The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km from the asteroid's center. Eros is roughly box-shaped, with dimensions 40 km 6 km 6 km. Assume Eros has a density (mass/volume) of about 2.3 103 kg/m3.
(a) What will be the period of NEAR as it orbits Eros?
(b) If Eros were a sphere with the same mass and density, what would its radius be?
(c) What would g be at the surface of this spherical Eros?
To answer these questions, we need to understand some basic principles of orbital motion and gravitational forces. Let's break down each question and explain how to find the answers.
(a) To find the period of NEAR as it orbits Eros, we can use Kepler's third law. This law states that the square of the period of an orbiting object is proportional to the cube of the semi-major axis of the orbit.
First, we need to calculate the semi-major axis of the orbit. The height of NEAR above Eros' center is given as 15 km. However, we need to add the radius of Eros itself to this distance to get the semi-major axis. Since Eros is roughly box-shaped, we take the longest dimension, which is 40 km, as the radius. Therefore, the semi-major axis is 15 km + 40 km = 55 km.
Now we can use Kepler's third law to find the period. The equation is T^2 = (4π^2 / GM) * a^3, where T is the period, G is the gravitational constant, M is the mass of Eros, and a is the semi-major axis.
First, we need to find the mass of Eros. The density of Eros is given as 2.3 * 10^3 kg/m^3, and we can use this to calculate the volume of Eros:
Volume = length * width * height = 40 km * 6 km * 6 km = 1440 km^3
To convert this volume to m^3, we multiply by (1000 m / 1 km)^3 = 1 * 10^9 m^3. Thus, the volume of Eros is 1440 * 10^9 m^3.
Next, we calculate the mass using the density:
mass = density * volume = 2.3 * 10^3 kg/m^3 * 1440 * 10^9 m^3 = 3.312 * 10^12 kg.
Substituting the values into the formula, we get:
T^2 = (4π^2 / G * 3.312 * 10^12 kg) * (55 km)^3 = (4π^2 / G * 3.312 * 10^12 kg) * (55,000 m)^3.
Simplifying the equation and taking the square root of both sides will give us the period (T) of NEAR as it orbits Eros.
(b) To find the radius of a spherical Eros with the same mass and density, we can use the formula for the volume of a sphere. The volume of a sphere is given by V = (4/3)πr^3, where r is the radius.
We already calculated the mass of Eros in the previous part, which is 3.312 * 10^12 kg. Using the density, we can solve for the radius by equating the volume of the sphere to the volume we previously calculated: (4/3)πr^3 = 1440 * 10^9 m^3.
Solving for r gives us the radius of the spherical Eros.
(c) To find the acceleration due to gravity (g) at the surface of the spherical Eros, we can use the formula for gravitational acceleration: g = GM / r^2, where G is the gravitational constant, M is the mass of Eros, and r is the radius of Eros (which we calculated in part (b)).
Substituting the values into the formula will give us the acceleration due to gravity at the surface of Eros.
It's important to note that these calculations involve several mathematical steps, and the values used are approximations based on the information given.