Astrophysics & Cosmology Equations Cheat Sheet

A printable reference covering stellar luminosity, magnitude, redshift, Hubble's law, Schwarzschild radius, and cosmology equations for grades 11-12.

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Astrophysics and cosmology use physics equations to describe stars, galaxies, black holes, and the expansion of the universe. This cheat sheet helps students connect observable quantities, such as brightness, wavelength, and distance, to physical properties. It is useful for solving problems involving stellar radiation, apparent magnitude, redshift, orbital motion, and cosmic expansion. The goal is to keep the most important equations organized for quick reference during review and practice. Core ideas include the inverse square law for light, the Stefan-Boltzmann law for stellar luminosity, and Kepler or Newton equations for orbital systems. Cosmology problems often use redshift, Hubble's law, and the scale factor to describe how the universe expands. Black hole calculations commonly involve escape velocity and the Schwarzschild radius. Careful unit use is essential because distances may appear in meters, parsecs, light-years, or megaparsecs.

Key Facts

The luminosity of a star is related to its radius and temperature by $L = 4\pi R^2\sigma T^4$ .
The observed flux from a source follows the inverse square law $F = \frac{L}{4\pi d^2}$ .
The apparent and absolute magnitude relation is $m - M = 5\log_{10}\left(\frac{d}{10\text{ pc}}\right)$ .
Redshift is defined by $z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}$ .
For small redshifts, recession speed is approximately $v \approx cz$ .
Hubble's law relates recession speed and distance by $v = H_0 d$ .
The Schwarzschild radius of a nonrotating black hole is $r_s = \frac{2GM}{c^2}$ .
Kepler's third law for two masses is $T^2 = \frac{4\pi^2 a^3}{G(M_1 + M_2)}$ .

Vocabulary

Luminosity: Luminosity is the total power emitted by a star or other object, measured in watts.
Flux: Flux is the power received per unit area from a source, usually measured in $\text{W m}^{-2}$ .
Redshift: Redshift is the fractional increase in wavelength of light, often caused by cosmic expansion or relative motion away from the observer.
Hubble Constant: The Hubble constant $H_0$ is the proportionality constant in $v = H_0 d$ that describes the present expansion rate of the universe.
Parsec: A parsec is an astronomical distance unit equal to about $3.09 \times 10^{16}\text{ m}$ or $3.26$ light-years.
Schwarzschild Radius: The Schwarzschild radius is the radius at which the escape speed from a mass equals the speed of light.

Common Mistakes to Avoid

Using apparent brightness as luminosity is wrong because flux depends on distance, while luminosity is the total power emitted by the source.
Forgetting to convert megaparsecs to parsecs or meters is wrong because formulas like $F = \frac{L}{4\pi d^2}$ require consistent units.
Applying $v \approx cz$ at very large redshift is wrong because the approximation works best for small redshifts and ignores full cosmological effects.
Reversing observed and emitted wavelength in $z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}$ is wrong because it changes the sign and meaning of the redshift.
Treating magnitude as a linear brightness scale is wrong because magnitude is logarithmic, so a difference of $5$ magnitudes corresponds to a factor of $100$ in brightness.

Practice Questions

1 A star has radius $R = 6.96 \times 10^8\text{ m}$ and temperature $T = 5800\text{ K}$ . Use $L = 4\pi R^2\sigma T^4$ with $\sigma = 5.67 \times 10^{-8}\text{ W m}^{-2}\text{K}^{-4}$ to estimate its luminosity.
2 A galaxy has redshift $z = 0.030$ . Using $v \approx cz$ and $c = 3.00 \times 10^8\text{ m/s}$ , find its approximate recession speed.
3 Using $H_0 = 70\text{ km s}^{-1}\text{Mpc}^{-1}$ , estimate the distance to a galaxy receding at $v = 2100\text{ km/s}$ .
4 Explain why two stars with the same luminosity can have different apparent brightnesses when viewed from Earth.

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