Understanding the Concepts
* Orbital Period: The time it takes for an object to complete one full orbit around another object.
* Gravitational Force: The force of attraction between any two objects with mass.
* Centripetal Force: The force that keeps an object moving in a circular path.
Applying the Concepts
1. Newton's Law of Universal Gravitation: The force of gravity between the spacecraft and the planet is given by:
```
F = G * (m1 * m2) / r^2
```
where:
* F is the gravitational force
* G is the gravitational constant (6.674 × 10^-11 m^3 kg^-1 s^-2)
* m1 is the mass of the spacecraft
* m2 is the mass of the planet
* r is the distance between their centers
2. Centripetal Force: The spacecraft is in orbit, meaning it's moving in a circle. The force keeping it in this path is the centripetal force:
```
F = (m1 * v^2) / r
```
where:
* v is the orbital speed of the spacecraft
3. Equating Forces: Since the gravitational force is what provides the centripetal force to keep the spacecraft in orbit, we can equate the two equations from above:
```
G * (m1 * m2) / r^2 = (m1 * v^2) / r
```
4. Orbital Speed and Period: We can relate the orbital speed (v) to the orbital period (T) using:
```
v = 2 * pi * r / T
```
5. Solving for the Planet's Mass:
* Substitute the expression for orbital speed (v) into the equation from step 3.
* Rearrange the equation to solve for the mass of the planet (m2).
Calculations
1. Convert Period to Seconds: 52 hours * 3600 seconds/hour = 187200 seconds
2. Substitute and Solve:
* G * (m1 * m2) / r^2 = (m1 * (2 * pi * r / T)^2) / r
* Simplify and solve for m2:
```
m2 = (4 * pi^2 * r^3) / (G * T^2)
```
3. Plug in the values:
* m2 = (4 * pi^2 * (5.2 * 10^7 m)^3) / (6.674 × 10^-11 m^3 kg^-1 s^-2 * (187200 s)^2)
* m2 ≈ 1.83 × 10^25 kg
Result
The mass of the unknown planet is approximately 1.83 × 10^25 kg.
