Why do Planets keep spinning?

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To answer to this question, we need to start from the physic concept of Angular Momentum.

Conservation of Angular Momentum

Considering 2 body, the angular momentum of body m_2 relative to m_1 is:

(1)   \begin{equation*} \vec{H}_{21} = \vec{r} \wedge m_2 \dot{\vec{r}} \end{equation*}

The path of body m2 with respect to m1
[Figure 1] Path of m_2 around m_1 lies in a plane

where \dot{\vec{r}} = v . Dividing for m_2 , it is defined the specific angular momentum.

(2)   \begin{equation*} \vec{h} = \frac{\vec{H_{21}}}{m_2} = \vec{r} \wedge \dot{\vec{r}} \end{equation*}

Now, we can find the derivative of \vec{h} :

(3)   \begin{equation*} \begin{align} \frac{d\vec{h}}{dt} &= \frac{d\vec{r}}{dt} \wedge \dot{\vec{r}} & &+ \vec{r} \wedge \frac{d\dot{\vec{r}}}{dt} \\ \\ &= \dot{\vec{r}} \wedge \dot{\vec{r}} & &+ \vec{r} \wedge \ddot{\vec{r}} \end{align} \end{equation*}

The first term is {\dot{r} \dot{r} \sin{\theta} = 0} , since \dot{r} \| \dot{r} .

Second term can be analyzed in this way:

    \begin{equation*}\begin{align} \vec{r} \wedge \ddot{\vec{r}} &= \vec{r} \wedge \Big( - \frac{\mu}{r^3} \vec{r} \Big) \\ \\ &= - \frac{\mu}{r^3} (\vec{r} \wedge \vec{r} ) \end{align} \end{equation*}

Since r \| r , also the second term is zero.

Therefore, angular momentum is conserved:

(4)   \begin{equation*} \dot{\vec{h}} = \frac{d\vec{h}}{dt} = 0 \end{equation*}

Conclusion

If the position vector and the velocity vector are parallel, then it follows from eq. (2) that the angular momentum is zero, and according to eq. (4), it remains zero at all points of the trajectory (rectilinear motion).

Considering a curvilinear trajectory, it is possible to state that the position vector and the velocity vector lie in the same plane (as illustrated in Figure 1). This means that the path of m_2 around m_1 lies in a single plane.

Since \dot{\vec{h}} = 0 \Rightarrow \vec{h} = cost , which explains why planets are forced to rotate around Sun. If there is a reduction in the position vector, there should be an increase in speed, in order to maintain the specific angular momentum constant.

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2 thoughts on “Why do Planets keep spinning?”

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