Precession of a spherical top in orbit around a rotating star

In summary, the conversation discusses L&L's solution to problem four of section §106, which involves finding the Lagrangian for a system of particles. The conversation delves into the first and second deviations, represented by ##\delta^{(1)}L## and ##\delta^{(2)}L##, respectively. The first deviation arises from a term involving the masses and velocities of the particles, while the second deviation accounts for the rotation of the star. The formula for ##h_{03}## is mentioned and there is a discussion about how to use it to arrive at the formula for ##\delta^{(2)}L##. There is some uncertainty and confusion about the exact calculations and approximations being made.
  • #1
etotheipi
Looking at L&L's solution to problem four of section §106. Lagrangian for a system of particles:\begin{align*}
L = &\sum_a \frac{m_a' v_a^2}{2} \left( 1 + 3\sum_{b}' \frac{km_b}{c^2 r_{ab}} \right) + \sum_a \frac{m_a v_a^4}{8c^2} + \sum_a \sum_b' \frac{km_a m_b}{2r_{ab}} \\

&- \sum_a \sum_b' \frac{km_a m_b}{4c^2 r_{ab}} \left[ 7 \mathbf{v}_a \cdot \mathbf{v}_b +(\mathbf{v}_a \cdot \mathbf{n}_{ab})(\mathbf{v}_b \cdot \mathbf{n}_{ab}) \right] - \sum_a \sum_b' \sum_c' \frac{k^2 m_a m_b m_c}{2c^2 r_{ab} r_{ac}}\end{align*}I think in the first approximation L&L are writing ##L = L_0 + \delta^{(1)}L + \delta^{(2)} L## with ##L_0 = \sum_a \frac{m_a v_a^2}{2}## and the first deviation ##\delta^{(1)}L## is arising from the term ##\sum_a \frac{m_a' v_a^2}{2} \sum_{b}' \frac{3km_b}{c^2 r_{ab}}##, where I guess they were considering something like\begin{align*}

\sum_{a \in \mathrm{top}} \sum_{b \in \mathrm{star}} \frac{3km_a m_b(\mathbf{V} + \boldsymbol{\omega} \times \mathbf{r})^2}{2c^2 r_{ab} } &\overset{\mathrm{continuum}}{\longrightarrow} \frac{3km'}{2c^2} \int_{\mathrm{top}} \frac{2}{R(\mathbf{r}')}(\mathbf{V} \cdot \boldsymbol{\omega} \times \mathbf{r}) dm\end{align*}having dropped the ##V^2## and neglecting anything in ##\omega^2##, also ##m'## is the mass of the star. It simplifies writing ##\int_{\mathrm{top}} x_i x_j dm = \frac{1}{2} I \delta_{ij}##. I don't know which term they used to write ##\delta^{(2)} L##; this second deviation is supposed to account for the rotation of the star; they say you can make use of the equation$$h_{03} = \frac{2kM'}{Rc^2} \sin^2{\theta}$$How do you use this? You're supposed to get$$\delta^{(2)} L = \frac{2k}{c^2} \int_{\mathrm{top}} \frac{\mathbf{M}' \cdot (\boldsymbol{\omega} \times \mathbf{r}) \times \mathbf{R}}{R^3} dm$$here ##\mathbf{M}'## is the angular momentum of the star. How do you arrive at this given the formula for ##h_{03}##? Very lost.
 
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  • #2
I wonder if it could be$$\delta^{(2)} L = -c \int h_{03} v(\mathbf{r}')^3 dm = - \frac{2k}{c} \int_{\mathrm{top}} \frac{M' v(\mathbf{r})^3 \sin^2{\theta}}{R} dm = \int_{\mathrm{top}} \frac{M' (\boldsymbol{\omega} \times \mathbf{r})^3 \sin^2{\theta}}{R} dm$$but given ##\mathbf{M}' \cdot (\boldsymbol{\omega} \times \mathbf{r}) \times \mathbf{R} = - \mathbf{M}' \cdot \mathbf{R} \times (\boldsymbol{\omega} \times \mathbf{r}) =-(\boldsymbol{\omega} \cdot \mathbf{M}')(\mathbf{R} \cdot \mathbf{r}) + (\boldsymbol{\omega} \cdot \mathbf{R})( \mathbf{r} \cdot \mathbf{M}')## it's not so clear how to show the two integrals are the same? Maybe there's some approximations being made?
 
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1. What is the precession of a spherical top in orbit around a rotating star?

The precession of a spherical top in orbit around a rotating star is a phenomenon that occurs when an object, such as a planet or moon, is in orbit around a rotating star. Due to the gravitational pull of the star and the rotation of the star, the object's orbit will gradually change over time, causing it to rotate or "precess" around the star in a slightly different direction than its original orbit.

2. What causes the precession of a spherical top in orbit around a rotating star?

The precession of a spherical top in orbit around a rotating star is primarily caused by the combined effects of gravity and rotational forces. The gravitational pull of the star causes the object to orbit around it, while the rotation of the star creates a centrifugal force that pulls the object in a slightly different direction, resulting in precession.

3. How is the precession of a spherical top in orbit around a rotating star measured?

The precession of a spherical top in orbit around a rotating star can be measured by observing the changes in the object's orbit over time. This can be done through precise measurements of the object's position and velocity, as well as tracking its movement relative to other objects in the same system.

4. What are the real-world applications of studying the precession of a spherical top in orbit around a rotating star?

Studying the precession of a spherical top in orbit around a rotating star can provide valuable insights into the dynamics of celestial bodies and their interactions with each other. This knowledge can be applied to better understand and predict the movements of planets, moons, and other objects in our solar system, as well as in other planetary systems.

5. Can the precession of a spherical top in orbit around a rotating star be affected by other factors?

Yes, the precession of a spherical top in orbit around a rotating star can also be influenced by other factors such as the shape and mass distribution of the object, the presence of other nearby objects, and external forces such as solar wind. These factors can cause variations in the precession rate and direction of the object's orbit.

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