How to calculate the velocity of a satellite?

Orbital velocity is defined as the velocity at which a body circles around another body. The more substantial the body in the centre of attraction is, the higher the orbital velocity for a given height or distance. For an artificial or natural satellite, orbital velocity can be interpreted as the velocity necessary to maintain it in its orbit. Space organizations rely on it heavily to figure out how to launch satellites. It aids scientists in determining the speeds at which satellites must rotate around a planet or celestial body in order to avoid colliding with it.

Formula

The orbital velocity of a satellite orbiting around the Earth is determined by its height above the Earth. More is the orbital velocity, the closer satellite is to the Earth. It is equal to the square root of the product of the gravitational constant and mass of the body divided by the radius of its orbit.

where,
G is the gravitational constant,
M is the mass of object at centre,
R is the radius of the orbit.

Derivation

The formula for orbital velocity is derived through the concepts of gravitational force and centripetal force.
Suppose a satellite of mass m and radius r is orbiting circularly around planet Earth at an altitude h from Earth surface. Let us say, the mass and radius of Earth is M and R respectively. This implies that,
=> r = R + h ……. (1)
Now, we know that to make the satellite revolve in its orbit, a centripetal force of mV2/r is required. This force is provided by the gravitational force existing between the satellite and the earth.
So, we have
=> mV2/r = GMm/r2
=> V2 = GM/r
Using (1), we have
=> V2 = GM/(R + h)
As (R + h) ≈ R, we get
This derives the formula for orbital velocity of an object or satellite revolving around a planet.

Sample Problems

Problem 1. Find the orbital velocity of an object revolving around the planet Earth if the radius of Earth is 6.5 × 106 m, the mass of Earth is 5.9722 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 6.5 × 106

M = 5.9722 × 1024
Using the formula we have,
V = √(GM/R)
= (6.67408 × 10-11)(5.9722 × 1024)/(6.5 × 106)
= 29.8 km/s

Problem 2. Find the orbital velocity of an object revolving around the planet Mercury if the radius of Mercury is 2439.7 km, the mass of Mercury is 0.33 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 2439.7
M = 0.33 × 1024
Using the formula we have,
V = √(GM/R)
= (6.67408 × 10-11)(0.33 × 1024)/(2439.7)
= 47.4 km/s

Problem 3. Find the orbital velocity of an object revolving around the planet Venus if the radius of Venus is 6051.8 km, the mass of Venus is 4.87 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 6051.8
M = 4.87 × 1024
Using the formula we have,
V = √(GM/R)
= (6.67408 × 10-11)(4.87 × 1024)/(6051.8)
= 35 km/s

Problem 4. Find the orbital velocity of an object revolving around the planet Mars if the radius of Mars is 3389 km, the mass of Mars is 0.642 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 3389
M = 0.642 × 1024
Using the formula we have,
V = √(GM/R)
= (6.67408 × 10-11)(0.642 × 1024)/(3389)
= 24.1 km/s

Problem 5. Find the orbital velocity of an object revolving around the planet Jupiter if the radius of Jupiter is 69911 km, the mass of Jupiter is 1898 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 69911
M = 1898 × 1024
Using the formula we have,
V = √(GM/R)
= (6.67408 × 10-11)(1898 × 1024)/(69911)
= 13.1 km/s

Problem 6. Find the orbital velocity of an object revolving around the planet Saturn if the radius of Saturn is 58232 km, the mass of Saturn is 568 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 58232
M = 568 × 1024
Using the formula we have,

V = √(GM/R)
= (6.67408 × 10-11)(568 × 1024)/(58232)
= 9.7 km/s

Problem 7. Find the orbital velocity of an object revolving around the planet Uranus if the radius of Uranus is 25362 km, the mass of Uranus is 86.8 × 1024 kg and the gravitational constant G is 6.67408 × 10-11 m3 kg-1 s-2.

Solution:

We have,
G = 6.67408 × 10-11
R = 25362
M = 86.8 × 1024
Using the formula we have,
V = √(GM/R)
= (6.67408 × 10-11)(86.8 × 1024)/(25362)
= 6.8 km/s

This Earth orbit calculator helps you determine the orbital speed and orbital period of Earth's satellites at a given height above the average earth sea level. The fundamental laws that control the motion of Earth's satellites around Earth apply to the motion of planets, the moon, the earth's only natural satellite, and other satellites in the skies. Read on to learn about how different attributes of the satellite rotation are determined by its distance from the Earth's surface.

A satellite is a small object that orbits another larger object. The earth is a satellite because it orbits the sun. We consider the moon a satellite because it orbits the earth. However, most people think of an artificial satellite when they say satellite. These spacecraft are launched into an orbit around the earth or any other celestial body. Sputnik 1 was Earth's first artificial satellite. It was launched into an elliptical low Earth orbit at the height of 939 km. It orbited for three weeks before its batteries ran out.

There are tens of thousands of artificial satellites orbiting the earth. Some photograph our globe, while others photograph other planets, the sun, and other celestial bodies. These images aid scientists in studying the earth, the solar system, and the universe. Other satellites broadcast television and make phone calls all around the world.

This Earth orbit calculator has two features to help you calculate the orbital speed and period of rotation of an Earth's satellite launched at a specific height from the surface of the earth.

According to Nicolaus Copernicus, earth and the other planets orbit the Sun in circles. He also discovered that as the distance from the Sun increased, so did the orbital periods. Kepler later found these orbits to be ellipses, but the orbits of most planets in the solar system are roughly circular. The Earth's orbital distance from the Sun fluctuates by only 2%. The eccentric orbit of Mercury, whose orbital distance changes by approximately 40%, is an exception.

Determining a satellite's orbital speed and period is significantly easier in circular orbits. We use this simplification in the following calculations. This tool focuses on objects orbiting the Earth, but you can apply our findings to other situations. Therefore, the orbital speed (see orbital velocity calculator) of an earth's satellite is given as follows:

$speed=G⋅ME(RE+h)\small \text{orbital speed} = \sqrt{\frac{G \cdot M_{E}}{(R_{E}+h)}}$

where:

$G$ – Earth's gravitational constant;
$M_E$ – Earth's mass;
$R_E$ – Earth's radius; and
$h$ – Perpendicular distance of the satellite from the surface of the earth.

You can simply calculate this using this Earth orbit calculator by selecting the option Speed of the satellite, entering the height of the object rotating the earth, and voila, you have the orbital speed of the earth's satellite!

You need a certain amount of speed to reach the orbit (and stay there): learn how to calculate it with our delta- $v$ calculator!

Similar to the earth's orbit, all of the earth's satellites orbit the earth at a certain height with a constant period of rotation.

Therefore, the orbital period of an earth's satellite is given as follows:

$period=2π(RE+h)3G⋅ME\small \text{orbital period} = 2\pi \sqrt{\frac{(R_{E}+h)^3}{G \cdot M_{E}}}$

You can select the option Period of satellite rotation; enter the height of the object rotating the earth; you now know the orbital period of the earth's satellite!

Let's try to determine the orbital speed and period for the International Space Station (ISS)

Since the ISS orbits at a height of 400 km above Earth's surface, the radius at which it orbits is RE + 400 km.

You can just plug in the height of the ISS from the earth's surface as 400 km, and the calculator will compute its orbital period as 1.54 hrs and orbital speed as 7.672 km/s.

To calculate the orbital speed of an earth's satellite, you need to know the gravitational constant (G), earth's mass (M), earth's radius (R), and the height of rotation of the satellite (h).

The orbital speed is calculated as:

The orbital period of a satellite depends on the mass of the planet being orbited and the distance of the satellite from the center of the planet.

We can obtain the orbital period of a satellite T from Newton's form of Kepler's third law.

where G is the gravitational constant and M is the mass of the earth.