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Physics LibreTexts

Resources: Formulae, Constants, and Conversions

  • Page ID
    45853
  • Formulae

    Relationship

    Initial Chapter

     = 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 10km 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

    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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