Solving this word problem in a way i can understand

Question

Please help. I do not understand how to solve this.

The point in a planet’s orbit where it is the closest is to the sun is called Perihelion and the point where it is farthest away is called Aphelion. Where the Sun is a focal point of the planets orbit. Suppose that Pluto is 7.376 𝑥109 km from the Sun at Aphelion. What is the distance between Mars and the Sun at Perihelion, given that the elliptical eccentricity of Pluto is 0.2488

Answer 1

If c is the distance from the center to the focus and a is the distance from the center to the major axis, then e=c/a. The distance at aphelion will be c+a.

From the question, you cannot answer the distance to Mars at perihelion, but you can answer the distance to Pluto at perihelion. This will, of course, be a-c

c/a = .2488

c = .2488a

c+a = 7.376 𝑥10^9 km

.2488a + a = 7.376 𝑥10^9 km

1.2488 a = 7.376 𝑥10^9 km

a = 5.906 * 10^9 km

c = 7.376 𝑥10^9 km - 5.906 * 10^9 km

c = 1.470 x 10^9 km

Then, the distance to perihelion is a - c = 5.906 * 10^9 km - 1.470 x 10^9 km

The distance to perihelion for Pluto is 4.436 x 10^9 km (or 4.437, depending upon when we round).

Answer 2

Unless you replace "What is the distance between Mars and the Sun at Perihelion" with "What is the distance between Pluto and the Sun at Perihelion," you don't have enough information here to solve this. Assuming that is what they intended and this was a typo, we first need to understand orbital eccentricity. In the simplest terms, it's a measure of how circular an orbit is. A perfect 0 is a perfect circle, anything over 0 and below 1 is an ellipse (a squished circle), exactly 1 is a parabola, and anything over 1 is a hyperbola. To find eccentricity we'd usually take the distance of the focus from the center (usually called c) divided by half the length of the longer side of the ellipse (called the semi-major axis, or a). In those terms, it's just c/a. We don't know the major axis, though, and we can't solve for any part of that equation with two unknowns (distance of focus from center AND major axis are unknown).

We can say that the offset from center (c) + the distance of the center to the aphelion (a) = c + a, and that's our total aphelion distance, c + a. The formula for Perihelion is a(1 - e) and aphelion = a(1 + e). So then we take 7.376 𝑥10⁹ km = a(1- 0.2488), 7.376 𝑥10⁹ km = 0.7512a, 9.81 x 10⁹ = a. Then we just plug in a & e into the equation, and a(1 + e) gives us 9.81 x 10⁹(1 + 0.2488) km, which comes out to 1.23 x 10¹⁰ km!