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The small moon Amalthea orbits Jupiter at about the same distance in kilometers at which the moon Mimas orbits Saturn. Here distance means distance from the center of the planet. However, Mimas takes almost twice as long to complete one orbit as Amalthea. From this observation, what can you conclude about how Jupiter and Saturn differ in mass. Explain your answer. [Note: Start from the ratio between the masses and remember that the mass of a planet is much larger than the mass of its moon.]

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The moon Amalthea orbits Jupiter more or less at the same distance at which the moon Mimas orbits Saturn. However, Mimas takes almost twice as long to complete one orbit as Amalthea. What you can conclude from this is that Jupiter has a much larger mass than Saturn. Therefore, a moon that orbits at the same distance in this planet is affected by this difference in mass. A moon at this distance in Saturn takes almost twice as long to orbit as a moon in Jupiter. The larger mass of Jupiter causes the moon to orbit faster.

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Answer:

Based on the information provided, it can be deduced that:

  • The mass of Jupiter is about four times higher than the mass of saturn.

Explanation:

The orbital period (name given to the time it takes for a celestial body to perform the full orbit with respect to another celestial body) is usually calculated by Kepler's third law (whose formula is in the attached file), where:

  • a = the semi-axis of the largest orbit (that is, the longest radius in the ellipse made by the celestial body).
  • μ = G * M
  • G = Gravitational constant (which is always the same)
  • M = the mass of the most massive body (in this case the planet).

So that the example is quite simple we will invent values without units, so that we can see the difference in mass, first, since it is inferred that jupiter is four times more massive than saturn, then a mass will be chosen for Jupiter of 8 and a mass of 2 for Saturn (which maintains this relationship), the gravitational constant 6.674 * 10 ^ (- 11) and the semi-major axis, since it is the same as mentioned in the statement, a value will be given of 2.5, replacing these values you have:

1. T= 2*(3,1416) square root (2.5^(3))/(6.674 * 10 ^ (- 11))* 2 (in the case of Saturn-Mimas).

  • T= 2149719.433

2. T= 2*(3,1416) square root (2.5^(3))/(6.674 * 10 ^ (- 11))* 8 (in the case of Jupiter-Amalthea).

  • T= 1074859.716

And the relationship between the two values obtained is:

  • 2149719.433 / 1074859.716 = 2

Since the time ratio is double, it is found that the ratio between the mass of Jupiter and that of Saturn is 4:1 approximately.

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