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Resources: Formulae, Constants, and Conversions

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Aug 21, 2021
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Formulae	Relationship	Initial Chapter
= c/ f	wavelength, frequency, and speed	Chapter 2
E = hf	energy and frequency
E_photon = E_electron	energy of photon emitted/absorbed by atom
E_n = -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	Chapter 3
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
F_g = mg	Weight and mass
a_c = v²/r	Centripetal acceleration
F_g = Gm₁m₂/r²	Newton's law of gravity
PE = mgh	Potential energy (non-relativistic)
KE = ½ mv²	Kinetic energy (non-relativistic)
E_final = E_initial	Conservation of energy
v_escape = (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/r^½	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
T_e / T_o = 1 + z = S_o / S_e	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	Chapter 16
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
k_B = 1.38 × 10^-23 J/K	Boltzmann's constant
t_planck ~ 10^-43 s	Planck time
t_planck ~ 4 × 10^-35 m	Planck length
m_e = 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 10⁸km	AU and meters and km
1 ly = 9.5 x 10¹⁵m = 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^-19J	eV and joules
1 degree = 60 arcmin	degrees and arcminutes
1 arcmin = 60 arcsec	arcminutes and arcseconds
1 angstrom (Å) = 1 x 10^-10meters	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 10²²m	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

Prefix	Meaning (in USA)	Exponent	Symbol
Tera	trillion	10¹²	T
Giga	billion	10⁹	G
Mega	million	10⁶	M
kilo	thousand	10³	k
centi	one-hundredth	10^-2	c
milli	one-thousandth	10^-3	m
micro	one-millionth	10^-6	μ
nano	one-billionth	10^-9	n
pico	one-trillionth	10^-12	p

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