Selected Questions - Answers From All Experts Astronomy Forum (Celestial Mechanics)

Question -

How do they find the positions of planets  (mathematically)?

Answer:

Determining the position of any planet in the orbital plane is not trivial
at all, nor can it be answered in a simple response format. The reason is
that whole courses in what is called celestial mechanics are devoted to
it.

Students cannot take this course without first having studied (or taken
courses in) calculus, advanced calculus and differential equations, as
well as advanced computer programming. Hence, if you haven't taken or been
exposed to such - it is impossible to convey what is necessary to compute
orbit positions!

Basically, position determination entails identifying - calculating the
seven elements that make up a given planetary orbit, including: the
eccentricity of orbit (how much it deviates from circularity); the
semi-major axes (or mean distance from the Sun); the longitude of the
ascending node, the inclination (i) of the orbit, the true anomaly, and
the mean anomaly, and the time of perihelion passage (T) or alternately
the 'mean motion' n = M/ (t - T).

Getting more specific, the diagram  below shows assorted orbital elements

for a planet of mass m2 (the Sun is m1) .

In the diagram, w   is the argument of the perihelion, W is the

longitude of the ascending node, f is the true anomaly and i is the

inclination of the orbit. The critical or key parameter

here is h, the angular momentum vector for the orbiting system. 

The true anomaly is perhaps most difficult to obtain since it requires one
take a Fourier expansion (again, this is taught in advanced calculus!) of
the difference (w - M), e.g. the mean anomaly (M) from the true anomaly.

The student must be  able to ascertain that  W  = M - w  (difference

between mean anomaly and argument of the perihelion) where M can

be obtained from a table based on observations, and w can be obtained

using a Fourier expansion of the mean anomaly, M:

w  = M + (2e – e³ / 4) sin M + 5 e²/4 sin 2M + ... etc.

One then needs to obtain the energy constants C1, C2, C3.

Where:

C1/ h = sin W sin (i)
C2/ h = - cos W sin (i)
C3/h = cos(i)

The magnitude h, of the angular momentum vector is:

h = r x r’ =

(y z’ - z y’)

(z x’ - x z’) = (C1 C2 C3)

(x y’ - y z’)

so (r x r’) = (C1/ h, C2/ h, C3/h)

Since the inclination of Earth's orbit to the ecliptic  (i) is known

(23.5 deg) and therefore cos(i) can be determined, then sin(i) can be

found as well.  Also h can be determined, since: h = C3 / cos(i) =

(GMm a (1 – e²)¹² where all the constants are known (a = semi-major axis

of orbit, e = eccentricity of orbit)

(The energy equation is: ½V ² - u/r = C, and the C's - energy integration

constants- are  found from this.)

Once W is known, C1 and C2 and C3 are known, and the student can

compute the position of a planet, say Jupiter, forty or so years in the future.

If you would like to find out more details of planetary position computation

and exactly how it's done - there is an excellent and detailed site (with tutorials)

at:

http://www.stjarnhimlen.se/comp/tutorial.html