Question

Interstellar cloud having a diameter of 10¹³ km and a rotation period of 10⁶ years collaps- ing to a sphere the size of the Sun (1.4 x 10⁶ km in diameter). We point out that if all the cloud’s angular momentum went into that sphere, the sphere would have a rotation period of only 0.6 second. Do the calculation to confirm this result.

Answer 1

Hello,

Let us understand that here disc shaped cloud rotating about its own axis is forming/collapsing into a sphere. Let us now look at things that have remained constant before as well as after the change from disc to sphere took place. Firstly its the 'mass' of the cloud as all the matter that was present in cloud form would just be present in the sphere. Secondly, we are considering from the question that the 'angular momentum' too remained same both in cloud form as well as in sphere form.

The angular momentum 'L' is given by L=I* ; where I- Moment of Inertia and -angular velocity

Hence we can equate the angular momentums before and after the change, giving us

L₁=L₂

==>I₁₁=I₂₂---------------> 1

Where I₁ is moment of inertia of a disc given by=1/4*M₁*R₁² ; I₂ is moment of inertia of a solid sphere given by=2/5*M₂*R₂² ;

The rotation period of disc cloud is given as 106 years. Hence its angular velocity is

₁=2/(106*365*24*60*60) rad/sec

Let us assume that rotation period of sphere is x sec. Then ₂=2/x.

Substituting above in 1 gives;

(1/4*M₁*R₁²)*(2/(106*365*24*60*60)) = (2/5*M₂*R₂²)*(2/x)

Since M₁=M₂ both cancel out and above gives

x=2*(0.7*106)²*4*106*365*24*60*60/(5*506.5²) = 0.6sec