A spherical body of radius $R$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $P(r)$ is the pressure at $r\,(<R)$, then the correct option(s) is (are)

Step 1: Find the gravitational field inside the sphere.

The mass enclosed within a radius $r$ (where $r < R$) for a constant density $\rho$ is given by the volume of the inner sphere.

$$M(r) = \frac{4}{3}\pi\rho r^{3}$$

The gravitational acceleration at radius $r$ depends only on this enclosed mass.

$$g(r) = \frac{GM(r)}{r^{2}}$$ $$g(r) = \frac{4}{3}\pi G\rho r$$

Step 2: Set up the hydrostatic equilibrium equation.

The pressure gradient balances the inward gravitational force.

$$\frac{dP}{dr} = -\rho g(r)$$

Substitute the expression for $g(r)$.

$$\frac{dP}{dr} = -\frac{4}{3}\pi G\rho^{2} r$$

Step 3: Integrate to find the pressure profile $P(r)$.

Integrate from radius $r$ to the surface $R$, using the boundary condition that pressure at the surface is zero, $P(R) = 0$.

$$\int_{P(r)}^{0} dP = -\int_{r}^{R} \frac{4}{3}\pi G\rho^{2} r \, dr$$ $$-P(r) = -\frac{2}{3}\pi G\rho^{2} (R^{2} - r^{2})$$ $$P(r) = \frac{2}{3}\pi G\rho^{2}(R^{2}-r^{2})$$

This can be written in terms of the central pressure $P_c = P(0)$.

$$P(r) = P_c\left(1 - \frac{r^{2}}{R^{2}}\right)$$ $$P_c = \frac{2}{3}\pi G\rho^{2}R^{2}$$

Step 4: Evaluate the pressure at the center.

Check the first option by evaluating the pressure at $r = 0$.

$$P(0) = P_c \neq 0$$

The pressure at the center is non-zero, so the first option is incorrect.

Step 5: Evaluate the ratio of pressures for the remaining options.

The ratio of pressures at any two distances $a$ and $b$ simplifies by canceling the constant factors.

$$\frac{P(a)}{P(b)} = \frac{R^{2}-a^{2}}{R^{2}-b^{2}}$$

Test the ratio for $a = \frac{3R}{4}$ and $b = \frac{2R}{3}$.

$$\frac{P\left(\tfrac{3R}{4}\right)}{P\left(\tfrac{2R}{3}\right)} = \frac{1-\tfrac{9}{16}}{1-\tfrac{4}{9}}$$ $$\frac{P\left(\tfrac{3R}{4}\right)}{P\left(\tfrac{2R}{3}\right)} = \frac{\tfrac{7}{16}}{\tfrac{5}{9}}$$ $$\frac{P\left(\tfrac{3R}{4}\right)}{P\left(\tfrac{2R}{3}\right)} = \frac{63}{80}$$

Test the ratio for $a = \frac{3R}{5}$ and $b = \frac{2R}{5}$.

$$\frac{P\left(\tfrac{3R}{5}\right)}{P\left(\tfrac{2R}{5}\right)} = \frac{1-\tfrac{9}{25}}{1-\tfrac{4}{25}}$$ $$\frac{P\left(\tfrac{3R}{5}\right)}{P\left(\tfrac{2R}{5}\right)} = \frac{\tfrac{16}{25}}{\tfrac{21}{25}}$$ $$\frac{P\left(\tfrac{3R}{5}\right)}{P\left(\tfrac{2R}{5}\right)} = \frac{16}{21}$$

Test the ratio for $a = \frac{R}{2}$ and $b = \frac{R}{3}$.

$$\frac{P\left(\tfrac{R}{2}\right)}{P\left(\tfrac{R}{3}\right)} = \frac{1-\tfrac{1}{4}}{1-\tfrac{1}{9}}$$ $$\frac{P\left(\tfrac{R}{2}\right)}{P\left(\tfrac{R}{3}\right)} = \frac{\tfrac{3}{4}}{\tfrac{8}{9}}$$ $$\frac{P\left(\tfrac{R}{2}\right)}{P\left(\tfrac{R}{3}\right)} = \frac{27}{32} \neq \frac{20}{27}$$

Answer: The correct options are:

• $P\left(\frac{3R}{4}\right) / P\left(\frac{2R}{3}\right) = \frac{63}{80}$
• $P\left(\frac{3R}{5}\right) / P\left(\frac{2R}{5}\right) = \frac{16}{21}$