# Orbital Velocity Calculator

Created by Davide Borchia
Last updated: Sep 07, 2022

Motions in space follow a small set of rules in a rather elegant fashion: with our orbital velocity calculator, you will learn some of them.

Follow us in a discovery of the fundamental laws of orbital mechanics: we will learn the fundamental parameters of an orbit, how to calculate orbital mechanic's most essential quantities, with a focus on the orbital speed: the equation and formulas to work it out in various situations and much more! To conclude, we will give you some examples of how to calculate the orbital velocity: the formula for a satellite around Earth and Earth around the Sun.

## Orbits: calculate the mechanics of planets, satellites, and all the rest

Every object acting only under the effect of the gravitational force moves in space following a restricted set of possible motions: the conics. These curves, obtained from slicing a cone with a plane, have two closed elements, ellipses and circles.

An ellipse is a closed figure defined by two quantities, the major and minor axis. We can calculate a measure of the distance of an ellipse from ideal circularity: we call it eccentricity. Here is its formula:

$e = \sqrt{1-\frac{b^2}{a^2}}$

Where:

• $e$ — The eccentricity;
• $a$ — The semi-major axis; and
• $b$ — The semi-minor axis.

Circular orbits are part of a perfectionist subset of elliptical orbits and are rarely encountered in Nature. They have zero eccentricity.

Once you have defined the parameters of your orbit, you need to know something about the players: the gravitational force acts mainly on masses; you need to know the value of the masses of the orbiting and central body.

For many applications, one of the two values is negligible. Think of a few hundred kilogram satellite vs. the Earth!

Together, the masses of the bodies in analysis define the gravitational parameter, which will be of utmost importance during the calculations of the orbital velocity formula, and much more. Starting from Newton's gravitational force equation, we calculate the gravitational parameter with the formula:

$\mu = G\cdot M\cdot m$

Where $G = 6.6743 \times 10^{-11}\ {\text{m}^3}/(\text{kg}\cdot\text{s}^2)$ is the gravitational constant.

## Formula for the orbital velocity at any given point in an orbit: the vis-viva equation

To calculate the orbital speed, we introduce the vis-viva equation

$v^2 = \mu \cdot \left(\frac{2 }{ r} - \frac{1}{a}\right)$

Where:

• $\mu$ — The gravitational parameter;
• $r$ — The distance between the bodies at the desired moment; and
• $a$ — The semi-major axis.
• $v$ — The orbital speed calculated at the distance $r$.

Notice that the result of the equation is the orbital velocity squared.

The vis-viva equation gives valid results for every point of an orbit. However, we often focus on specific points.

## How to calculate the orbital speed at the periapsis and the apoapsis

An elliptical orbit is characterized by two fundamental distances, the periapsis and the apoapsis. The periapsis is the point of the orbit where the two bodies are the closest to each other. The apoapsis is the point where the distance is maximum.

Thanks to Kepler's second law of planetary motion, we know that the orbital speed depends on the position in orbit. First, we calculate the distance at the apoapsis, $r_{\text{a}}$, and periapsis, $r_{\text{p}}$. To do so, we use a simple set of equations:

\begin{align*} r_{\text{a}} + r_{\text{p}} &= 2\cdot a\\ r_{\text{a}} \cdot r_{\text{p}} &=2\cdot b^2 \end{align*}
• Apply the vis-viva equation for the orbital speed using the value of the distances $r_{\text{a}}$ and $r_{\text{p}}$;
• The results of the formula for the orbital speed confirm Kepler's second law: at the periapsis, the body moves faster than at the apoapsis.

🙋 Learn more about the three Kepler's laws of orbital motion at our Kepler's third law calculator.

## The orbital speed formula in action: calculating the orbital velocity for a satellite orbiting Earth

Take an artificial satellite, any satellite is suitable for this example, but we will consider the International Space Station. With its $440$ tons, the ISS is the heaviest man-made object in Earth's orbit. However, its mass is negligible compared to the one of Earth:

$M_{\oplus} = 5.972 \times 10^{24}\ \text{kg}$

We can calculate the gravitational parameter by multiplying the mass of Earth and the gravitational constant: Kepler's third law.

\begin{align*} \mu &= 5.972 \!\times \!10^{24}\ \text{kg} \\ &\cdot 6.6743 \!\times \!10^{-11}\ \frac{\text{m}^3}{\text{kg}\cdot\text{s}^2}\\ & =3.986\!\times\!10^{14}\ \frac{\text{m}^3}{\text{s}^2} \end{align*}

The last data we need is the orbital height of the ISS. The semi-major axis of the station is $6738\ \text{km}$, but mind that the height over Earth's surface is on average only $420\ \text{km}$: to calculate the orbital speed, you need to use the distance between centers of mass!

Insert in our orbital velocity calculator the value of semi-major axis and eccentricity ($e=0.0002985$): you will find the values of periapsis and apoapsis. If you fill in the masses of the bodies, we will calculate the remaining quantities for you. Explore the results: the ISS has an orbital speed of around $7.7\ \text{km}/\text{s}$: more than $22$ times the speed of sound. To fly from San Francisco to New York, the station takes a bit less than 9 minutes!

🙋 Discover more about the fundamentals of orbital mechanics with our orbital period calculator and earth orbit calculator. And if you want to jump on a rocket, visit our space travel calculator!

## Use the equation for orbital velocity to calculate Earth's speed

Let's try a classic: how to calculate the orbital speed of Earth — among its other orbital parameters.
Earth has a semi-major axis $a = 149.60\times10^{6}\ \text{km}$ and a beautifully low eccentricity $e =0.0167$. From these parameters, we can find the distance at the periapsis and apoapsis:

\begin{align*} r_{\text{a}} &= 152.1\times10^{6}\ \text{km}\\ r_{\text{p}} &= 147.1\times10^{6}\ \text{km} \end{align*}

Now let's calculate the orbital speed at these points. Apply twice the vis-viva equation:

\begin{align*} v_{\text{a}} &= 29.3\ \frac{\text{km}}{\text{s}}\\ v_{\text{p}} &= 30.29\ \frac{\text{km}}{\text{s}} \end{align*}

We are speeding thorugh the cosmos at a neck-breaking speed of almost $90$ times the speed of sound.

Davide Borchia
Ellipse parameters
Semi-major axis
au
Semi-minor axis
au
Ecce
Masses
Star mass
Suns
Satellite mass
Earths
Standard gravitational parameter
cu mi
/ s²
Orbital parameters
Distance at apoapsis
au
Distance at periapsis
au
Velocity at apoapsis
mi/s
Velocity at periapsis
mi/s
Period
yrs
Total energy
* 10³³
J
Vis-viva equation
Distance
au
Velocity
mi/s
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