Orbital Motion Formula Sheet

Constants §

Universal Gravitation Constant §

$$G=6.67\times 10^{-11}{\text{ Nm}}^2{\text{/kg}}^2$$

Earth Constants §

Acceleration of Gravity On Earth §

$$g_e=9.8{\text{ m/s}}^2$$

Mass of the Earth §

$$m_e=5.98\times 10^{24}\text{ kg}$$

Radius of the Earth §

$$r_e=6.37\times 10^6\text{ m}$$

Conversion Constants §

Miles to Meters §

$$1\text{ mi}=1609\text{ m}$$

Formulas §

Note

Many of the values (i.e., kg, seconds) do not suit all use cases, but are just default SI units.

Newton’s Law of Universal Gravitation §

• $F_g$ = gravitational force, in Newtons
• $G$ = universal gravitation constant
• $m_1,m_2$ = masses of two objects, in kg
• $r$ = center to center distance between the objects, in meters

$$F_g=G\frac{m_1{m}_2}{r^2}$$

Weight §

• $W$ = weight, in Newtons
• $m$ = mass, in kilograms
• $g$ = acceleration of gravity, in $m/s^2$ (on Earth - $9.8$)

$$W=mg$$

Gravitational Field Strength §

On Earth §

• $g$ = gravitational field strength, in $m/s^2$
• $G$ = universal gravitation constant
• $m_e$ = mass of the Earth, in kg
• $r_e$ = radius of the Earth, in meters

$$g=\frac{G{m}_e}{{r_e}^2}$$

For large spherical bodies §

• $g$ = gravitational field strength, in $m/s^2$
• $G$ = universal gravitation constant
• $m$= mass, in kg
• $r$ = radius, in meters

$$g=\frac{Gm}{r^2}$$

Orbits §

Centripetality §

Centripetal Force §

• $F_c$ = centripetal force, in Newtons
• $m$ = mass, in kg
• $v$ = velocity, in $m/s$
• $r$ = radius, in meters

$$F_c=\frac{m{v}^2}{r}$$

Centripetal Acceleration §

(Rate of change of velocity due to centripetal acceleration)

• $a_c$ = centripetal acceleration, in $m/s^2$
• $v$ = velocity, in $m/s$
• $r$ = radius, in meters

$$a_c=\frac{v^2}{r}$$

Orbital Period §

• $2\pi r$ = distance traveled, in meters
• $T$ = orbital period
• $v$ = velocity, in $m/s$

$$2\pi r=vT$$

Altitude Incorporation §

• $r_{TOT}$ = total radius
• $r_{planet}$ = radius of planet
• $\text{altitude}$ = altitude

$$r_{TOT}=r_{planet}+\text{altitude}$$

Kepler’s Law of Harmony §

• $T_1,T_2$ = orbital periods of objects $1$ and $2$
• $r_1,r_2$ = radii of objects $1$ and $2$

$$\frac{{T_1}^2}{{r_1}^3}=\frac{{T_2}^2}{{r_2}^3}$$

Derived Kepler’s Law of Harmony §

Derived by setting $F_c$ = $F_g$ to find $v$, then substituting into Kepler’s Law of Harmony.

$$\frac{r^3}{T^2}=\frac{Gm}{4{\pi }^2}$$

Gravitational Potential Energy §

• ${PE}_g$ = gravitational potential energy, in Joules
• $G$ = universal gravitation constant
• $m_1,m_2$ = masses of objects, in kg
• $r$ = center-to-center distance between $m_1$ and $m_2$, in meters

$${PE}_g=G\frac{m_1{m}_2}{r}$$

Escape Speed §

Derived by setting $KE$ (kinetic energy) = $U$ (gravitational potential energy)

$$v=\sqrt{\frac{2G{m}_1}{r_1}}$$

Tidal Force §

• $m_1,m_2$ = masses of objects, in kg
• $a$ = distance, from top to bottom, in meters
• $r$ = center-to-center distance between $m_1$ and $m_2$, in meters

$$F_t=\frac{4G{m}_1{m}_{2}a}{r^3}$$