Question

After the NEAR spacecraft passed Mathilde, on several occasions rocket propellant was expelled to adjust the spacecraft's momentum in order to follow a path that would approach the asteroid Eros, the final destination for the mission. After getting close to Eros, further small adjustments made the momentum just right to give a circular orbit of radius 45 km (45 × 103 m) around the asteroid. So much propellant had been used that the final mass of the spacecraft while in circular orbit around Eros was only 550 kg. The spacecraft took 1.04 days to make one complete circular orbit around Eros. Calculate what the mass of Eros must be.

Answer 1

Answer:

$6.68\times 10^{15}\ kg$

Explanation:

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

R = Radius of orbit = 45 km

T = Time period = 1.04 days

Mass of Eros would be given by the following equation

$M=\dfrac{4\pi^2R^3}{GT^2}\\\Rightarrow M=\dfrac{4\pi^2\times (45\times 10^3)^3}{6.67\times 10^{-11}\times (1.04\times 24\times 3600)^2}\\\Rightarrow M=6.68\times 10^{15}\ kg$

The mass of Eros is $6.68\times 10^{15}\ kg$

Answer 2

The mass of Eros must be $M = 6.68\times 10^{15}kg$

Calculation of the mass of Eros:

Since

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

R = Radius of orbit = 45 km

T = Time period = 1.04 days

So, the mass of Eros should be

$F = \frac{GMm}{d^2} = m\omega^2 d\\\\\frac{GM}{d^3}= \frac{4\pi^2}{T^2}$

Here T is the period

Now the mass should be

$M = \frac{4\pi^2d^3}{GT^2}$

So,

$M = 6.68\times 10^{15}kg$

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