Orbital Velocity Formula: Definition, Equation, Formula and Derivation

Orbital Velocity

Edited By Vishal kumar | Updated on Jul 02, 2025 07:03 PM IST

Orbital velocity refers to the speed at which an object must travel to remain in a stable orbit around a celestial body, such as Earth. This fundamental concept in physics and space science is crucial for understanding how satellites, space stations, and spacecraft navigate the cosmos. When an object achieves orbital velocity, it counteracts the pull of gravity by continuously falling toward the planet while moving forward, effectively balancing its trajectory. This principle is not only pivotal for launching satellites that provide crucial communication and weather data but also for ambitious space missions exploring our solar system. In everyday life, the technology and principles derived from orbital velocity enhance our global positioning systems, and satellite TV, and even contribute to scientific research that drives advancements in technology and exploration.

This Story also Contains

What is Orbital Velocity?
Proof
Solved Examples Based on Orbital Velocity
Summary

What is Orbital Velocity?

Orbital velocity is the speed at which an object must travel to remain in a stable orbit around a celestial body, such as a planet or star. For an object to achieve this velocity, it must balance the gravitational pull of the body it is orbiting with its own forward motion. Essentially, the object is in constant free-fall towards the celestial body but moving forward fast enough that the curvature of its path matches the curvature of the body’s surface.

For the figure below

$
\begin{aligned}
& r_1=r_p=a-c \\
& r_2=r_a=a+c
\end{aligned}
$
If Eccentricity is given by

$
\text { (e) }=\frac{c}{a}
$

then The velocity of the planet at Apogee and Perigee in terms of Eccentricity is given by

$
\begin{aligned}
V_a & =\sqrt{\frac{G M}{a}\left(\frac{1-e}{1+e}\right)} \\
V_p & =\sqrt{\frac{G M}{a}\left(\frac{1+e}{1-e}\right)} \\
V_A & =\text { The velocity of the planet at apogee } \\
V_p & =\text { Velocity of perigee }
\end{aligned}
$

Proof

Let the mass of the sun be M and the mass of the planet be m

Applying the law of conservation of angular momentum at perigee and apogee about the sun
$
\begin{aligned}
& m v_p r_p=m v_a r_a \\
\Rightarrow & \frac{v_p}{v_a}=\frac{r_a}{r_p}=\frac{a+c}{a-c}=\frac{1+e}{1-e}
\end{aligned}
$
Applying the conservation of mechanical energy at perigee and apogee

Applying the conservation of mechanical energy at perigee and apogee

$
\begin{aligned}
& \frac{1}{2} m v_p^2-\frac{G M m}{r_p}=\frac{1}{2} m v_a^2-\frac{G M m}{r_a} \Rightarrow v_p^2-v_a^2=2 G M\left[\frac{1}{r_p}-\frac{1}{r_a}\right] \\
\Rightarrow & v_a^2\left[\frac{r_a^2-r_p^2}{r_p^2}\right]^2=2 G M\left[\frac{r_a-r_p}{r_a r_p}\right] \quad\left[\text { As } v_p=\frac{v_a r_a}{r_p}\right] \\
\Rightarrow & v_a^2=\frac{2 G M}{r_a+r_p}\left[\frac{r_p}{r_r}\right] \Rightarrow v_a^2=\frac{2 G M}{2 a}\left(\frac{a-c}{a+c}\right)=\frac{G M}{a}\left(\frac{1-e}{1+e}\right)
\end{aligned}
$
Thus the speeds of the planet at apogee and perigee are

$
v_a=\sqrt{\frac{G M}{a}\left(\frac{1-e}{1+e}\right)}, \quad v_p=\sqrt{\frac{G M}{a}\left(\frac{1+e}{1-e}\right)}
$

Solved Examples Based on Orbital Velocity

Example 1: The ratio of velocities of a planet at perigee and apogee $\frac{v_p}{v_a}$ is equal to

1) 1
2) $\frac{1-e}{1+e}$
3) $\frac{1+e}{1-e}$
4) $\frac{1+e^2}{1-e^2}$

Solution:

The velocity of the planet in terms of Eccentricity

$\begin{aligned} & V_a=\sqrt{\frac{G M}{a}\left(\frac{1-e}{1+e}\right)} \\ & V_p=\sqrt{\frac{G M}{a}\left(\frac{1+e}{1-e}\right)} \\ & V_A=\text { The velocity of the planet at apogee } \\ & V_p=\text { Velocityo fperigee } \\ & \text { wherein } \\ & \text { Eccentricity (e) }=\frac{c}{a} \\ & r_p=a-c \\ & r_a=a+c\end{aligned}$

By conservation of angular momentum,

$\begin{aligned} & m v_p r_p=m v_a r_a \\ & \Rightarrow \frac{v_p}{v_a}=\frac{r_a}{r_p}=\frac{a+c}{a-c}=\frac{1+e}{1-e}\end{aligned}$

Hence, the answer is the option (3).

Example 2: A satellite is in an all-optical orbit around planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances b/w the satellite and the planet at closest and farthest point is

1) $3:4$

2) $1:6$

3) $1:2$

4) $1:3$

Solution:

As the angular momentum remains conserved. So,
$
\begin{aligned}
& L_{\text {Nearest }}=L_{\text {Farthest }} \\
& m \times v_n \times r_n=m \times v_f \times r_f
\end{aligned}
$
Also,

$
V_f=\frac{V_n}{6}
$
As,

$
\begin{aligned}
& V_n \times r_n=V_f \times r_f \\
& V_n \times r_n=\frac{V_n}{6} \times r_f
\end{aligned}
$
So,

$
\frac{r_n}{r_f}=\frac{1}{6}
$

Hence, the answer is the option (2).

Summary

Orbital velocity is the speed required for an object to maintain a stable orbit around a celestial body, balancing gravitational pull with forward motion. It is determined using the formula v = $\sqrt{\frac{G M}{r}}$. where G is the gravitational constant, M is the mass of the celestial body, and r is the distance from its centre. For elliptical orbits, the velocities at apogee (farthest point) and perigee (closest point) are given by $V_a=\sqrt{\frac{G M}{a}\left(\frac{1-e}{1+e}\right)}$ and $V_p=\sqrt{\frac{G M}{a}\left(\frac{1+e}{1-e}\right)}$ respectively. Examples illustrate how these velocities relate to orbital parameters and conservation laws.

Frequently Asked Questions (FAQs)

1. How does the mass of an orbiting object affect its orbital velocity?

The mass of the orbiting object does not affect its orbital velocity. Orbital velocity depends on the mass of the central body and the distance of the orbit, but not on the mass of the orbiting object itself.

2. How does orbital velocity relate to the distance from the center of the Earth?

Orbital velocity decreases as the distance from the center of the Earth increases. This is because the gravitational force weakens with distance, so objects farther from Earth need less speed to maintain their orbit.

3. Can an object have an orbital velocity greater than escape velocity?

No, an object cannot have an orbital velocity greater than escape velocity. If an object's velocity exceeds escape velocity, it will leave orbit and escape the gravitational influence of the central body.

4. Why do satellites in lower orbits move faster than those in higher orbits?

Satellites in lower orbits move faster because they experience stronger gravitational pull from Earth. To maintain a stable orbit closer to Earth, they need higher velocities to counteract the stronger gravitational force.

5. What is the relationship between orbital period and orbital velocity?

Orbital period and orbital velocity are inversely related. As orbital velocity increases, the orbital period decreases. This is because faster-moving objects complete their orbits in less time.

6. What is the "vis-viva equation" and how is it used to calculate orbital velocity?

The vis-viva equation relates velocity, position, and orbital energy. It states that v² = GM(2/r - 1/a), where v is velocity, G is the gravitational constant, M is the central body's mass, r is the current distance, and a is the semi-major axis of the orbit. It's used to calculate velocity at any point in an elliptical orbit.

7. How does the mass distribution of a non-spherical body (like some asteroids) affect orbital velocity calculations?

For non-spherical bodies, the uneven mass distribution creates an irregular gravitational field. This can cause variations in orbital velocity and make orbits less stable. Precise orbital velocity calculations must account for these gravitational anomalies.

8. What is the "Oberth effect" and how does it relate to efficient changes in orbital velocity?

The Oberth effect states that using a rocket engine at high speed generates more useful energy than at low speed. In orbital mechanics, this means that velocity changes are most efficient when performed at the point of highest orbital velocity (usually at perigee).

9. What is orbital velocity?

Orbital velocity is the speed an object needs to maintain a stable orbit around another body, like a planet or star. It's the velocity that balances the gravitational pull of the central body, allowing the orbiting object to neither fall towards nor escape from it.

10. Why don't objects in orbit fall to Earth?

Objects in orbit don't fall to Earth because they're moving fast enough sideways that the curve of their fall matches the curve of the Earth. They're essentially in a constant state of free fall, but their horizontal velocity prevents them from getting closer to the Earth's surface.

11. What's the difference between escape velocity and orbital velocity?

Escape velocity is the minimum speed an object needs to break free from a planet's gravitational pull, while orbital velocity is the speed needed to maintain a stable orbit. Escape velocity is always √2 times greater than the orbital velocity at any given altitude.

12. How does the shape of an orbit affect orbital velocity?

In elliptical orbits, orbital velocity varies. It's fastest at the closest point to Earth (perigee) and slowest at the farthest point (apogee). In circular orbits, the velocity remains constant throughout the orbit.

13. How does Earth's rotation affect the launch of satellites into orbit?

Earth's rotation can assist in satellite launches. When launching eastward, the Earth's rotation provides an additional velocity boost, reducing the amount of fuel needed to achieve orbital velocity.

14. What is meant by "geosynchronous orbit"?

A geosynchronous orbit is one where a satellite's orbital period matches Earth's rotational period (about 24 hours). This means the satellite appears to stay fixed above a single point on Earth's equator.

15. How does air resistance affect orbital velocity in low Earth orbit?

Air resistance, even in the thin atmosphere of low Earth orbit, gradually slows down satellites. This decrease in velocity causes them to lose altitude over time, requiring occasional boosts to maintain their orbit.

16. What is a "transfer orbit" and how does it involve changes in orbital velocity?

A transfer orbit is an intermediate orbit used to move a satellite from one orbit to another. It often involves changing the satellite's velocity at specific points to raise or lower its orbit efficiently.

17. What is a "parking orbit" and why is it used in space missions?

A parking orbit is a temporary orbit where a spacecraft waits before moving to its final destination. It's often used in missions to other planets, allowing the spacecraft to adjust its trajectory or wait for the right alignment of planets before departure.

18. What is "orbital resonance" and how does it relate to orbital velocities?

Orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influences on each other due to their orbital periods being related by a ratio of small integers. This phenomenon can stabilize or destabilize orbits and affects the long-term evolution of orbital velocities.

19. What is the "Lagrange point" and how does it relate to orbital velocity?

Lagrange points are positions in space where the gravitational forces of two large bodies (like the Earth and Moon) balance each other. At these points, a smaller object can maintain its position relative to the two larger bodies with minimal orbital velocity adjustments.

20. How does atmospheric drag affect the orbital velocity of satellites in very low Earth orbit?

In very low Earth orbit, atmospheric drag significantly slows down satellites, reducing their orbital velocity. This loss of velocity causes the orbit to decay rapidly, requiring frequent boosts or resulting in the satellite's eventual re-entry into Earth's atmosphere.

21. What is a "frozen orbit" and how does it maintain stable orbital parameters?

A frozen orbit is designed to maintain nearly constant orbital elements (like eccentricity and argument of perigee) over time. It achieves this by carefully selecting the initial orbital velocity and other parameters to balance various perturbative forces.

22. How do space agencies use "Weak Stability Boundary" trajectories to minimize orbital velocity changes?

Weak Stability Boundary trajectories exploit areas where gravitational forces of different bodies nearly cancel out. By navigating these regions, spacecraft can make large changes in their orbit with minimal propulsion, effectively changing their orbital velocity very efficiently.

23. What is the "Oberth effect" and how does it relate to orbital velocity?

The Oberth effect is the phenomenon where a spacecraft gains more kinetic energy from its propulsion system when moving at high velocity. It's often used in orbital maneuvers to change orbits more efficiently by firing engines at the point of highest orbital velocity.

24. What is the "vis-viva equation" and how is it used in orbital mechanics?

The vis-viva equation relates the velocity of an orbiting body to its position in the orbit and the total energy of the orbit. It's used to calculate orbital velocity at any point in an elliptical orbit.

25. How do gravitational assists (slingshots) affect a spacecraft's orbital velocity?

Gravitational assists use a planet's gravity and motion to increase a spacecraft's velocity relative to the Sun. As the spacecraft swings by the planet, it gains kinetic energy and speed, allowing it to reach distant targets or change its orbit more efficiently.

26. What is the concept of "orbital energy" and how does it relate to velocity?

Orbital energy is the sum of kinetic and potential energy of an orbiting object. It remains constant in a stable orbit. As an object moves closer to Earth, its potential energy decreases and kinetic energy (velocity) increases, and vice versa.

27. How does the concept of orbital velocity apply to natural satellites like the Moon?

The Moon, like artificial satellites, orbits Earth at a specific orbital velocity. This velocity, combined with Earth's gravity, keeps the Moon in its elliptical orbit around our planet.

28. How does orbital velocity affect the apparent weight of astronauts in the International Space Station?

Astronauts in the ISS experience apparent weightlessness because they and the station are in constant free fall around Earth. Their orbital velocity creates a centrifugal effect that counteracts Earth's gravity, resulting in the floating effect we observe.

29. What is the "orbital velocity equation" and what factors does it depend on?

The orbital velocity equation is v = √(GM/r), where v is orbital velocity, G is the gravitational constant, M is the mass of the central body, and r is the orbital radius. It depends on the mass of the central body and the distance from its center.

30. How does orbital velocity relate to centripetal acceleration in circular orbits?

In circular orbits, the centripetal acceleration is provided by gravity. The orbital velocity is precisely what's needed to create the centripetal acceleration that keeps the object in orbit. This acceleration is given by v²/r, where v is the orbital velocity and r is the orbital radius.

31. How does the concept of orbital velocity apply to planets orbiting the Sun?

Just like satellites orbiting Earth, planets orbit the Sun at specific velocities determined by their distance from the Sun and the Sun's mass. Planets closer to the Sun have higher orbital velocities than those farther away.

32. What is "orbital decay" and how is it related to orbital velocity?

Orbital decay is the gradual decrease in a satellite's orbital altitude due to atmospheric drag or other forces. As a satellite's orbit decays, its orbital velocity increases slightly (due to being closer to Earth), but not enough to compensate for the energy lost to drag.

33. How do space agencies use the concept of orbital velocity to plan interplanetary missions?

Space agencies use orbital velocity calculations to determine launch windows, plan trajectory corrections, and design gravitational assists. Understanding how velocity changes throughout an orbit helps in planning fuel-efficient maneuvers for long-distance space travel.

34. What is the "Hohmann transfer orbit" and how does it optimize changes in orbital velocity?

The Hohmann transfer orbit is an elliptical orbit used to move between two circular orbits of different radii in the same plane. It minimizes the change in velocity needed to move between orbits, making it the most fuel-efficient method for orbital transfers.

35. How does the rotation of a planet affect the orbital velocity needed for a satellite to appear stationary?

For a satellite to appear stationary (geostationary), it must orbit at the same angular velocity as the planet's rotation. This requires a specific orbital velocity that depends on the planet's mass and rotational period, as well as the orbit's altitude.

36. How does the concept of orbital velocity apply to binary star systems?

In binary star systems, both stars orbit their common center of mass. Their orbital velocities depend on their masses and the distance between them, following the same principles as planets orbiting the Sun or satellites orbiting Earth.

37. What is the concept of "orbital inclination" and how does it affect orbital velocity?

Orbital inclination is the angle between the orbital plane and the equatorial plane of the central body. While inclination doesn't directly affect orbital velocity, changing inclination requires significant velocity changes, making it one of the most energy-intensive orbital maneuvers.

38. How does the "Oberth effect" allow for more efficient changes in orbital velocity?

The Oberth effect allows for more efficient velocity changes when a spacecraft is at its highest speed in an orbit (usually at perigee). Burning fuel at this point provides a greater change in kinetic energy and, therefore, a larger change in velocity for the same amount of fuel.

39. What is a "sun-synchronous orbit" and how is its orbital velocity determined?

A sun-synchronous orbit is a nearly polar orbit where the satellite passes over any given point of the Earth's surface at the same local solar time. Its orbital velocity is carefully calculated to ensure that the orbit precesses (rotates) at the same rate that the Earth moves around the Sun.

40. How does the concept of orbital velocity apply to artificial satellites around other planets?

The principles of orbital velocity apply to satellites around any celestial body. The orbital velocity needed depends on the mass of the planet and the orbital distance, following the same equation (v = √(GM/r)) used for Earth satellites.

41. What is "orbital eccentricity" and how does it affect orbital velocity throughout an orbit?

Orbital eccentricity measures how much an orbit deviates from a perfect circle. In more eccentric orbits, the orbital velocity varies more dramatically, being fastest at perigee (closest approach) and slowest at apogee (farthest point).

42. How do space agencies use "gravity assists" to change a spacecraft's orbital velocity?

Space agencies use gravity assists to change a spacecraft's velocity and trajectory by having it fly close to a planet. The spacecraft's path bends and its speed changes relative to the Sun, allowing it to gain or lose orbital energy without using fuel.

43. What is the relationship between orbital velocity and the "sphere of influence" of a planet?

The sphere of influence is the region where a planet's gravity dominates over the Sun's. As a spacecraft approaches a planet, its velocity relative to the planet becomes more important than its heliocentric velocity. Understanding this transition is crucial for planning orbital insertions.

44. What is a "Molniya orbit" and how does its orbital velocity vary?

A Molniya orbit is a highly elliptical orbit with an inclination of 63.4°. Its orbital velocity varies greatly, being very high at perigee and very low at apogee. This allows satellites to spend most of their time over high latitudes, useful for communications in polar regions.

45. How does the concept of "specific orbital energy" relate to orbital velocity?

Specific orbital energy is the sum of kinetic and potential energy per unit mass of an orbiting object. It's directly related to orbital velocity: as specific orbital energy increases, so does the object's ability to achieve higher orbits or escape velocity.

46. How does the concept of "delta-v" relate to changes in orbital velocity?

Delta-v represents the change in velocity needed to perform an orbital maneuver. It's a crucial concept in space mission planning, as it directly relates to the amount of fuel needed for orbital transfers, inclination changes, and interplanetary trajectories.

47. How does Earth's oblateness (non-spherical shape) affect satellite orbital velocities?

Earth's oblateness causes variations in gravitational force at different latitudes, which can affect orbital velocities. This leads to orbital precession and can be used advantageously in sun-synchronous orbits or must be compensated for in other types of orbits.

48. What is the concept of "orbital period" and how is it related to orbital velocity?

Orbital period is the time taken for an object to complete one orbit. It's inversely related to orbital velocity: higher velocity results in shorter periods. For circular orbits, the period T is given by T = 2π√(r³/GM), where r is the orbital radius.

49. How does the concept of "escape velocity" differ from orbital velocity?

Escape velocity is the minimum speed needed to break free from a body's gravitational influence, while orbital velocity is the speed needed to maintain a stable orbit. Escape velocity is always √2 times the circular orbital velocity at any given altitude.

50. What is a "hyperbolic trajectory" and how does its velocity compare to orbital velocity?

A hyperbolic trajectory is an open orbit where the object's velocity exceeds escape velocity. Its speed is always greater than the orbital velocity for a circular orbit at the same distance, allowing the object to escape the gravitational influence of the central body.