
A / 10,000K / H / 0.00
F / 7,000K / H, Ca, Fe / +0.42
G / 5,900K / Metals, weak H / +0.65
K / 5,200K / More metals / +1.54
M / 3,700K / 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

Light Gathering Power (LGP): LGPALGPB=DADB2 where D is diameter

Resolving power α=11.6D where D is diameter in cm, gives power in arcseconds

Magnification M=FOFE where FO, E is the focal length of the objective and eyepiece, respectively

Equations for Galaxies

Tully-Fisher relation:

For Sa, MB=-9.95logVmax+3.15

For Sb, MB=-10.2logVmax+2.71

For Sc, MB=-11.0logVmax+3.31

Spirals: logradius in kpc= -.249MB-4

Ellipticals: logσ=-.1MB+.2

Virial Theorem: total mass=2r