Newton's Law of Gravitation
Newton's Law of Universal Gravitation is perhaps the single most important physical law in the study of orbital mechanics, for the interaction between the spacecraft and the celestial bodies found on its way will determine the trajectory.
Any two bodies of masses m1, m2 separated by a distance d are attracted to each other with a force derived from the following expression :
Where G is the constant of Gravitation (G=6.67 E-11 N m2/kg2), m1 and m2 the mass of the objects. d is the distance that separates them. We see then that if the masses of the objects are big, the force will increase, and if they are at a short distance the force will increase again.
As we have said, this force is barely perceptible in normal cases. If this was not so, we would be hauled into other persons or buildings. However, when one or two of the objects possess an enormous mass, as in the case of the Earth, the force is important.
If we substitute in the formula our mass and the mass of the Earth respectively, and then the distance to the center of the Earth, we will find that the force found will correspond exactly to our weight.
This same formula explains why our weight is reduced when we ascend. As the distance to the center of the Earth is increased, the force decreases. Nevertheless, this decrease is hardly noticeable because the magnitude of the ascent is insignificant as compared to the radius of the Earth.
A person will weigh less in the Moon due to lower gravity. The force with which it pulls a person is 6 times less than in the Earth. The weight in any given planet depends on the gravity. However, the mass of an object is invariable. The mass measures the "quantity of matter" or substance in an object and that will obviously not change in another planet's environment.
Distribution of Gravity in Near Earth Space
In order to start thinking of regions in space in terms of their gravity, it is a good idea to map the distribution of gravity on Near Earth Space.
If the values are substituted in the formula for gravitation at a point x,y (taking the Earth as the origin of the coordinate system) then the value of gravity at any point x,y can be found by :
where G = 6.67 x 10^ -11 N m2 / kg
and MEarth = 5.98 x 10^24 kg
Another interesting visualization is that of the gravity distribution in the Earth-Moon system. The gravity of the Moon at a point x,y (again using the Earth as origin and placing the Moon on the x-axis) would be :
where R (mean distance from the Earth to the Moon)= 3.85 x 10^8m
and MMoon = 7.36 x 10^22 kg