Question

The artificial satellite Vanguard 1 was found to have a period of revolution of 134 minutes. It approached the Earth to distance of 660 km and receded to a distance of 4023 km. Calculate the mass of the Earth with respect to the Sun’s mass if the semi-major axis of the Earth’s orbit is 149·5 × 106 km, the radius of the Earth is 6372 km and the year contains 365·25 mean solar days.

Answer 1

Answer on Question #54957, Physics / Astronomy | Astrophysics

We need draw the artificial satellite is at perigee at A (the point in its orbit nearest the Earth's centre E) and at apogee at A' (when it is farthest from the Earth's centre):

The size of the major axis AA' of the ellipse is given by:

$$\mathrm {A A} ^ {\prime} = 4 0 2 3 + 2 \times 6 3 7 2 + 6 6 0 = 1 7 4 2 7 \mathrm {k m}$$

So that the semi-major axis $a1$ is of length $8713 \cdot 5 \, \text{km}$ . The period of revolution is $T1 = 134$ minutes. For the Earth orbit:

$$\mathrm {a} = 1 4 9. 5 \times 1 0 ^ {6} \mathrm {k m}$$ $$\mathrm {T} = 3 6 5. 2 5 \times 2 4 \times 6 0 = 5. 2 5 9 6 \times 1 0 ^ {5} \text { minutes}.$$

Letting $M$ and $m$ be the masses of Sun and Earth respectively, we have, using equation:

$$\frac {m}{M} = \left(\frac {1 3 4}{5 . 2 5 9 6 \times 1 0 ^ {5}}\right) ^ {2} \left(\frac {1 4 9 . 5 \times 1 0 ^ {6}}{8 7 1 3 . 5}\right) ^ {3} = 3 2 7 8 0 0$$

Answer: $\mathbf{m} / \mathbf{M} = 327800$

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