Asteroid satellite: While on its way to Jupiter in 1993, the Galileo spacecraft made a flyby of asteroid Ida. Images captured (Fig. P5.66) of Ida discovered that the asteroid has a tiny moon of its own, since given the name Dactyl. Measurements found that Ida to be about 55 × 24 × 22 km in size, and that Dactyl’s orbit period and radius are approximately 27 h and 94 km respectively. From this data determine Ida’s approximate
mass
density
To determine Ida's approximate mass and density, we can use the information about Dactyl's orbit. The relation between the orbital period (T) and the radius (r) of a satellite around a central body can be given by Kepler's Third Law:
T^2 = (4π^2/ GM) * r^3
Where:
T = orbital period of the satellite (27 hours in this case)
G = gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
M = mass of the central body (Ida in this case)
r = radius of the satellite's orbit (94 km in this case)
We can rearrange the equation to solve for the mass (M):
M = (4π^2/ G) * (r^3/ T^2)
Let's calculate the mass:
M = (4π^2/ G) * (94,000^3/ (27*3600)^2)
Using a calculator, we find:
M ≈ 3.61 x 10^17 kg
Now that we have the mass of Ida, we can calculate its density by dividing its mass by its volume. The volume of a rectangular-shaped asteroid can be approximated as:
Volume = length * width * height
Volume = 55 km * 24 km * 22 km
Converting km to meters:
Volume = 55,000 m * 24,000 m * 22,000 m
Calculating the volume:
Volume ≈ 2.76 x 10^13 m^3
Now we can calculate the density:
Density = mass / volume
Density ≈ (3.61 x 10^17 kg) / (2.76 x 10^13 m^3)
Using a calculator:
Density ≈ 13077 kg/m^3
Therefore, the approximate mass of Ida is 3.61 x 10^17 kg, and its approximate density is 13077 kg/m^3.