The Question is from neet-physics gravitation , and it was created by the author SUBHANI

Question

Two asteroids, A and B, orbit the Sun with periods $T_A$ and $T_B$ respectively, where $T_A = 2T_B$.  Let $a_A$ and $a_B$ be their respective semi-major axes.  If asteroid A has an orbital eccentricity of 0.2, what is the maximum possible eccentricity of asteroid B?

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [Option D]
## CONCEPT FRAMEWORK
	**Primary Concept**: Kepler's Third Law of Planetary Motion
	**Related Topics**: Orbital Period, Semi-major Axis, Eccentricity
	**Prerequisites**: Understanding of Kepler's laws, basic algebra
## SOLUTION PATHWAY
**Given Information**:
		$T_A = 2T_B$
		Eccentricity of A, $e_A$ = 0.2
		Semi-major axes are $a_A$ and $a_B$
**Method & Steps**:
	1.	Apply Kepler's Third Law: The square of the orbital period is proportional to the cube of the semi-major axis.
		Reasoning: Kepler's Third Law relates orbital periods and semi-major axes for objects orbiting the same central body.
		Formula/Rule: $T^2 \propto a^3$
	2.	Set up the ratio of the periods and semi-major axes for asteroids A and B.
		Working: $\frac{T_A^2}{T_B^2} = \frac{a_A^3}{a_B^3}$
		Key Point: We are given $T_A = 2T_B$, so $\frac{T_A^2}{T_B^2} = \frac{(2T_B)^2}{T_B^2} = 4$
	3.	Solve for the ratio of the semi-major axes.
		Working: $4 = \frac{a_A^3}{a_B^3} \Rightarrow \frac{a_A}{a_B} = \sqrt[3]{4}$
		Key Point:  The eccentricities do not affect the relationship between the periods and semi-major axes. Kepler's Third Law is independent of the eccentricity.
	4.	Determine the eccentricity of B.
		Result: The eccentricity of asteroid B is independent of its orbital period and semi-major axis. We are only given that the eccentricity of asteroid A is 0.2, and the question asks for the *maximum possible* eccentricity of asteroid B. The eccentricity of an ellipse can be any value between 0 and 1. The fact that $T_A = 2T_B$ only affects the semi-major axis, and the eccentricity is independent from semi-major axis. The maximum value for eccentricity is less than 1.
		Verification: The relationship between periods and semi-major axes does not constrain eccentricity, which can have any value between 0 and 1.
 ## OPTION ANALYSIS
[A] :x: 0.1 is an arbitrary value. There's no calculation that leads to 0.1.
[B] :x: 0.4 is an arbitrary value. There's no calculation that leads to 0.4.
[C] :x: 0.8 is an arbitrary value. There's no calculation that leads to 0.8.
[D] ✓: The eccentricity of an orbit can be any value between 0 and 1 (exclusive of 1, which would be a parabola). The orbital period and the semi-major axis are related by Kepler's 3rd law, but this does not constrain the eccentricity. Therefore, the eccentricity of asteroid B can be less than 1.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing eccentricity with semi-major axis. Eccentricity is independent from semi-major axis.
		2.[Error 2]: Trying to relate the eccentricities of the two asteroids when they are independent.
	**Quick Tips**:
		1.[Helpful shortcut]: Understand that Kepler's Third Law only relates periods and semi-major axes.
		2.[Memory aid]: Remember that eccentricity is a measure of how elliptical an orbit is, and can have any value between 0 and 1.

SUBHANI · Answer

0.1

SUBHANI · Answer

0.4

SUBHANI · Answer

0.8

SUBHANI · Answer

The correct Option is "Less than 1"

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