Mercury has a semi-major axis of 0.367AU and an eccentricity of e=0.206. Calculate its distance to the Sun at perihelion. Remember to use units of AU in your answer.
let c be the distance from the center to the focus
let a be the semi-major axis
e = c/a
c = e*a = .206*.367 = .0756
distance at perihelion = a-c = .367 - .0756 = 0.291
oh, yeah. AU (duh)
semi-major axis = a = 0.367 au
closest distance = (1-e)*a = 0.794 a
= 0.291 au
To calculate the distance of Mercury from the Sun at perihelion, we need to multiply the semi-major axis (a) of Mercury by the eccentricity (e) and subtract that from the semi-major axis.
Distance at perihelion = a - a * e
Distance at perihelion = 0.367AU - 0.367AU * 0.206
Distance at perihelion = 0.367AU - 0.075482AU
Distance at perihelion = 0.291518AU
Therefore, the distance of Mercury from the Sun at perihelion is approximately 0.291518 AU.
To determine the distance of Mercury from the Sun at perihelion, we first need to understand the concept of perihelion and eccentricity.
Perihelion refers to the point in Mercury's orbit where it is closest to the Sun. Eccentricity, denoted by "e," reflects how elongated an orbit is. In this case, e=0.206 suggests that Mercury's orbit is slightly elongated.
Now, let's proceed with the calculation:
1. Start with the semi-major axis, which is given as 0.367 AU. The semi-major axis represents half the length of the longest diameter of an ellipse (the orbit, in this case).
2. To find the distance at perihelion (r_peri), we need to subtract the product of the semi-major axis and the eccentricity from the semi-major axis itself. Mathematically, it can be expressed as:
r_peri = semi-major axis - (semi-major axis x eccentricity)
Plugging in the values:
r_peri = 0.367 AU - (0.367 AU x 0.206)
r_peri = 0.367 AU - 0.075402 AU
r_peri = 0.291598 AU
Therefore, the distance from Mercury to the Sun at perihelion is approximately 0.291598 AU.