A planet is revolving in a circular orbit with a time period $T$ around the center of a star solely under the gravity of the star. Suppose the distance between the star and the planet is halved. The individual radii of the star and the planet are also halved, keeping their uniform mass densities unchanged. What will be the time period of the new orbit of the planet?
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For any gravitational system, if all linear dimensions (orbital radius and object sizes) are scaled by a factor $k$ while densities remain constant, the orbital time scale remains invariant ($T \propto \sqrt{\frac{r^3}{M}} \propto \sqrt{\frac{r^3}{r^3}} = \text{const}$).
Step 1: Understanding the Question:
We are given a system of a planet orbiting a star in a circular path.
The orbital distance, along with the physical sizes (radii) of the star and planet, are halved.
The densities of both bodies remain constant. We need to find how these changes affect the orbital period of the planet. Step 2: Key Formula or Approach:
Kepler's Third Law states that the orbital period $T$ of a planet around a star of mass $M$ in a circular orbit of radius $r$ is:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
The mass of a spherical star with radius $R_{star}$ and density $\rho$ is:
\[ M = \frac{4}{3}\pi R_{star}^3 \rho \]
Step 3: Detailed Explanation:
• Let us express the initial mass of the star as:
\[ M = \frac{4}{3}\pi R_{star}^3 \rho \]
• When the radius of the star is halved ($R'_{star} = \frac{R_{star}}{2}$) while keeping density $\rho$ constant, the new mass $M'$ is:
\[ M' = \frac{4}{3}\pi \left( \frac{R_{star}}{2} \right)^3 \rho = \frac{M}{8} \]
• The new orbital distance is also halved:
\[ r' = \frac{r}{2} \]
• Let us calculate the new orbital period $T'$:
\[ T' = 2\pi \sqrt{\frac{(r')^3}{G M'}} \]
• Substituting the values of $r'$ and $M'$:
\[ T' = 2\pi \sqrt{\frac{\left( \frac{r}{2} \right)^3}{G \left( \frac{M}{8} \right)}} = 2\pi \sqrt{\frac{\frac{r^3}{8}}{G \frac{M}{8}}} = 2\pi \sqrt{\frac{r^3}{GM}} = T \]
• Thus, the orbital period of the planet remains completely unchanged. Step 4: Final Answer:
The new time period of the orbit is $T$.
