Animated Solution for Physics - Gravitation: A double star system consists of two stars $A$ and $B$ which have time periods $T_A$ and $T_B$. Radius $R_A$ and $R_B$ and mass $M_A$ and $M_B$. Choose the correct option.

Imagine looking deep into the cosmos and witnessing a cosmic dance: two stars, bound by their mutual gravity, revolving around each other. This is a binary star system (or double star system).

Unlike our solar system, where a relatively light Earth orbits a massive, nearly stationary Sun, a binary system consists of two bodies of comparable masses. Neither star is stationary; instead, both orbit around a single, invisible point called their common center of mass.

Let us explore the physics behind this system and understand why their orbital time periods must be identical.

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For any stable binary system, the two stars must always lie on opposite sides of their common center of mass $O$. If they did not, the center of mass would accelerate, violating the conservation of momentum for an isolated system.

Let the masses of the two stars be $M_A$ and $M_B$, and their distances from the center of mass $O$ (their orbital radii) be $R_A$ and $R_B$ respectively.

By the definition of the center of mass:

This is our first crucial equation. It tells us that the heavier star orbits closer to the center of mass, while the lighter star orbits further away.

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The distance between the two stars is the sum of their individual distances from the center of mass:

According to Newton's Law of Universal Gravitation, the mutual gravitational force of attraction between them is:

This gravitational force acts along the line joining the two stars, pulling each star toward the center of mass $O$. Thus, for both stars, this mutual gravitational force acts as the necessary centripetal force keeping them in circular orbits.

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Let us write the centripetal force equations for both stars in terms of their angular velocities ${\omega }_A$ and ${\omega }_B$.

For Star $A$:

For Star $B$:

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Since the gravitational force $F_g$ experienced by both stars is a mutual action-reaction pair (Newton's Third Law), we can equate the centripetal expressions directly:

We can rearrange this equation as:

Now, recall our center of mass constraint: $M_A{R}_A=M_B{R}_B$. Since these terms are equal, they cancel out beautifully from both sides of the equation!

This leaves us with:

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Equality of Time Periods

The time period of revolution $T$ is related to the angular velocity $\omega$ by:

Since we have proven that ${\omega }_A={\omega }_B$, it immediately follows that:

This means that both stars must have exactly the same time period of revolution, regardless of their individual masses or orbital radii.

This makes perfect intuitive sense: for the two stars to always remain on opposite sides of the center of mass, they must complete one full orbit in the exact same amount of time. If one star completed its orbit faster than the other, the system would lose its alignment and destabilize.

Therefore, the correct option is (d).