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 .
- The observed flux from a source follows the inverse square law .
- The apparent and absolute magnitude relation is .
- Redshift is defined by .
- For small redshifts, recession speed is approximately .
- Hubble's law relates recession speed and distance by .
- The Schwarzschild radius of a nonrotating black hole is .
- Kepler's third law for two masses is .
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 .
- 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 is the proportionality constant in that describes the present expansion rate of the universe.
- Parsec
- A parsec is an astronomical distance unit equal to about or 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 require consistent units.
- Applying 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 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 magnitudes corresponds to a factor of in brightness.
Practice Questions
- 1 A star has radius and temperature . Use with to estimate its luminosity.
- 2 A galaxy has redshift . Using and , find its approximate recession speed.
- 3 Using , estimate the distance to a galaxy receding at .
- 4 Explain why two stars with the same luminosity can have different apparent brightnesses when viewed from Earth.
Understanding Astrophysics & Cosmology Equations
Astronomers rarely touch the objects they study. They infer properties from light collected by telescopes. A spectrum spreads that light into wavelengths and reveals dark or bright lines from particular atoms.
These lines act like labels. Hydrogen, helium, calcium, and other elements each leave recognisable patterns. If the whole pattern shifts toward longer wavelengths, the source is moving away relative to the observer.
A shift can come from cosmic expansion, ordinary motion through space, or strong gravity near a compact object. Students should first identify which situation a problem describes before choosing a redshift relationship.
Brightness measurements need careful language. Luminosity is the total power emitted by an object. Flux is the power arriving at each square metre of a detector.
A nearby low luminosity star may look brighter than a very luminous distant star. This distinction explains why astronomers use absolute magnitude, which compares stars as if they were placed at one standard distance. The magnitude scale is logarithmic rather than linear.
A difference of five magnitudes represents a factor of one hundred in received flux. Lower magnitude numbers mean brighter objects, which feels backwards at first because the system began with an old visual ranking of stars.
Temperature gives another route to understanding stars. A hotter surface emits much more energy per unit area than a cooler one. The temperature effect is especially strong because the emitted power per unit area depends on temperature raised to the fourth power.
Radius matters too because a larger star has more radiating surface. This helps explain why a cool red giant can be highly luminous while a hot white dwarf can be faint overall.
Colour gives a useful temperature clue, but dust between Earth and a star can make the light look redder and dimmer. Real observations therefore need corrections for absorption and scattering by interstellar dust.
Hubble's law is most reliable for galaxies at sufficiently large distances, where local gravitational motions are a smaller part of the measurement. Nearby galaxies can move toward or away from us because of their interactions with neighbouring galaxies, so their motion does not always follow the general expansion neatly. At very large redshift, the simple link between redshift and speed is no longer accurate.
The light has travelled while the universe expanded, and cosmologists use models containing matter, radiation, and dark energy to relate redshift to distance and time. This is why a basic Hubble calculation is useful in class but has stated limits.
Black hole equations require similar attention to assumptions. The Schwarzschild radius describes an ideal black hole with no rotation and no electric charge. It marks the event horizon, not a solid surface.
An object crossing that boundary cannot send information back to distant observers. The radius grows directly with mass, so a more massive black hole has a larger horizon.
Orbital equations likewise assume gravity is the dominant force and often treat bodies as point masses. In every calculation, write units at each step, convert them before substitution, and check whether the final size, time, or speed is physically reasonable.