Astrophysics and geophysics extend classical and modern physical principles across the macroscopic cosmos: from elliptical planetary orbits and tidal Roche limits to stellar nucleosynthetic lifetimes, cosmic distance ladders, and terrestrial gravitational anomaly fields.
This cheat sheet organizes 10 foundational physical laws and governing equations in astrophysics and geophysics, categorized into three core domains: Celestial Mechanics, Stellar & Galactic Astrophysics, and Terrestrial Geophysics, complete with mathematical formulations, orbital mechanics, and observational applications.
🔗 Physics Equations Suite: Classical Mechanics · Electromagnetism · Fluid & Thermal · Optics & Acoustics · Nuclear & Particle · Relativity & Cosmology · Quantum Mechanics
1. Celestial Mechanics (3 Items)
01 · Kepler's First Law
Law of Ellipses
$$r = \frac{p}{1 + e\cos\theta} \quad (0 \le e < 1)$$
- ①Physical Essence: The orbit of every planet is an ellipse with the Sun located at one of the two foci, shattering ancient dogmas of uniform circular motion.
- ②Formula Notation: $r$ is radial distance from the primary focus, $p = a(1-e^2)$ is semi-latus rectum, $a$ is semi-major axis, $e$ is orbital eccentricity ($0 \le e < 1$), $\theta$ is true anomaly.
- ③Theoretical Significance: Laid the geometric foundation for modern celestial mechanics, directly leading to Newton's discovery of the inverse-square law of universal gravitation.
- ④Boundary Conditions & Applications: Most major planets have low eccentricities (Earth $e \approx 0.0167$), while Mercury ($e \approx 0.2056$) and comets (Halley $e \approx 0.967$) follow highly eccentric paths. Applied in Hohmann transfer orbits and interplanetary gravity-assist trajectories.
02 · Kepler's Second Law
Law of Equal Areas
$$\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{1}{2}r^2\frac{\mathrm{d}\theta}{\mathrm{d}t} = \frac{L}{2m} = \text{const}$$
- ①Physical Essence: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time; planets accelerate at perihelion and decelerate at aphelion.
- ②Formula Notation: $\mathrm{d}A/\mathrm{d}t$ is areal velocity, $L = m r^2 \dot{\theta}$ is orbital angular momentum, $m$ is planet mass.
- ③Theoretical Significance: The direct physical manifestation of conservation of angular momentum under a central force field (zero external torque $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = 0$).
- ④Boundary Conditions & Applications: Strictly holds for any central force potential $V(r)$. Governs orbital timing calculations, perihelion transit velocities, and binary pulsar orbital models.
03 · Roche Limit
Tidal Disruption Threshold
$$d \approx 2.44 R \left(\frac{\rho_M}{\rho_m}\right)^{1/3}$$
- ①Physical Essence: The minimum orbital distance to which a celestial body, held together only by its own self-gravity, can approach a primary body before tidal forces tear it apart.
- ②Formula Notation: $d$ is Roche distance, $R$ is radius of primary, $\rho_M$ is mean density of primary, $\rho_m$ is mean density of satellite ($2.44$ for fluid satellites; $\approx 1.26$ for rigid bodies).
- ③Theoretical Significance: Governs the boundary between stable natural satellites and planetary ring formation; explains the origin of Saturn's rings.
- ④Boundary Conditions & Applications: Comets penetrating inside a planet's Roche limit fragment violently (e.g., Comet Shoemaker-Levy 9 colliding with Jupiter in 1994). Essential for tidal disruption event (TDE) models around supermassive black holes.
2. Stars & Galaxies (4 Items)
04 · Stellar Luminosity Formula
Radiative Power Equation
$$L = 4\pi R^2 \sigma T_{\text{eff}}^4$$
- ①Physical Essence: The total electromagnetic power radiated by a spherical star equals its surface area multiplied by blackbody flux ($\sigma T_{\text{eff}}^4$).
- ②Formula Notation: $L$ is stellar bolometric luminosity (W), $R$ is stellar radius (m), $\sigma$ is Stefan-Boltzmann constant, $T_{\text{eff}}$ is effective surface temperature (K).
- ③Theoretical Significance: The core equation linking stellar radius, temperature, and luminosity on the Hertzsprung-Russell (H-R) diagram.
- ④Boundary Conditions & Applications: Assumes isotropic blackbody emission. Allows precise determination of red giant and white dwarf radii when combined with spectroscopic temperature measurements.
05 · Distance Modulus
Cosmic Distance Ladder Metric
$$\mu = m - M = 5\log_{10}\left(\frac{d}{10\text{ pc}}\right) = 5\log_{10} d - 5$$
- ①Physical Essence: Connects apparent brightness and intrinsic luminosity via the inverse-square law of radiation, establishing a cosmic distance measurement scale.
- ②Formula Notation: $m$ is apparent magnitude, $M$ is absolute magnitude (luminosity at standard 10 pc distance), $d$ is luminosity distance in parsecs (pc).
- ③Theoretical Significance: Fundamental yardstick for the astronomical distance ladder, enabling extragalactic distance determinations.
- ④Boundary Conditions & Applications: Requires interstellar dust extinction corrections ($m - M = 5\log d - 5 + A_V$). Used with Cepheid variable stars and Type Ia standard candle supernovae to measure cosmic expansion.
06 · Mass-Luminosity Relation
Main-Sequence Scaling Law
$$\frac{L}{L_\odot} \approx \left(\frac{M}{M_\odot}\right)^\alpha \quad (\alpha \approx 3.5 \sim 4.0)$$
- ①Physical Essence: For main-sequence stars, core thermonuclear fusion rates scale non-linearly with mass; a small mass increase yields an astronomical surge in luminosity.
- ②Formula Notation: $L_\odot$ and $M_\odot$ are solar luminosity and solar mass; scaling exponent $\alpha \approx 3.5$ for intermediate-mass stars ($2 M_\odot < M < 20 M_\odot$).
- ③Theoretical Significance: Derived from Eddington's stellar radiative equilibrium and opacity models; dictates the structure of the main sequence.
- ④Boundary Conditions & Applications: Applies strictly to hydrogen core-burning main-sequence stars. Breaks down for degenerated white dwarfs, neutron stars, and extreme giants.
07 · Main-Sequence Lifetime
Nuclear Fuel Exhaustion Timescale
$$\tau_{\text{MS}} \approx 10^{10} \left(\frac{M}{M_\odot}\right)^{-2.5} \text{ years}$$
- ①Physical Essence: Massive stars consume their hydrogen core fuel at furious rates, resulting in dramatically shorter lifespans than low-mass stars.
- ②Formula Notation: $\tau_{\text{MS}} \propto M / L \propto M / M^{3.5} = M^{-2.5}$; Sun's total main-sequence lifespan is approximately 10 billion years ($10^{10}\text{ yr}$).
- ③Theoretical Significance: Explains stellar population demographics, globular cluster turnoff ages, and galactic nucleosynthesis cycles.
- ④Boundary Conditions & Applications: A $20 M_\odot$ massive O-type star lives only $\sim 10$ million years before exploding as a core-collapse supernova, whereas a $0.1 M_\odot$ red dwarf burns fuel stably for trillions of years.
3. Geophysics (3 Items)
08 · Gravitational Acceleration & Field
Planetary Gravity Formulation
$$g = \frac{GM}{r^2} \implies g(h) \approx g_0 \left(1 - \frac{2h}{R_\oplus}\right)$$
- ①Physical Essence: Gravitational acceleration on a planetary surface is dictated by planetary mass and radius; decreases radially outward according to the inverse-square law.
- ②Formula Notation: $G$ is universal gravitational constant, $M$ is planetary mass, $r$ is distance from center, $g_0 \approx 9.81\text{ m/s}^2$ is standard surface gravity.
- ③Theoretical Significance: Governs atmospheric retention, orbital capture, and geophysical gravity surveying.
- ④Boundary Conditions & Applications: Sensitive micro-gravimeters measure free-air and Bouguer gravity anomalies to locate subterranean mineral deposits, magma chambers, and oil reservoirs.
09 · Earthquake Magnitude & Energy
Richter & Gutenberg-Richter Scale
$$\log_{10} E = 4.8 + 1.5 M$$
- ①Physical Essence: Logarithmic scaling linking seismic wave energy release to earthquake magnitude; an increase of 1.0 in magnitude multiplies released seismic energy by $10^{1.5} \approx 31.62$.
- ②Formula Notation: $E$ is radiated seismic energy in Joules (J), $M$ is earthquake magnitude. An increase of 2.0 in magnitude represents an exact 1000-fold increase in energy.
- ③Theoretical Significance: Quantitative physical bridge connecting seismograph wave amplitudes to tectonic rupture strain release.
- ④Boundary Conditions & Applications: Distinguishes Magnitude (a single global metric of total radiated energy) from Seismic Intensity (varying local levels of shaking and structural damage). Critical for building code standards and tsunami early warning networks.
10 · Coriolis Force
Geostationary Deflection Force
$$\mathbf{F}_c = -2m\, (\boldsymbol{\omega} \times \mathbf{v})$$
- ①Physical Essence: An apparent inertial force arising in rotating reference frames, acting perpendicular to both the rotation axis and the velocity vector of the moving mass.
- ②Formula Notation: $\mathbf{F}_c$ is Coriolis force vector, $m$ is moving mass, $\boldsymbol{\omega}$ is Earth's angular rotation vector ($7.292 \times 10^{-5}\text{ rad/s}$), $\mathbf{v}$ is relative horizontal velocity.
- ③Deflection Law: Horizontal motion deflects to the **right in the Northern Hemisphere** and to the **left in the Southern Hemisphere**; vanishes at the equator and peaks at the geographic poles.
- ④Boundary Conditions & Applications: Governs global atmospheric Hadley cells, jet streams, cyclonic storm rotation (hurricanes/typhoons), and ocean gyres. Required trajectory correction in long-range ballistic missile guidance.
4. Astronomical & Geophysical Constants & Units
| Quantity | Symbol | SI Base or Derived Unit | Conventional Astrophysical Unit | Equivalence / Value |
|---|---|---|---|---|
| Solar Mass | $M_\odot$ | Kilogram ($\text{kg}$) | Solar mass baseline | $1\ M_\odot \approx 1.988 \times 10^{30}\ \text{kg}$ ($\approx 333,000\ M_\oplus$) |
| Solar Luminosity | $L_\odot$ | Watt ($\text{W}$) | Solar radiative power | $1\ L_\odot \approx 3.828 \times 10^{26}\ \text{W}$ |
| Solar Radius | $R_\odot$ | Meter ($\text{m}$) | Solar radius baseline | $1\ R_\odot \approx 6.957 \times 10^8\ \text{m} \approx 695,700\ \text{km}$ |
| Earth Mass | $M_\oplus$ | Kilogram ($\text{kg}$) | Planetary mass baseline | $1\ M_\oplus \approx 5.972 \times 10^{24}\ \text{kg}$ |
| Earth Mean Radius | $R_\oplus$ | Meter ($\text{m}$) | Mean volumetric radius | $R_\oplus \approx 6371\ \text{km}$ (Equatorial $6378\text{ km}$, Polar $6357\text{ km}$) |
| Astronomical Unit | $\text{AU}$ | Meter ($\text{m}$) | Mean Sun-Earth distance | $1\text{ AU} \equiv 149,597,870,700\ \text{m} \approx 1.496 \times 10^{11}\ \text{m}$ |
| Parsec | $\text{pc}$ | Meter ($\text{m}$) | Stellar parallax baseline | $1\text{ pc} \approx 3.086 \times 10^{16}\ \text{m} \approx 3.262\text{ light-years}$ |
| Gravity Acceleration Unit | $\text{Gal}$ | Meter per second squared ($\text{m/s}^2$) | Gal and milliGal ($\text{mGal}$) | $1\text{ Gal} \equiv 1\ \text{cm/s}^2 = 0.01\ \text{m/s}^2$; $1\text{ mGal} = 10^{-5}\ \text{m/s}^2$ |
| Stefan-Boltzmann Constant | $\sigma$ | $\text{W}/(\text{m}^2\cdot\text{K}^4)$ | Blackbody constant | $\sigma \approx 5.670 \times 10^{-8}\ \text{W}/(\text{m}^2\cdot\text{K}^4)$ |
| Earth Angular Velocity | $\omega_\oplus$ | Radian per second ($\text{rad/s}$) | Rotational frequency | $\omega_\oplus \approx 7.292 \times 10^{-5}\ \text{rad/s}$ (Sidereal day $86164\text{ s}$) |