A test particle is moving in a circular orbit in the gravitational field produced by a mass density $$\rho(r) = \frac{K}{r^2}$$. Identify the current relation between the radius R of the particle's orbit and its period T:

We have a continuous spherically-symmetric mass distribution whose density varies with the radial distance as $$\rho(r)=\dfrac{K}{r^{2}}$$, where $$K$$ is a constant. To find the relation between the orbital radius $$R$$ and the period $$T$$ of a light test particle moving in a circular orbit, we first determine the mass enclosed within any radius $$r$$.

The formula for mass enclosed inside a sphere of radius $$r$$ for a density distribution $$\rho(r)$$ is

$$M(r)=\displaystyle\int_{0}^{r}4\pi r'^{2}\,\rho(r')\,dr'.$$

Substituting $$\rho(r')=\dfrac{K}{r'^{2}}$$ gives

$$M(r)=\int_{0}^{r}4\pi r'^{2}\left(\dfrac{K}{r'^{2}}\right)dr'$$ $$=4\pi K\int_{0}^{r}dr'$$ $$=4\pi K\,r.$$

Thus the total mass inside the orbit grows linearly with radius:

$$M(r)=4\pi K\,r.$$

Next, the magnitude of the gravitational field at distance $$r$$ (for a spherically symmetric mass distribution) is given by Newton’s law of gravitation,

$$g(r)=\dfrac{G\,M(r)}{r^{2}},$$

where $$G$$ is the gravitational constant. Substituting $$M(r)=4\pi K\,r$$ yields

$$g(r)=\dfrac{G\,(4\pi K\,r)}{r^{2}}=\dfrac{4\pi G K}{r}.$$

Now consider a particle of mass $$m$$ in a circular orbit of radius $$R$$ and speed $$v$$. The required centripetal force is supplied by gravity, so we equate

$$\dfrac{m v^{2}}{R}=m\,g(R)=m\left(\dfrac{4\pi G K}{R}\right).$$

Canceling the common factor $$m$$ and the radius $$R$$ on both sides, we get

$$v^{2}=4\pi G K.$$

Thus $$v$$ is a constant (independent of $$R$$):

$$v=\sqrt{4\pi G K}.$$

The time period $$T$$ of the circular orbit is related to the speed $$v$$ through the usual relation for one full revolution,

$$T=\dfrac{\text{circumference}}{\text{speed}}=\dfrac{2\pi R}{v}.$$

Substituting the constant value of $$v$$ obtained above, we have

$$T=\dfrac{2\pi R}{\sqrt{4\pi G K}}.$$

Since the denominator is a constant, this expression shows a direct proportionality:

$$\dfrac{T}{R}=\dfrac{2\pi}{\sqrt{4\pi G K}}=\text{constant}.$$

Hence $$T/R$$ remains the same for any circular orbit in this density field, while all other listed combinations vary with $$R$$. This matches option C.

Hence, the correct answer is Option C.