Question

The Apollo 8 mission in 1968 included a circular orbit at an altitude of \(111 \mathrm{~km}\) above the Moon's surface. What was the period of this orbit? (You need to look up the mass and radius of the Moon to answer this question!)

Short answer

Answer: The period of the orbit of Apollo 8 around the Moon is approximately 6,925 seconds.

Step-by-step solution

Search for the mass and radius of the Moon

The mass of the Moon (\(M_{moon}\)) is approximately \(7.342 \times 10^{22}\) kilograms, and the radius of the Moon (\(R_{moon}\)) is around \(1,737.4\) kilometers.

Convert altitude to meters and find total distance from Moon's center

The altitude of the orbit is given as \(111\mathrm{~km}\). Let's convert it to meters by multiplying it by 1000. Altitude in meters = \(111,000\mathrm{~m}\) Now, we will add the Moon's radius to the altitude to find the total distance (\(R_{total}\)) from the center of the Moon to the orbiting Apollo 8 spacecraft. \(R_{total}\) = \(R_{moon} + \) altitude So, \(R_{total} = 1,737,400 + 111,000 = 1,848,400\mathrm{~m}\)

Calculate the gravitational force

To calculate the gravitational force acting on Apollo 8, we will use the universal law of gravitation formula: \(F_{grav} = \frac{G \times M_{moon} \times m}{R_{total}^2}\) where \(F_{grav}\) is the gravitational force, \(G\) is the universal gravitational constant (\(6.674 \times 10^{-11} Nm^2/kg^2\)), \(M_{moon}\) is the mass of the Moon, \(m\) is the mass of the Apollo 8 spacecraft, and \(R_{total}\) is the total distance from the center of the Moon. NOTE: We don't need to find the exact value of the gravitational force, as it will cancel out with the mass of Apollo 8 in the next step when calculating the velocity.

Calculate the velocity of Apollo 8 spacecraft

To find the velocity of Apollo 8, we will equate the gravitational force with the centripetal force acting on the spacecraft: \(F_{grav} = F_{centripetal}\) \(\frac{G \times M_{moon} \times m}{R_{total}^2} = m \times \frac{v^2}{R_{total}}\) Since the mass (\(m\)) of Apollo 8 cancels out from both sides, we can find the velocity (\(v\)) as follows: \(v^2 = \frac{G \times M_{moon} \times R_{total}}{R_{total}^2}\) \(v = \sqrt{\frac{G \times M_{moon}}{R_{total}}}\) Now, plug in the values to calculate the velocity: \(v = \sqrt{\frac{6.674 \times 10^{-11} Nm^2/kg^2 \times 7.342 \times 10^{22} kg}{1,848,400\mathrm{~m}}}\) \(v \approx 1,680\mathrm{~m/s}\)

Calculate the period of the orbit

To find the period of the orbit, we will use the formula connecting the period (\(T\)) with the velocity and total distance from the Moon's center: \(T = \frac{2 \pi R_{total}}{v}\) Now, plug in the values to calculate the period of the orbit: \(T = \frac{2 \pi \times 1,848,400\mathrm{~m}}{1,680\mathrm{~m/s}}\) \(T \approx 6,925\mathrm{~s}\) The period of the orbit of Apollo 8 around the Moon is approximately \(6,925\) seconds.

Key concepts

Universal Law of Gravitation

Understanding the universal law of gravitation is pivotal to solving problems related to celestial motion. Initially formulated by Sir Isaac Newton, it states that every mass attracts every other mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

This law can be represented by the formula:
\[F_{grav} = \frac{G \times M_1 \times M_2}{r^2}\]
where:\

• \(F_{grav}\) is the gravitational force between the two masses,
• \(G\) is the gravitational constant (\(6.674 \times 10^{-11} N(m/kg)^2\)),
• \(M_1\) and \(M_2\) are the masses of the two objects, and
• \(r\) is the distance between the centers of the two masses.

When approaching problems involving the motion of a spacecraft, such as Apollo 8, around a celestial body like the Moon, this law allows us to calculate the gravitational pull exerted by the Moon on the spacecraft, which is essential for determining the orbit characteristics.

Centripetal Force

The term 'centripetal' comes from Latin words that mean 'seeking the center'. Centripetal force is required for an object to move in a circular path and is directed towards the center around which the object is moving. This force is not a separate force of nature but rather the net force causing the circular motion. For a spacecraft in orbit, like Apollo 8, this force is provided by the gravitational pull of the celestial body it orbits.

The relationship between centripetal force \(F_{centripetal}\) and the velocity of an object is given by the equation:
\[F_{centripetal} = m \times \frac{v^2}{R_total}\]
where:\

• \(m\) is the mass of the orbiting object,
• \(v\) is its velocity, and
• \(R_{total}\) is the radius of the circular orbit.

In the case of the Apollo 8 mission, we can use the equivalence of gravitational and centripetal forces to find the required velocity and hence, with the orbital radius, calculate the orbital period.

Gravitational Force Calculation

The calculation of gravitational force comes into play when determining the velocity necessary for a spacecraft to maintain orbit around a celestial body like the Moon. The steps in the provided solution employ gravitational force calculation to reach an understanding of orbital mechanics.

By equating the gravitational force \(F_{grav}\), which acts as the centripetal force keeping Apollo 8 in orbit, with the centripetal force formula, we eliminate the need for the spacecraft's mass in calculating its velocity, which simplifies the problem immensely:
\[\frac{G \times M_{moon}}{R_{total}} = \frac{v^2}{R_{total}}\]
After finding the velocity, the period \(T\) for one complete orbit can be found using the relationship:
\[T = \frac{2 \pi R_{total}}{v}\]
This process demonstrates the practical application of gravitational force calculation in predicting and analyzing the motion of satellites and spacecrafts in orbit.