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Astrophysics Formulas
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Drake Equation
N = R_* \times f_p \times n_e \times f_l \times f_i \times f_c \times L. It is used to estimate the number of active, communicative extraterrestrial civilizations in the Milky Way galaxy.
Schwarzschild Radius
r_s = \frac{2GM}{c^2}. It defines the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface would equal the speed of light.
Hubble's Law
v = H_0 \times d. It relates the distance to a galaxy (d) with the velocity (v) at which it is receding from us due to the expanding universe, where is the Hubble constant.
Kepler's Third Law
P^2 = \frac{a^3}{M}. It expresses the relationship between the period (P) of a planet's orbit and its semi-major axis (a) with the mass (M) of the central body.
Stefan–Boltzmann Law
P = \sigma A T^4. It states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (P) is directly proportional to the fourth power of the black body's thermodynamic temperature T.
Einstein's Field Equations
G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}. It describes the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
Chandrasekhar Limit
M_{ch} = \frac{\omega_3^0 \cdot (\pi)^{\frac{2}{3}} \cdot (\hbar c)^{\frac{3}{2}}}{(G)^{\frac{3}{2}} \cdot (m_H)^2}. It is the maximum mass of a stable white dwarf star beyond which it would collapse into a neutron star or black hole.
Friedmann Equations
H^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}. These equations govern the expansion of space in homogeneous and isotropic models of the universe within the context of General Relativity.
Wien's Displacement Law
\lambda_{max} = \frac{b}{T}. It relates the temperature of a black body to the peak wavelength of its emitted spectrum.
Virial Theorem
2\langle K \rangle = -\langle U \rangle. It relates the average kinetic energy (K) of particles in a system to the average potential energy (U), and is used in astrophysics to analyze the stability of star clusters and galaxies.
Planck's Law
B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}. It describes the electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T.
Tolman-Oppenheimer-Volkoff (TOV) Equation
\frac{dp}{dr} = -\frac{G}{r^2} \frac{(\rho + \frac{p}{c^2})(m + \frac{4\pi r^3 p}{c^2})}{1 - \frac{2Gm}{rc^2}}. It represents the equilibrium condition for a spherically symmetric mass of isotropically emitting fluids of stars in general relativity.
Saha Ionization Equation
\frac{n_{i+1} n_e}{n_i} = \frac{2}{\Lambda^3} e^{-\frac{E_{ion}}{k_B T}}. It is used to describe the physical conditions of stars by determining the level of ionization at different temperatures.
Ideal Gas Law
PV = nRT. In astrophysics, it is used to describe the state of a hypothetical ideal gas to understand the behavior of stellar atmospheres.
Parseval's Theorem
\int_{-\infty}^{\infty} |f(x)|^2 dx = \int_{-\infty}^{\infty} |F(k)|^2 dk. It states that the total energy of a wave described by function f(x) is equal to the total energy in its Fourier transform F(k), used in astrophysics for signal processing.
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