Answer 1:
That is a great question! You have hit upon the
Nbody problem in physics, a mathematically
fascinating and complex area of study. In fact,
there are entire graduatelevel courses offered
for how to solve problems with N > 2 things in it!
The particular problem you have mentioned is
the threebody gravitational problem.
Although interactions between bodies could be
other than gravitational (these are also
investigated at the quantum level, but these
interactions are instead Coulomb forces, among
other interactions that are important at the
quantum level), we'll focus on the planets since
it's the most familiar. You are indeed correct in
pointing that there would be gravitational forces
between all bodies involved (i.e., Earth, Jupiter,
and the sun). Of course, we have measured what the
orbits of the sun, Jupiter, and earth look like
(including the other planets as well), but the
problem is definitely a lot more interesting than
that!
Can this be measured?
In theory yes. This can be captured using
Newton's Laws of Gravitation. That is, in
perhaps the form you first see it
click
here to see the equation
The above equation is actually a differential
equation for the position of a planet as a
function of time. Velocity is the rate of change
of your position (i.e., its derivative);
acceleration is the rate of change of velocity,
and thus acceleration is the double derivative of
position. That is
click
here, please
If you solve the differential equation, you'll
(hypothetically) find an analytical solution for
position as a function time (i.e. an orbit
of sorts). With three bodies, you end up with
differential equations that are coupled with each
other. For example, making planet 3 heavier
affects not only its interaction with planet 1 but
also with planet 2, and this reflected
mathematically with differential equations. I'm
not sure how comfortable you are with differential
equations, but if you are curious about the math,
you can get a taste of it here
; it also has the restricted three body
problem, which matches fairly closely with the
problem you proposed.
What happens?
It turns out there is no analytical solution to
the general case of a three body problem.
There have been many studies that look at certain
cases. In the more general case, anything could
happen. And I mean that quite literally. The path
that any arbitrary body in a threebody system is
dependent on the initial conditions, and might not
even be a stable orbit (i.e., it goes flying off
into the distance), which is characteristic of any
system that can be described by chaos
theory (an entire mathematical branch in its
own right). What does chaos in math look
like? You can get a taste by looking at the
diversity of possible orbits in a threebody
problem with this Java applet
here . This is still an area of active
research. In fact, just last year, some physicists
announced the discovery of
13 new solution classes for the three body
problem!
The problem you mentioned is an example. For
most intents and purposes, the mass of the Earth
can be considered negligibly small to either that
of the sun or Jupiter. The interactions between
the sun and Jupiter will essentially look like
that of a twobody problem (assuming there are no
other planets in our solar system). The orbit of
the Earth depends on where it starts out initially
relative to the sun and Jupiter. One example can
be found this
set of Java applets that has certain initial
conditions set for you to see what happens, which
happens to have the restricted threebody problem,
though with the earth, moon, and a satellite.
Perhaps with initial positions closer to what our
solar system looks like, you can watch this video
here of the the same
kind of restricted threebody problem .
The ThreeBody Problem Elsewhere
I mentioned that the Nbody problem is applicable
to the quantum scale, particularly for studying
solids at the atomic level. This is indeed what I
do everyday! With a solid, things become more
complicated. In a solid, you need to consider
Coulomb interactions between each of the nuclei
with electrons, nuclei with nuclei, and electrons
with electrons. Electrons also have additional
quantum behaviors (e.g., obey Pauli exclusion) and
so add more complication. Imagine having to solve
the Nbody problem for not just three but 1 X 10
^{26} things! That's a lot of things, but
still barely one mole of anything! Quite far from
modeling a solid. It turns out, this is
mathematically intractable and not worth the time
to solve. What people have done over the years are
very clever approximations and reformulations of
the problem that let us imitate a macroscopic
solid with just a few hundred atoms. This is a
common theme in the Nbody problem (i.e., using
alternate methods). In fact, problems with four or
more bodies become mathematically intractable to
solve.
Hope this helps!
Best,

Answer 2:
You are absolutely right in thinking that there
would be gravity between all of the interacting
bodies (sun, earth and Jupiter). All objects take
part in the gravitational force as long as they
have mass. When dealing with more than two
objects, the gravitational force between each must
be added together along their defined directions.
For example if you line up Jupiter, Earth, and the
sun all in a line, earth would feel two opposing
forces, one from the sun (from the right) and one
from Jupiter (from the left).
The equation for calculating the gravitational
force is defined as:
F = G x m_{1} x m_{2} /
r^{2}
where G is the gravitational constant,
m_{1} and m_{2} are
the mass of the two
interacting bodies you are looking at, and
r is the distance between the two objects.
It is important to note the dependence of the
force on the mass of the two bodies interacting,
as well as the distance between them. Much higher
masses give higher gravitational forces, where
larger distances give lower gravitational forces.
I challenge you to use this equation and look up
some values for the masses of the planets and the
sun, as well as their distances to each other.
When you have the three individual forces
calculated (between
1. earth and sun,
2. earth and jupiter,
3. jupiter and sun),
compare their magnitudes and directions and see
which ones are largest and most relevant.
I hope this helps you to realize why the
planets stay in orbit around the sun and planets
do not get pulled too strongly by other planets
that would drastically change their orbit around
the sun.
Thanks for the great question!
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