The distance between the Earth and Mars when the two panets are at opposition varies greatly because of the large eccentricity of Mar's orbit. The perihelion distance of a planet is given by rmin=a(1-e)and the aphelion distance by rmax=a(1+e) where a is the semimajor axis and e is the orbital eccentricity. Find the smallest and largest opposition distances assuming the Earth's orbit is a circle.

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The eccentricity of Mar's orbit is .0934.

The semi-major axis is 141,643,675 miles.

Therefore, the perihelion distance is
r(p) = 141,643,675(1-.0934) = 128,414,418.
and
r(a) = 141,643,675(1+.0934) = 154,871,931.

The mean radius of the earth's orbit is
r = 92,960,242 miles.

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To find the smallest and largest opposition distances between the Earth and Mars, we can use the given formulas for perihelion and aphelion distances in Mars' orbit. However, since we are assuming Earth's orbit to be a circle, we can simplify the calculation by considering the average distance between the two planets during opposition.

During opposition, Earth and Mars are on the same side of the Sun, with Earth closer to the Sun. This means that the two planets are at their minimum distance from each other. Let's calculate the smallest opposition distance.

Given:
Earth's orbit is a circle, so a (semimajor axis) = average distance between Earth and Sun.
Mars' orbit is an ellipse, so rmin (perihelion distance) = smallest distance between Mars and Sun.
Mars' eccentricity e = orbital eccentricity = (rmax - rmin) / (rmax + rmin)

We need to find the smallest opposition distance, which is the distance between the Earth and Mars during opposition.

Step 1: Find Earth's average distance from the Sun (semimajor axis):
Since Earth's orbit is a circle, the average distance between Earth and Sun is the same as the radius of Earth's orbit. According to NASA, this distance is approximately 1 Astronomical Unit (AU) or about 149.6 million kilometers.

a (semimajor axis) = 149.6 million kilometers

Step 2: Find Mars' perihelion distance (rmin):
Given the formula rmin = a(1 - e), where a is Earth's semimajor axis and e is Mars' eccentricity.

Assuming the eccentricity of Mars' orbit is 0.0934 (NASA's value for Mars' orbital eccentricity), we can compute rmin:

rmin = a(1 - e)
= 149.6 million kilometers * (1 - 0.0934)
≈ 136.1 million kilometers

Therefore, during opposition, the smallest distance between Earth and Mars is approximately 136.1 million kilometers.

To find the largest opposition distance, we use the formula for Mars' aphelion distance (rmax):

rmax = a(1 + e)
= 149.6 million kilometers * (1 + 0.0934)
≈ 163.1 million kilometers

Therefore, during opposition, the largest distance between Earth and Mars is approximately 163.1 million kilometers.