Mathematical Formulae for Astronomy 1
Each of the following formulae is in this format:
Name of Formula: Mathematical Formula
Variable = name of variable (units in words, units abbreviation)
Variable of Physical Constant = name of physical constant = number (units)
CHAPTER 1
Conversions of Distances
1 Astronomical unit (AU) = 1.495979 x 1011 (meters, m)
1 light-year (ly) = 9.46053 x 1015 (meters, m)
1 parsec (pc) = 3.085678 x 1016 (meters, m)
Scientific Notation
CHAPTER 4
Kepler’s Third Law: P2 = a3
P = orbital period (years, y)
a = semi-major axis or average distance to Sun (Astronomical Units, AU)
Newton’s Second Law: F = ma
F = force (Newtons = kilograms meters per second per second, N = kg•m / s2)
m = mass (kilograms, kg)
a = acceleration (meters per second per second, m / s2)
Newton’s Law of Universal Gravitation: Fgrav = G(m1m2) / r2
Fgrav = force due to gravity (Newtons, N)
G = Gravitational constant = 6.67 x 10-11 (N•m2 / kg2)
m1 = mass of object 1 (kilograms, kg)
m2 = mass of object 2 (kilograms, kg)
r = radial distance between two object’s centers (meters, m)
Newton’s form of Kepler’s Third Law: P2 = [4π2 / G (m1+m2)] a3
P = orbital period (seconds, s)
π = 3.1415927….
G = Gravitational constant = 6.67 x 10-11 (N•m2 / kg2)
m1 = mass of object 1 (kilograms, kg)
m2 = mass of object 2 (kilograms, kg)
a = semi-major axis or average distance between m1 and m2 (meters, m)
CHAPTER 5
Wavelength – Frequency relationship: ln = c
l = wavelength (meters, m) (Greek letter lambda)
n = frequency (Hertz = 1 / seconds, Hz = 1 / s) (Greek letter nu not a “v”)
c = speed of all electromagnetic radiation = 299,792,458 (meters per second, m / s)
Wein’s Law: lmax = c / T
c = Wein’s constant = 0.0029 (meters × Kelvin = m•K)
lmax = wavelength of maximum emission (meters, m)
T = temperature (Kelvin, K)
Energy Flux: F = L / 4pR2
F = energy flux (joules / second / square meters = watts / square meters, W / m2)
L = luminosity (joules / second = watts, W)
π = 3.1415927….
R = radius (meters, m)
Stefan – Boltzmann’s Law: F = sT4
F = energy flux (joules / second / square meters, W / m2)
s = Stefan-Boltzmann’s constant = 5.67 x 10-8 (W / m2•K4)
T = temperature (Kelvin, K)
Photon Energy: E = hc / l = hn
E = energy (joules, J)
h = Planck’s constant = 6.625 x 10-34 (J•s)
l = wavelength (meters, m)
n = frequency (Hertz = 1 / seconds, Hz = 1 / s)
Bohr formula for Electron Energy Level transitions: 1 / l = R(1 / N2 – 1 / n2)
l = wavelength (meters, m)
R = Rydberg’s constant = 1.097 x 107 (1 / m)
N = energy level number of the inner (lower) orbit
n = energy level number of the outer (higher) orbit
Redshift / Doppler Effect: (l - l0) / l0 = vradial / c
z = redshift (+ numbers means away, - numbers means towards, no units)
l = wavelength of moving object (meters, m)
l0 = wavelength of non-moving object (meters, m)
vradial = radial velocity (meters / second, m / s)
c = speed of all electromagnetic radiation = 299,792,458 (meters / second, m / s)
CHAPTER 18
Energy – Mass conversion: E = mc2
E = energy (joules, J)
m = mass (kilograms, kg)
c = speed of all electromagnetic radiation = 299,792,458 (meters / second, m / s)
CHAPTER 19
Stellar Parallax: d = 1 / p
d = distance (parsecs, pc)
p = parallax angle (arcseconds, “)
Luminosity – Brightness relationship: b = L / 4πd2
b = apparent brightness (watts / square meters, W / m2)
L = luminosity (watts, W)
π = 3.1415927….
d = distance (meters, m)
Radial Velocity – Wavelength: vr / c = (l - l0) / l0
vr = radial velocity (meters / second, m / s)
c = speed of all electromagnetic radiation = 299,792,458 (meters / second, m / s)
l = wavelength of moving object (meters, m)
l0 = wavelength of non-moving object (meters, m)
Distance Modulus: m – M = 5log(d) – 5 OR d = 10[(m-M+5) / 5]
m = apparent magnitude (no units)
M = absolute magnitude (no units)
d = distance (meters, m)
Stefan-Boltzmann Law combined with Energy Flux: L / 4pR2 = sT4
L = luminosity (watts, W)
π = 3.1415927….
R = radius (meters, m)
s = Stefan-Boltzmann’s constant = 5.67 x 10-8 (W / m2•K4)
T = temperature (Kelvin, K)
Kepler’s Third Law for Binary Star Systems: m1 + m2 = a3 / P2
m1 = mass of object 1 (Solar Masses, M8)
m2 = mass of object 2 (Solar Masses, M8)
P = orbital period (years, y)
a = semimajor axis (Astronomical units, AU)
CHAPTER 24
Schwarzschild Radius: RSch = 2GM / c2
RSch = Schwarzschild radius (meters, m)
G = Gravitational constant = 6.67 x 10-11 (N•m2 / kg2)
M = mass (kilograms, kg)
c = speed of all electromagnetic radiation = 299,792,458 (meters / second, m / s)
CHAPTER 26
Hubble Law: v = H0d
v = velocity (kilometers / second, km / s)
H0 = Hubble’s number = 71 (kilometers per second per Megaparsec, km / s / Mpc)
d = distance to galaxy (Megaparsecs, Mpc)
Redshift / Doppler Effect: z = (l - l0) / l0
z = redshift (+ numbers means away, - numbers means towards, no units)
l = wavelength of moving object (meters, m)
l0 = wavelength of non-moving object (meters, m)
CHAPTER 28
Age estimate of the Universe: T0 = d / v = 1 / H0
T0 = age estimate of the universe (seconds, s)
d = distance (meters, m)
v = velocity (meters / second, m / s)
H0 = Hubble’s number (kilometers per second per Megaparsec, km / s / Mpc)
CHAPTER 30
Frank Drake Equation: N = R* fp ne fl fi fc L
N = number of technologically advanced civilizations (no units)
R* = Rate of Sun-like stars forming (no units)
fp = fraction of stars with planets (no units, between 0 and 1)
ne = number of planets that are Earth-like (no units)
fl = fraction of Earth-like planets with life (no units, between 0 and 1)
fi = fraction of life that has become intelligent (no units, between 0 and 1)
fc = fraction of intelligent species technologically civilized (no units, between 0 and 1)
L = lifetime of a civilization (no units, between 0 and 1)
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