Expert: Philip Stahl - 10/20/2006

Question
Hi. I am studying for an astronomy test that is in a day and a half and am having difficulty with one of the review questions.  It would be greatly appreciated if you could show me how to figure this out!

Question:
  - The moon you have discovered takes 30 days to go around Jupiter and is 19 Jupiter diameters from the planet.  What is Jupiter's mass based on this info?

P=_______________yrs
'a'=_____________AU
Mj=______________solar masses

That is the question he asks on the review. Thanks so much for your time!

Answer
Hello,

In this problem you are basically using Kepler's 3rd or 'harmonic law' to obtain Jupiter's mass from Jupiter's moon's motion - knowing the mass of the Sun (1 solar mass).

From the harmonic law we know that the period squared is proportional to the semi-major axis of the orbit cubed:

P^2  ~  a^3

Where P is in years and a in astronomical units or AU. (1 AU = 1.496 x 10^8 km)

(Note: when we use AU and yrs. we are implicitly comparing the elements of the unknown system (e.g. moon going round Jupiter) with the known system of the Earth going round the Sun.)

For Earth, we know P = 1 year,   a = 1 AU

Keep this for reference.

For the moon going round Jupiter,

P(m) =  30 days

Or, in the units we need:

P(m) = 30 days/  365 days/year =  0.0822 yrs.

The distance or semi-major axis for the moon's orbit is 19 Jupiter diameters, or

a(m) =  19 x  (1.43 x 10^5 km) =  2.717 x 10^6 km

where the quantity in brackets is the known *equatorial diameter* of Jupiter, found from a table of planetary data.

The value above is a(m) = 0.018 AU since

(2.717 x 10^6 km)/ (1.496 x 10^8 km/ AU) = 0.018 AU

Now, according to the use of the 3rd law, the relations between period and semi-major axis obtaining for the Earth-Sun ought to be in the same proportion as the relation between period and semi-major axis for the moon of Jupiter and Jupiter. Thus, we can write the proportion:

M(S)/ M(J)  =   1/ (a^3/ P^2)

Now, M(S) is the mass of the Sun, and M(J) is the mass for Jupiter.

The numerator on the right side is simply the ratio of a^3 to P^2 for Earth (look at the values put up earlier, 9th line down:

(1 AU)^3/  (1 yr.) ^2  =   1  AU^3/yr^2

The denominator is for the moon of Jupiter's values, or:

(a^3)/ P^2  =  (0.018 AU)^3/  (0.0822 yr)^2

=  0.00086  AU^3/yr^2

Now, put all these values back into the proportion we obtained:

M(S)/ M(J)  =  1/ 0.00086

Note that the units of AU and yrs. cancel out between the numerator (values for Earth (1 AU^3/yr^2) and the denominator (values for moon of Jupiter, e.g. (0.018 AU)^3/  (0.0822 yr)^2)

What we have then at the end, is a simple proportion from which we can find the mass of Jupiter (M(J):

Cross multiplying and setting the terms equal:

M(J)  =  0.00086 M(S)

In other words, the mass of Jupiter is about 8.6 x 10^-4 of the Sun's.

Or, since the Sun = 1 solar mass (by definition) then Jupiter must be 0.00086 solar masses.

Hopefully, this helps!