Question

Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the 𝓍-direction and 2b in the y-direction is (𝓍²/a²) + (y² /b²) = 1.
(a) Let d² denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ ―a, a] to show that the average value of d² is (a² + 2b²) /3 .
Diagram of an elliptical orbit with the Sun at one focus, labeled axes, and the Earth positioned along the orbit.

Diagram of an elliptical orbit with the Sun at one focus, labeled axes, and the Earth positioned along the orbit.

Accepted Answer

Step 1: Begin by recalling the equation of the ellipse: (𝓍²/a²) + (y²/b²) = 1. This represents the elliptical orbit of the planet with semi-major axis 'a' and semi-minor axis 'b'. Step 2: The square of the distance from the planet to the center of the ellipse is given by d² = 𝓍² + y². Substitute y² from the ellipse equation into this expression: y² = b²(1 - 𝓍²/a²). Thus, d² = 𝓍² + b²(1 - 𝓍²/a²). Step 3: Simplify the expression for d²: d² = 𝓍² + b² - (b²𝓍²/a²). Combine terms to get d² = (1 - b²/a²)𝓍² + b². Step 4: To find the average value of d² over the interval [−a, a], use the formula for the average value of a function: (1/(a - (-a))) ∫[−a, a] d² d𝓍. This simplifies to (1/(2a)) ∫[−a, a] [(1 - b²/a²)𝓍² + b²] d𝓍. Step 5: Break the integral into two parts: (1/(2a)) [∫[−a, a] (1 - b²/a²)𝓍² d𝓍 + ∫[−a, a] b² d𝓍]. Evaluate each integral separately, noting that ∫[−a, a] 𝓍² d𝓍 = (2/3)a³ and ∫[−a, a] d𝓍 = 2a. Combine results to show that the average value of d² is (a² + 2b²)/3.

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Concetti chiave

Elliptical Orbits

Average Value of a Function

Integration