Question

An asteroid is discovered to have a tiny moon that orbits it in a circular path at a distance of \(100 . \mathrm{km}\) and with a period of \(40.0 \mathrm{~h}\). The asteroid is roughly spherical (unusual for such a small body) with a radius of \(20.0 \mathrm{~km} .\) a) Find the acceleration of gravity at the surface of the asteroid. b) Find the escape velocity from the asteroid.

Short answer

Answer: The acceleration of gravity at the asteroid's surface is approximately \(1.85\times10^{-3}\mathrm{m ~s^{-2}}\), and the escape velocity from the asteroid is approximately \(107\mathrm{~m ~s^{-1}}\).

Step-by-step solution

(Step 1: Calculate the mass of the asteroid using the moon's orbit information)

We can use the given information about the moon's orbit to help us calculate the mass of the asteroid. According to Kepler's Third Law of Planetary Motion, the square of the period of the orbit is proportional to the cube of the semi-major axis. The formula for this relationship is: $$T^2 = \frac{4\pi^2}{G(M+m)}a^3$$ where \(T\) is the orbital period, \(a\) is the distance between the celestial bodies, \(G\) is the gravitational constant, and \(M\) and \(m\) are the masses of the two celestial bodies (in this case, the asteroid and its moon). In our case, \(m\) is negligible compared to \(M\), so we can approximate the equation as: $$T^2 = \frac{4\pi^2}{GM}a^3$$ We are given the orbital period \(T = 40.0 \mathrm{~h}=144000\mathrm{~s}\) and distance \(a = 100 \mathrm{~km}=1\times10^{5}\mathrm{~m}\). We can use these values to calculate the mass \(M\) of the asteroid. Rearrange the equation to solve for \(M\): $$M = \frac{4\pi^2a^3}{GT^2}$$

(Step 2: Calculate the mass of the asteroid using the values)

Now we plug in the values to calculate the mass of the asteroid: $$M = \frac{4\pi^2(1\times10^{5}\mathrm{~m})^3}{(6.674\times10^{-11}\mathrm{m^3~kg^{-1}~s^{-2}})(144000\mathrm{~s})^2}$$ $$M \approx 1.11\times10^{12}\mathrm{~kg}$$

(Step 3: Calculate gravitational acceleration at the asteroid's surface)

Now that we have the mass of the asteroid, we can calculate the acceleration of gravity at the surface using the formula: $$g=\frac{GM}{r^2}$$ where \(r\) is the radius of the asteroid (\(20.0\mathrm{~km}=2\times10^{4}\mathrm{~m}\)). Plugging in the values: $$g=\frac{(6.674\times10^{-11}\mathrm{m^3~kg^{-1}~s^{-2}})(1.11\times10^{12}\mathrm{~kg})}{(2\times10^{4}\mathrm{~m})^2}$$ $$g\approx 1.85\times10^{-3}\mathrm{m ~s^{-2}}$$ So, the acceleration of gravity at the surface of the asteroid is approximately \(1.85\times10^{-3}\mathrm{m ~s^{-2}}\).

(Step 4: Calculate the escape velocity from the asteroid)

Finally, we calculate the escape velocity using the formula: $$v_{esc}=\sqrt{\frac{2GM}{r}}$$ Plugging in the known values: $$v_{esc}=\sqrt{\frac{2(6.674\times10^{-11}\mathrm{m^3~kg^{-1}~s^{-2}})(1.11\times10^{12}\mathrm{~kg})}{2\times10^{4}\mathrm{~m}}}$$ $$v_{esc}\approx 107\mathrm{~m ~s^{-1}}$$ The escape velocity from the asteroid is approximately \(107\mathrm{~m ~s^{-1}}\).

Key concepts

Kepler's Third Law

Kepler's Third Law of Planetary Motion is a fascinating principle that describes the motion of celestial bodies. This law states that the square of the orbital period ( T^2 ) of a planet is directly proportional to the cube of the semi-major axis ( a^3 ) of its orbit. In simpler terms, the farther a planet or moon is from the body it orbits, the longer it takes to complete one orbit. This relationship can be expressed as:

• \( T^2 = \frac{4\pi^2}{G(M+m)}a^3 \)
• Here, \( T \) is the orbital period, \( a \) is the semi-major axis, \( G \) is the gravitational constant, and \( M \) and \( m \) are the masses of the two orbiting bodies.

For small celestial bodies like moons orbiting asteroids, the moon's mass is negligible relative to the asteroid, simplifying calculations. By knowing the period and the distance of the moon's orbit, we can determine the mass of the asteroid it's orbiting. This principle is instrumental in studying objects in our solar system and helps to calculate vital information like mass without direct observation.

Gravitational Acceleration

Gravitational acceleration is the force that a celestial body exerts on objects at or near its surface, causing them to fall or remain in orbit. This acceleration is dictated by the mass of the body and its radius. Newton's law of universal gravitation provides the framework for calculating gravitational acceleration on a celestial body such as an asteroid:

• \( g = \frac{GM}{r^2} \)
• Here, \( g \) is the gravitational acceleration, \( G \) is the gravitational constant, \( M \) is the mass of the celestial body, and \( r \) is the radius.

Asteroids, even though much smaller than planets, exert a gravitational pull which can be significant when in close proximity. In the case of our asteroid example, the gravitational acceleration was found to be \( 1.85 \times 10^{-3} \mathrm{m/s^2} \), which is much weaker than Earth's gravity but strong enough for its moon to stay in orbit.

Escape Velocity

Escape velocity is the speed needed for an object to break free from the gravitational pull of a celestial body without further propulsion. It depends on the mass and radius of the body. This critical velocity ensures that any object can overcome gravitational forces and move into space, and is calculated using:

• \( v_{esc} = \sqrt{\frac{2GM}{r}} \)
• \( v_{esc} \) is the escape velocity, \( G \) is the gravitational constant, \( M \) is the mass, and \( r \) is the radius of the celestial body.

For extremely small bodies like asteroids, the escape velocity is relatively low compared to planets. As calculated for the asteroid in the exercise, the escape velocity is approximately \( 107 \mathrm{~m/s} \). This low speed means less energy is required to send objects off the asteroid's surface into space, affecting how we might plan space missions.

Celestial Mechanics

Celestial Mechanics is a branch of astronomy that deals with the motion of celestial bodies governed by laws like Newton's theory of gravitation. It involves analyzing how bodies such as planets, moons, asteroids, and comets move through space. It covers several core aspects:

• Understanding gravitational interactions to predict orbits and trajectories.
• Calculating influences of gravitational accelerations and escape velocities.

By applying the principles of celestial mechanics, we can predict how moons can stably orbit smaller bodies like asteroids, and how much energy is needed to escape their gravitational influence. These insights are vital for forming strategies to explore our solar system and possibly beyond, ensuring safe space travel and successful deployment of spacecraft.