when earth and mars are closest to each other in their orbits the angular separation between mars's moon phobos and the surface of mars is observed to be 22 arc seconds. if phobos orbits mars every .31910 days what is the average density of mars expresses in kilograms per cubic meter? i need to know the steps to solve the problem, also.

To calculate the average density of Mars, we need to follow these steps:

Step 1: Find the diameter of Phobos:
The angular separation of Phobos from the surface of Mars can help us find the diameter of Phobos. The formula to calculate the diameter of an object is D = (2 × r × θ) / 360, where D is the diameter, r is the distance to the object, and θ is the angular separation in degrees.

Given:
- Angular separation between Phobos and Mars = 22 arc seconds = 0.00611 degrees
- Let's assume the distance from Mars to Phobos is 9,376 km (this will be refined in Step 3)

Using the formula, we can calculate the diameter of Phobos:
D = (2 × r × θ) / 360
D = (2 × 9376 × 0.00611) / 360
D ≈ 31.83 km

Step 2: Find the volume of Phobos:
The volume of Phobos can be calculated using the formula V = (4/3)πr^3, where V is the volume and r is the radius.

Since we know the diameter (D) from Step 1, we can calculate the radius (r) as half of the diameter: r = D/2.
r = 31.83 km / 2
r = 15.915 km

Plugging in the values, we can calculate the volume of Phobos:
V = (4/3)πr^3
V ≈ (4/3) × 3.14159 × (15.915 km)^3
V ≈ 11,699 km^3 (approx)

Step 3: Find the distance from Mars to Phobos:
By researching, we can find that the average distance from Mars to Phobos is approximately 9,377 km.

Step 4: Calculate the average density of Mars:
The average density of Mars can be found using the formula Density = Mass / Volume.

To calculate mass, we need to determine the acceleration due to gravity on Phobos. On the surface of Phobos, the gravitational acceleration can be calculated using the formula g = G × M / r^2, where G is the gravitational constant, M is the mass of Mars, and r is the distance from Mars to Phobos.

Using average values:
- Gravitational constant (G) ≈ 6.67430 × 10^-11 m^3 kg^-1 s^-2
- Mass of Mars (M) ≈ 6.39 × 10^23 kg
- Distance from Mars to Phobos (r) ≈ 9,377 km = 9,377,000 m

Using the formula, we can calculate the acceleration due to gravity on Phobos:
g = G × M / r^2
g ≈ (6.67430 × 10^-11 m^3 kg^-1 s^-2) × (6.39 × 10^23 kg) / (9,377,000 m)^2
g ≈ 7.124 m/s^2 (approx)

Now, to calculate the mass of Phobos, we can use the formula:
m = g × V
m ≈ (7.124 m/s^2) × (11,699 km^3 × 10^9 m^3 / 1 km^3)
m ≈ 8.350 × 10^19 kg (approx)

Finally, we can calculate the average density of Mars using the formula:
Density = Mass / Volume
Density = (6.39 × 10^23 kg) / (1.631 × 10^6 km^3 × 10^9 m^3 / 1 km^3)
Density ≈ 3,922 kg/m^3 (approx)

Therefore, the average density of Mars is approximately 3,922 kg/m^3.