# Orbital Velocity Calculator

By Dominik Czernia, PhD candidate
Last updated: May 24, 2021

The orbital velocity calculator is an advanced tool that you can use to find parameters of planet motion on elliptical orbit (or on circular orbit). Do you want to learn what is the orbital velocity of Earth or what is Jupiter's orbital period? With our orbital velocity calculator you can quickly estimate:

• orbital radius of the planets,
• orbital energy of the planets,
• orbital speed of the planets,
• orbital period of the planets.

You surely know that planets orbit around stars, but have you ever wondered what is an elliptical orbit? As the name suggests, planets do not move around a circle, but on an ellipse. Use ellipse calculator to learn what is an ellipse and how to estimate all the parameters of an ellipse. With our elliptical orbit definition, you will be able to find what is Earth's elliptical orbit including its apoapsis and periapsis distances and Earth's orbital velocity at those points. The apoapsis and periapsis are respectively the farthest and the closest points at which the Earth can come to the Sun (in many calculations we often use just an average orbital radius of Earth). In this certain case we can call them aphelion and perihelion since Earth orbits around the Sun.

Our orbital velocity calculator makes use of several astrophysical relations. For example, to describe the geometrical properties of Earth's elliptical orbit you only need to know the semi-major and semi-minor axes or the eccentricity of Earth's orbit. If you want to find more details, like Mars' sidereal period or its satellite speed, you should also enter the masses of the star (e.g., Sun) and the satellite (e.g., Mars, Moon). We can compute it with the Kepler laws and the orbital velocity equation. Finally, with the so-called vis-viva equation, you can find out what is, for example, the orbital velocity of the Earth.

At the end of this orbital velocity calculator you can find the planet data table with orbital parameters of different planets of the Solar System. Are you interested in problems of astrophysics? Try our escape velocity calculator or rocket equation calculator!

## What is elliptical orbit? - elliptical orbit definition

Elliptical orbit (or elliptic orbit) is a Kepler orbit with an eccentricity between one and zero (an orbit is circular when its eccentricity equals zero). The orbital eccentricity is a parameter that characterizes the shape of the orbit. The higher is its value, the more flattened ellipse becomes. It is linked to the other two important parameters: semi-major axis and semi-minor axis (see figure below) with the following eccentricity formula:

`e = √(1 - b²/a²)`

where

• `e` is the eccentricity,
• `a` is the semi-major axis,
• `b` is the semi-minor axis.

Knowing the above parameters, we can determine the closest possible distance of the satellite (planet) to a star - periapsis, and the farthest possible distance of the satellite (planet) to a star - apoapsis. Now, when you have learned what is an elliptical orbit, we can estimate distances to a star at periapsis and apoapsis with the following elliptical orbit definition:

`ra + rp = 2 * a` and `ra * rp = b²`

where

• `ra` is the distance between star and satellite in apoapsis,
• `rp` is the distance between star and satellite in periapsis.

The only thing we need do is to solve the above system of orbit equations. You can, of course, use our orbital velocity calculator to do it! For example, you can analyze Earth's elliptical orbit. The semi-major axis of Earth's orbit is `a = 1 a.u.` (`1 a.u.` is one astronomical unit which is an average distance between the Earth and the Sun), and the semi-minor axis of Earth's orbit is `b ≈ 0.99986 a.u.`. Therefore eccentricity `e = 0.0167`, distance at aphelion (apoapsis) `ra = 1.0167 a.u.` and distance at perihelion (periapsis) `rp = 0.9833`.

## Vis-viva equation and orbital velocity equations (apoapsis and periapsis)

In astrophysics, the vis-viva equation allows us to model the motion of orbiting satellites. It provides orbital speed of a satellite at a given point of an elliptic orbit as well as an orbital velocity of a satellite in periapsis and apoapsis. The vis-viva equation is as follows:

`v² = μ * (2 / r - 1 / a)`

where

• `v` is the relative satellite speed (compared to the speed of a star),
• `μ` is the standard gravitational parameter `μ = G * (M + m)`,
• `r` is the distance between the star and the satellite,
• `a` is the semi-major axis of the elliptical orbit,
• `G` is the gravitational constant `G = 6.674 * 10 ^(-11) N * m² / kg²`,
• `M` is the mass of the star,
• `m` is the mass of the satellite.

The standard gravitational parameter `μ` is expressed in SI units `m³/s²`, but in the scientific literature and spacecraft navigation we usually use the units of `km³/s²`. With above orbital speed formula you can also estimate the satellite speed at apoapsis (`r = ra`) and periapsis (`r = rp`). With orbital speed calculator you can compute that the orbital velocity of the Earth at periapsis is `vₚ = 30.29 km/s` and at apoapsis is `vₐ = 29.295 km/s`.

## Kepler laws, the orbital period of planets and the orbital energy of planets

You have learned, from the previous sections, how to use orbit equation to estimate, e.g., the orbital radius of the Earth. But how can you calculate orbital period of planets? The answer derives from the Kepler laws of planetary motion:

1. First law of Kepler - the law of orbits. Every planet of the Solar System moves around the Sun on an elliptical orbit. It is important to remember that the Sun is not at the center but at one of the two foci points of the elliptical orbit. Check our ellipse calculator if you want to find out what is focus point.
2. Second law of Kepler - the law of areas. At equal intervals, a line segment between a planet and the Sun sweeps out equal areas (area speed is constant).
1. Third law of Kepler - the law of periods. You can obtain the simplified form of the third law of Kepler by comparing the centripetal force to the gravitational force. The full corresponding formula states that:

`T² = 4 * π² * a³ / μ`

where

• `T` is the orbital period of a satellite,
• `a` is the semi-major axis of an ellipse,
• `μ` is the standard gravitational parameter `μ = G * (M + m)`,
• `G` is the gravitational constant `G = 6.674 * 10 ^(-11) N * m² / kg²`,
• `M` is the mass of the star,
• `m` is the mass of the satellite.

We encourage you to try our orbital velocity and calculate the orbital period of the Earth (`a = 1 a.u.`). You will see that it equals precisely one year.

You can use our calculator to compute the orbital energy of planets too. Under standard assumptions, the total orbital energy of planets is negative, and it is independent of the eccentricity of the ellipse. It can be calculated with the following orbital energy formula:

`E = - G * M * m / (2 * a)`

You should note that the orbital energy of the Earth (and other planets too) is large (`~ 10³³ J`). Check our energy conversion calculator if you want to find out what are other orders of magnitude of energies that can be found in nature.

## Planet data table - Orbital radius of Earth

Are you looking for Neptune's orbital period? Or maybe you're interested in Jupiter's orbital speed velocity? If you want to quickly estimate orbital properties of different planets in the Solar System, you can use our orbital velocity calculator alongside the below planet data table.

planet semi-major axis (astronomical units) eccentricity mass (Earth's masses)
Mercury 0.387 0.2056 0.0553
Venus 0.723 0.0068 0.815
Earth 1 0.0167 1
Mars 1.524 0.0934 0.107
Jupiter 5.203 0.0484 317.83
Saturn 9.537 0.0542 95.159
Uranus 19.191 0.0472 14.536
Neptune 30.069 0.0086 17.147
Pluto 39.482 0.2488 0.0021

With the information from the above planet data table, you can compute every orbital parameter provided by our orbital speed calculator. Try with different planets! Just remember that the distance between a planet and the Sun is expressed in astronomical units (average orbital radius of the Earth) and the mass is in units of Earth's masses.

Dominik Czernia, PhD candidate
Ellipse parameters
Semi-major axis
au
Semi-minor axis
au
Eccentricity
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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