Calculating net gravitational force on Uranus

I want to calculate the net/total gravitational force which is being exerted on the planet Uranus. There are many objects in our solar system like the Sun, other planets, moons of other planets, moons of the planet Uranus, asteroid belt also has many objects. So to calculate the net force on Uranus, should I consider all these objects?

As we know, the gravitational force between two bodies is given by F = G $$\frac{m1 m2}{r^2}$$, where m1 and m2 are the masses of the two bodies and r is the distance between them. Sun is far away from Uranus, but the mass of the Sun is also more. So on what basis should I choose which objects will contribute significantly and which objects contribution will be negligible while calculating the net force exerting on the planet Uranus? Individually calculating the actual force on the planet Uranus due to each of the above mentioned objects and then adding them all will be very lengthy problem. Is there any better way of doing it?

• Are you looking for the actual forces at a given instant, or a rough estimate based on some assumed distance (like James K's answer)? For the forces at a given instant, you will need an ephemeris to compute the locations of all of the objects for the given time. Aug 17, 2022 at 14:17
• What would be the proper way to consider "net gravitational force"? Would it be a simple addition of all total forces or should directionality come into consideration? As an overly simplified example consider two large and equal masses M separated by distance X and exactly right between them at X/2 is the third body. Would the net gravity at the third body be 0 or 2 x ((g x M1 x M3)/X^2). Aug 17, 2022 at 18:09

In the particular case of Uranus, (and considering Uranus and its moons a single system) You only need to consider the Sun + perturbations by the other outer planets.

As a more general procedure, consider the acceleration from each other planet at minimum distance. Here is the calculation for Uranus (in SI units):

Object GM r (min) a = GM/r²
Sun 1.33E+20 2.73E+12 1.78E-05
Jupiter 1.27E+17 1.92E+12 3.46E-08
Saturn 3.79E+16 1.23E+12 2.53E-08
Neptune 6.84E+15 1.47E+12 3.17E-09
Mars 4.28E+13 2.48E+12 6.96E-12
Earth 3.99E+14 2.58E+12 5.99E-11
Ceres 6.26E+10 2.29E+12 1.20E-14

The right column shows the acceleration of Uranus as a result of each body. You can see that the Sun is by far the largest contributor. The other giant planets have secondary effects, everything else is a couple of orders of magnitude less. In principle, you could get more accurate results if you included perturbations by the Earth, Mars etc. But you should then also consider other factors, like the fact that the Sun isn't a perfect sphere.

Of course, any cut-off point is in some sense arbitrary, but in the case of modelling the orbit of Uranus, it would be natural to only consider the effects of the outer planets.

• You could also do the same calculations for its moons and see that several of them dominate individually.
– pela
Aug 17, 2022 at 12:13