A very long (length L) cylindrical galaxy is made of uniformly distributed mass and has radius R (R<<L). A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is T and its distance from the galaxy's axis is r, then

The gravitational potential due to cylindrical mass distribution can be found by comparison with electric field due to cylindrical charge distribution.Applying gauss law for cylindrical charge distribution, consider a cylindrical Gaussian surface of length L:∮→E.d→A=Qϵ0E2πrL=λLϵ0E=λ2πϵ0rSimilarly gravitational field =E=2Gλrwhere14πε0=Gfrom similarityAs the Gravitational field is inversely proportional to distance from center, The centripetal force isFcentripetal=mE=m2Gλr.. (i)Fcentripetal=mω²r.. (ii)substituting value of centripetal force in equation (i) from equation (ii)mω²r=m2Gλrω=1r√2GλT=2πω=r√2Gλ⇒T∝√rhence correct answer is option B