 = c/ f |
wavelength, frequency, and speed |
Chapter 2 |
| E = hf |
energy and frequency |
Ephoton =  Eelectron |
energy of photon emitted/absorbed by atom |
| En = -13.6/n2 |
energy levels of atom |
| T = 2.9e-3/ |
temperature and peak wavelength |
F =  T4 |
brightness and temperature |
A =  r2 |
area and radius |
Chapter 3 |
 = 1.22 /D |
resolution |
| v = d/t |
speed, distance, and time |
Chapter 4 |
| z = / |
redshift |
| v = cz |
velocity and redshift |
| t = E/L |
stellar lifetime |
Chapter 5 |
| E = mc2 |
energy and mass |
| t ~ m-2 |
stellar lifetime and mass |
| L ~ m3 |
stellar mass and luminosity |
| d = 1/a |
parallax angle and distance |
Chapter 6 |
| d = S/ |
small angle formula |
| F= L/4d2 |
inverse square law |
| F = ma |
Newton's second law |
Chapter 7 |
| Fg = mg |
Weight and mass |
| ac = v2/r |
Centripetal acceleration |
| Fg = Gm1m2/r2 |
Newton's law of gravity |
| PE = mgh |
Potential energy (non-relativistic) |
| KE = ½ mv2 |
Kinetic energy (non-relativistic) |
| Efinal = Einitial |
Conservation of energy |
| vescape = (2GM/R)½ |
Escape velocity |
| v ∝ r |
Rotation of rigid disk |
Chapter 8 |
| v ∝ 1/r |
Rotation of water around drain |
| v ∝ constant |
Rotation of cars on roundabout |
| v ∝ 1/r½ |
Keplerian rotation (planets) |
| v2 = GM/r |
Relationship of enclosed mass to velocity and distance |
| M = ρV |
Mass, density, and volume |
 = 1/√(1-v2/c2) |
gamma factor |
Chapter 9 |
| t’ = t |
time dilation |
| L’ = L/ |
length contraction |
| d2 = x2 + y2 |
Pythagorean Theorem |
| s2 = x2 – c(t)2 |
spacetime interval |
| E = mc2 |
mass and rest energy |
| E = E0 |
total energy and rest energy |
| \begin{equation} g = \frac{GM}{R^2} \end{equation} |
Surface gravity |
Chapter 10 |
| \begin{equation} d=v_0t+\frac{1}{2}at^2 \end{equation} |
Distance and acceleration |
| \begin{equation} v=at \end{equation} |
Velocity and acceleration |
| \begin{equation} t=\frac{t_0}{\left(1-\frac{gH}{c^2}\right)} \end{equation} |
Time dilation (weak field approximation) |
| \begin{equation} f=f_0\left(1-\frac{gH}{c^2}\right) \end{equation} |
Gravitational redshift (weak field approximation, frequency, photon traveling upward) |
| \begin{equation} \lambda=\frac{\lambda_0}{\left(1-\frac{gH}{c^2}\right)} \end{equation} |
Gravitational redshift (weak field approximation, wavelength) |
| \begin{equation} f=f_0{\sqrt{1-\frac{2GM}{Rc^2}}} \end{equation} |
Gravitational redshift (full expression, frequency) |
| \begin{equation} \lambda=\frac{\lambda_0}{\sqrt{1-\frac{2GM}{Rc^2}}} \end{equation} |
Gravitational redshift (full expression, wavelength) |
| \begin{equation} d^2=\left(\Delta x\right)^2 + \left(\Delta y\right)^2 \end{equation} |
Pythagorean theorem |
| \begin{equation} d^2=\left(R\Delta\theta\right)^2+\cos^2\theta\left(R\Delta\alpha\right)^2 \end{equation} |
Distance on a sphere |
| \begin{equation} d^2=\left(\Delta x\right)^2 + \left(\Delta y\right)^2 +\left(\Delta z\right)^2 \end{equation} |
Pythagorean Theorem in 3-D |
| \begin{equation} s^2=\left(\Delta x\right)^2+\left(\Delta y\right)^2 + \left(\Delta z\right)^2-\left(c\Delta t\right)^2 \end{equation} |
Spacetime interval in flat space |
| \begin{equation} s^2=\left(1-\frac{2GM}{rc^2}\right)^{-1}\left[\left(\Delta x\right)^2+\left(\Delta y\right)^2 + \left(\Delta z\right)^2\right]-\left(1-\frac{2GM}{rc^2}\right)\left(c\Delta t\right)^2 \end{equation} |
Spacetime interval in spherically curved space |
| \begin{equation} \theta = \frac{2GM}{bc^2} \end{equation} |
Angle of deflection of light |
| \begin{equation} P_{gw}=\frac{2}{5}\left(\frac{GM}{Rc^2}\right)^5\left(\frac{m}{M}\right)^2\left(\frac{c^5}{G}\right) \end{equation} |
Power emitted by gravitational waves |
| \begin{equation} {\rm\bf{G}}={8\pi} G ~{\rm\bf{T}}/c^4 \end{equation} |
Einstein equation |
| \begin{equation} R_S = \frac{2GM}{c^2} \end{equation} |
Schwarzschild radius |
Chapter 11 |
| \begin{equation} s^2=\left(1-\frac{R_{S}}d\right)^{-1} (\Delta d)^2 - \left( 1-\frac{R_{S}}{d}\right) (c\Delta t)^2 \end{equation} |
Spacetime interval in a spherically symmetric space (Schwarzschild interval) |
| \begin{equation} T_{bh}=\frac{1.23 \times 10^{23}}{M} \end{equation} |
Temperature of a black hole |
| \begin{equation} \Delta E \Delta t \geq \frac{h}{4\pi} \end{equation} |
Uncertainty principle |
| \begin{equation} L=\frac{3.56\times10^{32}}{M^2} \end{equation} |
Luminosity of a blackhole |
| \begin{equation} t \approx 2.5\times 10^{-16} M^3 \end{equation} |
Evaporation time |
| \begin{equation} \frac{\Delta m}{\Delta t}=-\frac{2L}{c^2} \end{equation} |
Accretion rate |
| \begin{equation} L_{edd} = 6.3 M_{BH} \end{equation} |
Eddington luminosity |
| \begin{equation} \alpha = \frac{4GM}{bc^2} \end{equation} |
Deflection angle (full) |
Chapter 12 |
| \begin{equation} \theta_E=\sqrt{\left(\frac{4GM(b)}{c^2}\right)\left(\frac{D_{LS}}{D_{LO}D_{SO}}\right)} \end{equation} |
Einstein radius |
| \begin{equation} \theta^2-x\theta-\theta_E^2=0 \end{equation} |
Lens equation |
| \begin{equation} m = \frac{1}{\left[1-\left(\frac{\theta_E}{\theta}\right)^4\right]} \end{equation} |
Magnification for a point-mass lens |
| \begin{equation} v = H_0 d \end{equation} |
Hubble law |
Chapter 13 |
| \begin{equation} v = cz \end{equation} |
Cosmological redshift |
| \begin{equation} d_{\rm physical}(t) = d_{\rm comoving}(t) S(t) \end{equation} |
Comoving coordinates |
| \begin{equation} 1+z=\frac{S(t_{\rm observed})}{S(t_{\rm emitted})} \end{equation} |
Ratio of scale factors |
| \begin{equation} t = \frac{1}{H_0} \end{equation} |
Hubble time (age) |
| \begin{equation} H^2 - \frac{8 \pi G \rho}{3} = - \frac{k c^2}{S^2} \end{equation} |
Friedman equation |
| d = cz/H0 |
distance and redshift |
Chapter 14 |
| Te / To = 1 + z = So / Se |
Temperature, redshift, and scale factor |
Chapter 15 |
| E ~ kT |
energy and temperature |
Chapter 16 |
| T ~ mc2/k |
temperature of Universe and mass of particle in reaction |
|
Constants
|
Name |
| c = 3 x 108 m/s = 3 x 105 km/s |
speed of light |
| h = 6.63 x 10-34 J s = 4.136e-15 eV s |
Planck’s constant |
| G = 6.67 x 10-11 N m2/kg2 |
Universal gravitational constant |
| kB = 1.38 × 10-23 J/K |
Boltzmann's constant |
| tplanck ~ 10-43 s |
Planck time |
| tplanck ~ 4 × 10-35 m |
Planck length |
| me = 9.1 × 10-31 kg |
mass of an electron |
|
Conversions
|
Units |
| 1 km = 1000 m |
km and meters |
| 1 km = 0.6 mi |
km and miles |
| 1 AU = 1.5 x 1011 m = 1.5 x 108 km |
AU and meters and km |
| 1 ly = 9.5 x 1015 m = 9.5 x 1012 km |
light-years and meters and km |
| 1 ly = 6.3 x 104 AU |
light-years and AU |
| 1 eV = 1.6 x 10-19 J |
eV and joules |
| 1 degree = 60 arcmin |
degrees and arcminutes |
| 1 arcmin = 60 arcsec |
arcminutes and arcseconds |
| 1 angstrom (Å) = 1 x 10-10 meters |
angstroms and meters |
| 1 pc = 3.26 ly |
parsecs and light-years |
| 1 N = 0.2248 pounds |
newtons and pounds |
| 1 kpc = 3.086 x 1019 m |
kiloparsecs and meters |
| 1 solar mass = 2 × 1030 kg |
solar masses and kg |
| 1 Mpc = 3.09 x 1022 m |
megaparsecs and meters |
| 1 radian = 2.06 x 105 arcsecond |
radians and arcsec |
| 1 Mpc = 3.09 × 1019 km |
megaparsecs and km |
|
Units (abbreviation)
|
Type of quantity |
| meters (m) |
length (SI) |
| kilograms (kg) |
mass (SI) |
| second (s) |
time (SI) |
| meters per second (m/s) |
speed (SI) |
| kelvin (K) |
temperature (SI) |
| miles (mi) |
length |
| astronomical unit (AU) |
length |
| year (yr) |
time |
| light-year (ly) |
length |
| light-minutes |
length |
| light-seconds |
length |
| g/cm3 |
density |
| solar masses |
mass |
| hertz (Hz) = cycles/s = 1/s = s-1 |
frequency (SI) |
| joules (J) |
energy (SI) |
| electron volts (eV) |
energy |
| watts (W) = J/s |
power |
| radians |
angle |
| degrees |
angle |
| arcmin |
angle |
| arcsec |
angle |
| angstrom (Å) |
length |
| parsec (pc) |
length |
| m/s2 |
acceleration (SI) |
| newton (N) = kg m/s2 |
force (SI) |
| joules (J) = N m |
energy (SI) |
| μK micro Kelvin = 10-6 K |
temperature |
