The mean distance of Earth from the Sun is 93 million miles. If the aphelion of Earth is 94.5 million miles, what is the perihelion? Find an equation for the orbit of Earth around the Sun.
The correct answer and explanation is :
Solution:
To solve the problem, we need to understand the basic properties of Earth’s elliptical orbit around the Sun. The orbit of Earth is an ellipse, and the Sun is located at one of the foci of the ellipse. The mean distance from Earth to the Sun is called the semi-major axis, which is the average of the perihelion (the closest point to the Sun) and aphelion (the farthest point from the Sun).
Given:
- The mean distance of Earth from the Sun = 93 million miles (this is the semi-major axis, ( a )).
- The aphelion distance = 94.5 million miles.
- We are asked to find the perihelion distance.
Step 1: Use the properties of the ellipse
The semi-major axis, ( a ), is the average of the perihelion (( r_p )) and aphelion (( r_a )):
[
a = \frac{r_p + r_a}{2}
]
We know ( a = 93 ) million miles and ( r_a = 94.5 ) million miles, so we can substitute these values into the equation:
[
93 = \frac{r_p + 94.5}{2}
]
Step 2: Solve for the perihelion distance
Multiply both sides of the equation by 2:
[
186 = r_p + 94.5
]
Now, subtract 94.5 from both sides:
[
r_p = 186 – 94.5 = 91.5 \, \text{million miles}
]
Thus, the perihelion distance of Earth is 91.5 million miles.
Step 3: Equation for the orbit
The orbit of Earth around the Sun is elliptical, and the equation of an ellipse in polar coordinates (where the Sun is at one focus) is given by:
[
r = \frac{a(1 – e^2)}{1 + e \cos(\theta)}
]
Where:
- ( r ) is the distance from the Sun at any point in the orbit.
- ( a ) is the semi-major axis (93 million miles).
- ( e ) is the eccentricity of the ellipse.
- ( \theta ) is the true anomaly (the angle from the closest approach).
Step 4: Find the eccentricity ( e )
The eccentricity ( e ) of an ellipse is defined as:
[
e = \frac{r_a – r_p}{2a}
]
Substitute ( r_a = 94.5 ), ( r_p = 91.5 ), and ( a = 93 ):
[
e = \frac{94.5 – 91.5}{2 \times 93} = \frac{3}{186} = 0.0161
]
Thus, the eccentricity ( e ) of Earth’s orbit is approximately 0.0161.
Final Equation of the Orbit:
The equation of Earth’s orbit around the Sun can be written as:
[
r = \frac{93(1 – (0.0161)^2)}{1 + 0.0161 \cos(\theta)}
]
This equation describes the elliptical orbit of Earth with a semi-major axis of 93 million miles and an eccentricity of approximately 0.0161.
Explanation:
The orbit of Earth around the Sun is not a perfect circle, but an ellipse, where the Sun is located at one focus. The perihelion is the point of the orbit closest to the Sun (91.5 million miles), and the aphelion is the farthest point (94.5 million miles). The average distance from the Sun, or the semi-major axis, is 93 million miles. This elliptical shape of Earth’s orbit results in varying distances between the Earth and the Sun throughout the year, and the eccentricity value quantifies how “elongated” the orbit is. The orbit equation takes into account both the average distance and the shape of the ellipse to model the Earth’s motion around the Sun.