If we throw a stone in the air, it will fall another time due to the gravitational force of the Earth. If we throw it more strongly, it will arrive higher. If we repeat this process (with enough energy), it would come a moment in which the stone would find itself in a constant fall, that to say, it would be in orbit. However, the stone wouldn’t have escaped the terrestrial gravitational field yet. To do it, it would need to go to an even greater speed, in order to escape from the Earth’s orbit and not to fall again on its surface.
The escape velocity is the one that permits a propelled object to get out of the gravitational field of a planet, star or any other celestial body. Once exceeded, the celestial body can’t retain anything. This speed varies in function of every planet, moon, or star. The more mass the object from which we want to escape has, the greater its escape velocity will be.
From Newton’s equations, we deduce the equation (explained later) that lets us calculate the escape velocity of a body. In the case of the Earth, it is 11,2 km/s, and it is the speed rockets have to reach to be able to send probes to other planets. In the case of the Moon, the escape velocity is much lower, at only 2.4 km/s, and in the case of Jupiter, the required speed to get out of its gravitational field is 59,6 km/s.
Escape velocity’s equation
To calculate the required speed that any object has to reach to get out, for example, of the gravitational field of a planet, we use two masses: M, which represents the planet’s mass, and m, which is the mass of the propelled object. It is considered that the kinetic energy of this object has to equalise the potential gravitational energy which ties it to the planet from which it wants to escape. Any kinetic energy that the object reached, that was equal to or greater than the gravitational force of the planet, will let it get out of its gravitational field.
In the following equations, m represents the object’s mass, and M the celestial body’s mass. From Newton’s equations, we deduce that:
isolating from here the v(esc):
where:
- v(esc) is the escape velocity.
- G is the universal gravitational constant.
- M is the mass of the celestial body (planet, moon, star…) from which the object wants to escape.
- r is the distance between the masses centre of the body and the point from where we are calculating the v(esc). (If we find ourselves on the surface, r will be equal to the body’s radius).
From this equation, we deduce that the greater the celestial body’s mass would be, the greater the escape velocity will be. But the bigger the radius (r), the smaller will be the required speed to exit the gravitational field of the body. So, the escape velocity depends on how strong the gravity is from where the object is thrown. The higher up the object is in orbit, the lower the escape velocity.
The escape velocity doesn’t depend on the mass of the launched object. What, in fact, will be required the more massive the object is will be a more powerful rocket, but the speed it will have to reach is the same.
The escape velocity and the atmosphere
The smaller a celestial body might be, the lower its escape velocity will be, and this determines the atmosphere of planets, moons… As I said previously, the escape velocity of the moon is 2,4 km/s, and this explains why the moon hasn’t an atmosphere, as molecules acquire easily greater speeds than the one of escape and the Moon can’t hold back them. In the case of the Earth, this phenomenon takes place with the lightest molecules, like the ones of hydrogen and helium, which escape from the terrestrial atmosphere and make the Earth lose weight.
However, the Sun is so massive, and the escape velocity so big (620 km/s), that even hydrogen and helium can’t escape. These two elements (but mostly hydrogen), are the ones that, subjected to enormous pressures at the centre of the Sun, fusion themselves, releasing in the process very big quantities of energy. This process is known as nuclear fusion, and it is from where the Sun extracts its energy.
Obviously, the mass isn’t the only factor that determines the atmosphere of a planet or moon. Solar and cosmic rays, which are electrically charged particles, can evaporate and send to space the atmospheres of many celestial bodies. This phenomenon takes place on Mars, where each time there is less atmosphere for this reason.
Unsurpassable escape velocity
As we have seen, escapes velocities are diverse and vary in each celestial body. However, there are so massive places in the universe, where not even going at the speed of light, which is the fastest in the universe, we could escape. These places are the dark holes. Even if they emitted light, this one would be captured and wouldn’t get out.
Once surpassed what we name the event horizon, there would be no option to go back. At that moment, it would be impossible to get out of the dark hole. In the centre of one of these beasts, there is a place named singularity, where the deformation of space-time would be infinite.
Knowing the mass and the escape velocity of anybody, we can know the radius that the body would have to occupy to be converted into a dark hole. In the case of the Earth, it would convert into a dark hole if all its mass was concentrated in a sphere of 2 centimetres of diameter.
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