Can the moon crash onto Earth ? Can it escape its gravity ?
Is it a coincidence that the moon revolves around Earth ? What if its speed suddenly increases by a certain amount due to some cause, will it escape Earth’s gravity? What if it decreases, will it crash into Earth ?
To answer this question, we will see what astrophysicists and engineers usually do with satellites.
To revolve around Earth, a satellite is put into a low orbit, a medium orbit, or a high orbit depending on its mission. This choice is not definitive. For example, a space agency can decide to move a communication satellite from a low orbit into a higher orbit in order to serve more customers.
To move from orbit 1 to orbit 2, an efficient technique in terms of energy consumption is used. It is called a Hohmann transfer.
In a Hohmann transfer, the satellite is accelerated in orbit 1 at the point of exit A, this acceleration decreases the effect of gravity on it, hence the satellite goes to a higher intermediate orbit which we call the transfer orbit. Both the point at which the satellite is accelerated and the amount of acceleration are determined such that the orbit of transfer intersects with orbit 2.
When the satellite is at point B, which is the point of intersection of the intermediate orbit with orbit 2, it is again accelerated by the amount that will keep it in orbit 2.
This technique optimizes fuel consumption by making the satellite exit orbit 1 tangentially and land in orbit 2 tangentially as well. The tangent exit and enter only require to change the magnitude of velocity of the satellite at points A and B without changing its direction. Changing the direction of the satellite would require more energy because of the centrifugal force that will continuously tend to deflect the satellite outwards.
Like when it is in orbit 1 and orbit 2, the motion of the satellite in the intermediate orbit is also due to gravitational force and does not require to use its engines. The energy consumed during the whole operation is the energy consumed during the accelerations at point A and point B.
Answering our question
From this example, the answer to our question is clearer. If the moon is suddenly accelerated, it is not necessary that it will escape earth’s gravity. A priori, the acceleration will only take it into a higher orbit. The same thing applies to its deceleration, it may only move to a lower orbit.
However, there is a specific velocity, that if the moon reaches, it will escape Earth’s gravity. We call this the escape velocity and it has the following formula:
where R is the distance between the earth’s center and the moon’s center, M is Earth’s mass, and G is the gravitational constant.
If we consider the mean of the distance between the moon and Earth, which is 384,400km then the minimum velocity needed for the moon to escape Earth is 1,4km/s.
The previous formula is not very precise but it’s a good approximation. It is obtained by applying the principle of conservation of energy in an initial and a final point.
The initial point is where the moon escapes, and the final point is where the moon will be very far from Earth that its distance from it can be considered to be infinite.
When the moon gains the escape velocity at a distance d = 384,400km, it has a kinetic energy equal to:
where m is the moon’s mass.
Its potential energy (that comes from Earth’s gravity) is:
When the moon escapes Earth and becomes at an infinite distance from it (theoretically), its potential energy is equal to 0 as Earth is no more applying its gravity on it. Its speed becomes also negligible. Why? Because the farther the object is from Earth, the lower is the gravitational force applied to it, and hence the lower is its acceleration according to Newton’s second law of motion. This is a well known paradox that says when a satellite accelerates its speed decreases. This is because when it accelerates, it goes to a higher orbit in which it feels less gravity and hence it slows down. Therefore, at the final point the kinetic energy is null as well.
Applying the conservation of energy principle, we have:
which gives:
Conclusion
Indeed, it is not easy that the moon quits its orbit around Earth. However, if we consider the theory of the Big Bang to be true, then the presence of the moon around Earth remains one of the mysteries for which scientists only speculate the causes!
Bibliography
- https://en.wikipedia.org/wiki/Hohmann_transfer_orbit
- https://earthobservatory.nasa.gov/features/OrbitsCatalog
- https://en.wikipedia.org/wiki/Escape_velocity
- https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/cami/library/online_libraries/aerospace_medicine/tutorial/media/III.4.1.5_Maneuvering_in_Space.pdf
