The Question is from neet-physics gravitation keplers-laws, and it was created by the author Mr. ANIL KUMAR.G

Question

Two planets P1 and P2 orbit a star S in elliptical orbits with semi-major axes a1 and a2 respectively, where a1 > a2.  If T1 and T2 are their respective periods, and at some point both planets are simultaneously at their respective aphelia, after how many periods of P1 will they simultaneously be at their respective aphelia again?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option B
## CONCEPT FRAMEWORK
	**Primary Concept**: Orbital Periods and Ratios
	**Related Topics**: Kepler's Third Law, Rational Numbers
	**Prerequisites**: Understanding of orbital mechanics, periods, and rational numbers
## SOLUTION PATHWAY
**Given Information**:
	* Two planets, P1 and P2, orbiting a star S.
	* Semi-major axes: a1 and a2, where a1 > a2.
	* Orbital periods: T1 and T2.
	* Both planets are initially at their aphelia simultaneously.
**Method & Steps**:
	1.  The planets will be at their aphelia simultaneously again when a whole number of periods of each planet have passed. This means that for some integers 'm' and 'n', we must have mT1 = nT2.
		Reasoning: This ensures that each planet completes a whole number of orbits, returning to their respective aphelia. 
		Formula/Rule: Condition for simultaneous aphelia: m*T1 = n*T2
	2.  Rearranging the equation from step 1, we get T1/T2 = n/m.  This implies that the ratio of the periods must be a rational number. If T1/T2 is irrational, they will never line up again at their aphelia (or any other point in their orbits).
		Working: Rearranging m*T1 = n*T2 gives T1/T2 = n/m.
		Key Point: The ratio of periods must be rational for them to align again.
	3.  If the ratio T1/T2 = p/q, where p and q are co-prime integers, then after 'p' periods of P1, the time elapsed will be pT1.  For P2, the time elapsed is pT1. Since T1/T2 = p/q, we have pT1 = qT2, so after p periods of P1, it will be q periods of P2, meaning both will be back at their aphelia simultaneously.
		Result: The planets will simultaneously be at their aphelia again after 'p' periods of P1.
		Verification: Substituting the rational ratio confirms the condition for simultaneous aphelia.
 ## OPTION ANALYSIS
[A] :x: After one period of P1. This is incorrect because it does not account for the ratio of the periods and the possibility of requiring more than one period of P1.
[B] ✓: When $\frac{T_1}{T_2}$ is a rational number, represented as $\frac{p}{q}$ (p and q are co-prime integers), it will be after 'p' periods of P1. This is the correct answer as explained in the solution pathway.
[C] :x: After 'q' periods of P2. While 'q' periods of P2 does correspond to the same time as 'p' periods of P1, the question is specifically asking after how many periods of P1.
[D] :x: They will never be simultaneously at their aphelia again unless $a_1 = a_2$. This is incorrect because simultaneous aphelia is possible if the ratio of the periods is rational, regardless of the semi-major axes.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.  Assuming simultaneous aphelia always happens after one period: This overlooks the fact that the periods are different and can lead to miscalculations. 
		2.  Confusing periods of P1 and P2: It is essential to use the correct period in calculation. 
	**Quick Tips**:
		1.  Focus on the ratio of periods: The key to solving this problem is understanding the condition T1/T2=n/m, which indicates a rational ratio for alignment. 
		2.  Use the given rational ratio: When T1/T2=p/q, it implies pT1=qT2, which helps in determining the number of periods.

Mr. ANIL KUMAR.G · Answer

After one period of P1.

Mr. ANIL KUMAR.G · Answer

The correct Option is "When $\frac{T_1}{T_2}$ is a rational number, represented as $\frac{p}{q}$ (p and q are co-prime integers), it will be after 'p' periods of P1."

Mr. ANIL KUMAR.G · Answer

After 'q' periods of P2.

Mr. ANIL KUMAR.G · Answer

They will never be simultaneously at their aphelia again unless $a_1 = a_2$.

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