# 16.B: Some Equations and Constants

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## Physical Constants

Table B.1: 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

Table B.2: Moments of inertia, all about axes of symmetry through the center of mass.
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

Table B.3: Characteristics of the Sun, Earth and Moon.
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) $$1.4 \cdot 10^{3}$$ $$5.5 \cdot 10^{3}$$ $$3.3 \cdot 10^{3}$$
Table B.4: Properties of a number of solar system objects. Equatorial radii and masses are compared to those of Earth (see Table B.3). Orbital properties are around primary (the sun for (dwarf) planets, the planet for moons). Orbital radii and periods for planets again compared to Earth, for moons in kilograms and days. Rotation period for all objects in days. Inclination and axial tilt in degrees. Data from NASA planetary fact sheets [31].
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

$\nabla f(\boldsymbol{r})=\nabla f(x, y, z)=\left(\begin{array}{c} \partial_{x} f \\ \partial_{y} f \\ \partial_{z} f \end{array}\right)=\left(\frac{\partial f}{\partial x} \hat{x}+\frac{\partial f}{\partial y} \hat{y}+\frac{\partial f}{\partial z} \hat{z}\right)$

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:

$v_{x}=\frac{u+v_{x}^{\prime}}{1+u v_{x}^{\prime} / c^{2}} \quad \text { (longitudinal) } , v_{y}=\frac{1}{\gamma(u)} \frac{v_{y}^{\prime}}{1+u v_{x}^{\prime} / c^{2}} \quad \text { (transversal) }$

This page titled 16.B: Some Equations and Constants is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Timon Idema (TU Delft Open) via source content that was edited to the style and standards of the LibreTexts platform.