16.B: Some Equations and Constants
- Page ID
- 17467
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Physical Constants
| Name | Symbol | Value |
|---|---|---|
| Speed of light | \(c\) | \(3.00 \cdot 10^8 m/s\) |
| Elementary charge | \(e\) | \(1.60 \cdot 10^{-19} C\) |
|
Electron mass |
\(m_e\) | \(9.11 \cdot 10^{-31} kg = 0.511 MeV/c^2\) |
| Proton mass | \(m_p\) | \(1.67 \cdot 10^{-27} kg = 938 MeV/c^2\) |
| Gravitational constant | \(G\) | \(6.67 \cdot 10^{-11} N \cdot m^2 /kg^2\) |
| Gravitational acceleration | \(g\) | \(9.81 m/s^2\) |
| Boltzmann's Constant | \(k_B\) | \(1.38 \cdot 10^{-23} J/K\) |
| Planck's Constant |
\(h\) \(\hbar = h /2 \pi\) |
\(6.63 \cdot 10^{-34} J \cdot s\) \(1.05 \cdot 10^{-34} J \cdot s\) |
Moments of Inertia
| Object | Moment of Inertia |
|---|---|
| Thin stick (length L) | \(\frac{1}{12} M L^2\) |
| Ring of hollow cylinder (radius R) | \(M R^2\) |
| Disk or solid cylinder (radius R) | \(\frac{1}{2} M R^2\) |
| Hollow sphere (radius R) | \(\frac{2}{3} M R^2\) |
| Solid sphere (radius R) | \(\frac{2}{5} M R^2\) |
| Rectangle (size \(a \times b\)), perpendicular axis | \(\frac{1}{12} M (a^2 + b^2)\) |
| Rectangle (size \(a \times b\)), axis parallel to side b | \(\frac{1}{12} M a^2\) |
Solar System Objects
| Sun | Earth | Moon | |
|---|---|---|---|
| Mass (kg) | \(1.99 \cdot 10^{30}\) | \(5.97 \cdot 10^{24}\) | \(7.35 \cdot 10^{22}\) |
| Mean radius (m) | \(6.96 \cdot 10^{8}\) | \(6.37 \cdot 10^{6}\) | \(1.74 \cdot 10^{6}\) |
| Orbital period (s) |
\(6 \cdot 10^{15}\) (200 My) |
\(3.16 \cdot 10^{7}\) (365.25 days) |
\(2.36 \cdot 10^{6}\) (27.3 days) |
| Mean orbital radius (m) | \(2.6 \cdot 10^{20}\) | \(1.50 \cdot 10^{11}\) | \(3.85 \cdot 10^{8}\) |
| Mean density (kg/m3) | \(1.4 \cdot 10^{3}\) | \(5.5 \cdot 10^{3}\) | \(3.3 \cdot 10^{3}\) |
| Name | Symbol | Equatorial radius | Mass | Mean orbit radius | Orbital period | Inclination | Orbital eccentricity | Rotation period | Confirmed moons | Axial tilt |
|---|---|---|---|---|---|---|---|---|---|---|
| Mercury |  |
0.382 | 0.06 | 0.39 | 0.24 | 3.38 | 0.206 | 58.64 | 0 | 0.04 |
| Venus |  |
0.949 | 0.82 | 0.72 | 0.62 | 3.86 | 0.007 | -243.02 | 0 | 177.36 |
| Earth |  |
1 | 1 | 1 | 1 | 7.25 | 0.017 | 1 | 1 | 23.44 |
| Moon |  |
0.272 | 0.0123 | 384399 | 27.32158 | 18.29-28.58 | 0.0549 | 27.32158 | 0 | 6.68 |
| Mars |  |
0.532 | 0.107 | 1.52 | 1.88 | 5.65 | 0.093 | 1.03 | 2 | 25.19 |
| Ceres | 0.0742 | 0.00016 | 2.766 | 4.599 | 10.59 | 0.08 | 0.3781 | 0 | 4 | |
| Jupiter |  |
11.209 | 317.8 | 5.2 | 11.86 | 6.09 | 0.048 | 0.41 | 69 | 3.13 |
| Io | 0.285 | 0.015 | 421600 | 1.769 | 0.04 | 0.0041 | 1.769 | 0 | 0 | |
| Europa | 0.246 | 0.008 | 670900 | 3.551 | 0.47 | 0.009 | 3.551 | 0 | 0 | |
| Ganymede | 0.423 | 0.025 | 1070400 | 7.155 | 1.85 | 0.0013 | 7.155 | 0 | 0 | |
| Callisto | 0.378 | 0.018 | 1882700 | 16.689 | 0.2 | 0.0074 | 16.689 | 0 | 0 | |
| Saturn |  |
9.449 | 95.2 | 9.54 | 29.46 | 5.51 | 0.054 | 0.43 | 62 | 26.73 |
| Titan | 0.404 | 0.023 | 1221870 | 15.945 | 0.33 | 0.0288 | 15.945 | 0 | 0 | |
| Uranus |  |
4.007 | 14.6 | 19.22 | 84.01 | 6.48 | 0.047 | -0.72 | 27 | 97.77 |
| Oberon | 0.119 | 0.00051 | 583519 | 13.46 | 0.1 | 0.0014 | 13.46 | 0 | 0 | |
| Neptune |  |
3.883 | 17.2 | 30.06 | 164.8 | 6.43 | 0.009 | 0.67 | 14 | 28.32 |
| Triton | 0.212 | 0.00358 | 354759 | 5.877 | 157 | 0.00002 | 5.877 | 0 | 0 | |
| Pluto |  |
0.186 | 0.0022 | 39.482 | 247.9 | 17.14 | 0.25 | 6.39 | 5 | 119.59 |
| Charon | 0.095 | 0.00025 | 17536 | 6.387 | 0.001 | 0.0022 | 6.387 | 0 | unknown | |
| Haumea | 0.13 | 0.0007 | 43.335 | 285.4 | 28.19 | 0.19 | 0.167 | 2 | unknown | |
| Makemake | 0.11 | unknown | 45.792 | 309.9 | 28.96 | 0.16 | unknown | 1 | unknown | |
| Eris | 0.18 | 0.0028 | 67.668 | 557 | 44.19 | 0.44 | unknown | 1 | unknown |
Equations
B.4.1 Vector Derivatives
Gradient:
\[\nabla f(\boldsymbol{r})=\nabla f(x, y, z)=\left(\begin{array}{c}
\partial_{x} f \\
\partial_{y} f \\
\partial_{z} f
\end{array}\right)=\left(\frac{\partial f}{\partial x} \hat{x}+\frac{\partial f}{\partial y} \hat{y}+\frac{\partial f}{\partial z} \hat{z}\right)\]
Divergence:
\[\nabla \cdot \boldsymbol{v}=\left(\partial_{x}, \partial_{y}, \partial_{z}\right) \cdot\left(\begin{array}{c}
v_{x} \\
v_{y} \\
v_{z}
\end{array}\right)=\frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}\]
Curl:
\[\nabla \times \boldsymbol{A}=\left(\partial_{x}, \partial_{y}, \partial_{z}\right) \times\left(\begin{array}{c}
A_{x} \\
A_{y} \\
A_{z}
\end{array}\right)=\left(\begin{array}{c}
\partial_{y} A_{z}-\partial_{z} A_{y} \\
\partial_{z} A_{x}-\partial_{x} A_{z} \\
\partial_{x} A_{y}-\partial_{y} A_{x}
\end{array}\right)\]
B.4.2 Special Relativity
Lorentz transformations for the coordinates of a frame S' that moves with a speed u in the positive x-direction of frame S:
\[\begin{align}
x^{\prime} &=\gamma(u)\left(x-\frac{u}{c} c t\right) \\
c t^{\prime} &=\gamma(u)\left(c t-\frac{u}{c} x\right) \\
\gamma(u) &=\frac{1}{\sqrt{1-(u / c)^{2}}}
\end{align}\]
Velocity addition in a relativistic system:
\[v_{x}=\frac{u+v_{x}^{\prime}}{1+u v_{x}^{\prime} / c^{2}} \quad \text { (longitudinal) } , v_{y}=\frac{1}{\gamma(u)} \frac{v_{y}^{\prime}}{1+u v_{x}^{\prime} / c^{2}} \quad \text { (transversal) }\]