A body orbiting around a planet is in equilibrium between two forces: the centrifugal force and the gravitational one.
where Fc is the centrifugal force, Fg is the gravitational force, m2 is the mass of the orbiting body, m1 is the mass of the planet, v is the speed of the body while orbiting, r is the distance of the body from the center of the planet, G is the gravitational constant:
The minus sign is because the force is attractive. The orbiting body is characterized by a potential energy called “gravitational”, caused by the field in which it is submerged. When speaking about a potential energy, mind immediately runs to Bernoulli’s theorem, that states that, for a liquid, the sum of potential, kinetic and pressure energy is constant. You
could think to the water contained in a basin that is situated on the top of a mountain, water that is forced to pass into a conduct, transforming, due to the altitude, the initial potential in kinetic and pressure energy. It will be later collected into a turbine in order to transform energy into electricity.
Bernoulli’s theorem is an application of the principle of conservation of energy. The energy changes in form but its total amount doesn’t change. In the case of a forced conduct, the potential energy of the water in altitude is transformed into kinetic energy and into pressure energy. 𝑈 = 𝑚 ∙ 𝑔 ∙ ℎ is the formula for potential energy while Ec=1/2 mv^ 2 is the kinetic energy. The principle of energy conservation could be, in the same way, applied to a body orbiting in the gravitational field of a planet. In a gravitational field the potential energy is expressed by the general formula:
The expression 𝑈 = 𝑚 ∙ 𝑔 ∙ ℎ (to which we all were used at school) is a particular case of the foregoing more general expression, and can be applied only in the case of h<<R where R is the Earth radius (this means that it is valid only at soil level). The energy conservation principle for a body in a gravitational field is expressed by the relation:
The total amount of energy is the sum of kinetic and potential energy. Thus, according to this relation, a body in free fall in the gravitational field of a planet will convert its potential energy into kinetic energy, but it will maintain constant the sum of the two and produce an increase in speed. The total amount of energy will remain the same. The opposite could
not be possible: please, consider a body that, with a certain starting potential, (but without possessing any kinetic energy) increases its potential even without receiving any external addition of energy. This result will be obtained only by diminishing the kinetic term.
In fact, in order to increase the potential energy, the kinetic has to decrease, but when this energy is already zero… it can’t become negative.
Thus, in a hydroelectric power plant, water is driven nightlong up on altitude to the eventual lake by spending electric energy that, during the night, has a lower cost. However, the necessity is to spend further energy in order to obtain water again and in a greater quantity of potential energy. It will be used daylong to produce electrical current (to be sold at a higher price). Another example could be relating to a chute on which you can slide downwards without any effort but, when going up the opposite direction, you have to add a good amount of energy with respect to the conservation law.
Let’s imagine a similar situation with respect to an orbiting body moving in an orbital direction only, and not in any whatsoever radial direction. The body, thus, possesses potential energy only, being its speed perpendicular to the radial direction, on which the gravitational
force is acting upon, and, incidentally, considering all factors, this datum cannot have any influence on calculation.
Consider now a meteorite that happens to hit the orbiting body in a direction tangential to the orbit, going, in this manner, to increase the speed of the satellite we are taking into consideration (but let us suppose with a very small increase). Wishing to make a comparison with the forced conduct, imagine you were trying to launch a little amount of water from
a bucket upward in the conduct just by impressing to the water a small kinetic energy (that won’t, anyway, be sufficient to win the gravitational force).
Similarly water rises a little through the conduct but, then, it necessarily falls down again. This will be due to the fact there is not force enough to pull water to the altitude of the lake. In the same way, the new speed acquired by the considered satellite will be v’= v+Δv, where Δv is very small. Since, however, v’>v, the centrifugal force grows a little, according
to the relation:
The gravitational force, on the other hand, will remain the same. The equilibrium will be lost, when the satellite acquires a force
sufficient to drag it away from the planet. The resulting force will originate a speed in the radial direction, in such a manner that the kinetic energy, moving in the radial direction, would start increasing and the body would start departing from the planet. Since the distance grows, as far as the body is departing from the planet, Fc’ decreases in proportion to 1/r.
In the same time the gravity force
will decrease faster, in proportion to 1/r2. The body would accelerate more and more and the kinetic energy would grow very fast, no energy by the exterior being added (or very low energy). At the same time, since the body would be departing from the planet, the potential energy would grow… In the same manner the kinetic energy would increase, and so the total energy.
And here the paradox starts. The body should immediately stop orbiting around the planet and be trapped in another orbit, just because, departing from the original planet, the potential energy would increase, making the kinetic decrease, according to the conservation energy principle. But, however, the centrifugal force, continuing to be higher than the
gravitational force, the body should keep departing with a spiral movement from the planet. Actually it would be creating energy from nothing, not respecting even the conservation of mechanical energy.
Objection: the potential energy doesn’t grow but decreases when the satellite departs from the planet. In fact it has to be considered zero at an infinite distance from the planet itself.
Answer: you have to consider the sign of the energy that, in this case, is minus. It is a negative energy that continuously grows till zero. Let’s calculate, just to give a practical example, the total energy of a rocket of 10000kg that moves at the speed of 500 m/sec and is departing from the planet at a radius r1=500 km and r2=1000 km. There is no propulsion: the rocket is departing from the Earth due to a previous thrust and, since it is moving in the void, nothing is slowing its speed:
The potential energy, considering the mass of the Earth being M=5.97E24 kg,
at 500 km height: U = -Gm1m2/r = -7.96E12 Joule
at 1000 km height: U = -Gm1m2/r = -3.98 E12 Joule
7.96 seems to be greater than 3.98 but it is not due to the minus sign. The total
energy will be at 500 km:
And at 1000 km:
Etot= Ec+U = -3.978 E12 Joule . That is much bigger. The total amount of energy is not conserved.