Question

understanding mars geological and climatic history is crucial for future exploration. equally important is understanding planets orbital mechanics, which govern their movement around the sun. using keplers laws and the data provided, which claim is best supported by the data in the table regarding the orbital mechanics of mars? 1. mars has a shorter orbital period than earth because it has a lower mass. 2. jupiters longer orbital period compared to earth is due to its greater average distance from the sun. 3. earths orbital period is longer than mars because earth has a higher gravitational field strength near the planets surface. 4. mars and jupiter have similar orbital periods because they both have higher masses than earth.

Explanation:

Step1: Recall Kepler's third law

Kepler's third law states that \( T^{2}\propto a^{3}\), where \(T\) is the orbital period and \(a\) is the semi - major axis (related to the average distance from the Sun).

Step2: Analyze each option

• Option 1:

Mars has an orbital period of \(1.88\) years (longer than Earth's \(1.00\) year). Also, orbital period is related to the semi - major axis (distance from the Sun) by Kepler's laws, not directly to mass. So this option is incorrect.

• Option 2:

Jupiter has a semi - major axis of \(5.204\) AU (much larger than Earth's \(1.000\) AU). By Kepler's third law (\(T^{2}\propto a^{3}\)), a larger semi - major axis (average distance from the Sun) leads to a longer orbital period. Jupiter's orbital period (\(11.86\) years) is longer than Earth's (\(1.00\) year). This option is correct.

• Option 3:

Earth's orbital period (\(1.00\) year) is shorter than Mars' (\(1.88\) years). Also, gravitational field strength near the planet's surface is not related to the orbital period of a planet around the Sun (Kepler's laws relate orbital period to semi - major axis). So this option is incorrect.

• Option 4:

Mars has an orbital period of \(1.88\) years and Jupiter has \(11.86\) years (not similar). Also, Mars has a mass of \(0.642\times 10^{24}\) kg (less than Earth's \(5.972\times 10^{24}\) kg). So this option is incorrect.

Answer:

1. Jupiter’s longer orbital period compared to Earth is due to its greater average distance from the Sun.