Evolution Equations From Tidal Torque+Power¶
The orbital energy and angular momentum are:
\[\begin{split}E&=&-\frac{GMM'}{2a}\\
L&=&\frac{MM'}{M+M'}a^2\Omega\sqrt{1-e^2}
=GMM'\sqrt{\frac{(1-e^2)MM'}{(-2E)(M+M')}}\end{split}\]
Hence:
\[\begin{split}\dot{a}&=&a\frac{-\dot{E}}{E}\\
\dot{e}&=&\frac{(M+M')}{G^2(MM')^3}\frac{(\dot{E}L+2E\dot{L})L}{e}\\\end{split}\]
Now consider a single zone subject to tidal torques. We will use the following variables:
- \(\mathbf{S}\) and \(S\): the spin angular momentum vector of the zone and its absolute value
- \(\mathbf{L}\) and \(L\): the orbital angular momentum vector and its absolute value
- \(\theta\): angle between the angular momentum vector of the zone and the angular momentum of the orbit (inclination).
- \(\omega\): the argument of periapsis of the orbit with a plane of reference perpendicular to \(\mathbf{S}\).
- \(\mathbf{T}\): the tidal torque vector acting on this zone (and of course with a negative sign on the orbit).
- \(\mathbf{\tilde{T}}\): the negative of the tidal torques on the orbit due to other zones.
- \(\mathbf{\mathscr{T}}\): the torque on this zone due to coupling to other zones (e.g. due to differential rotation coupling or mass exchange).
- \(\mathbf{\hat{p}}\): a unit vector along the periapsis of the orbit
- \(\mathbf{\hat{z}}\): a unit vector along \(\mathbf{S}\).
- \(\mathbf{\hat{y}}\): a unit vector along the ascending node of the orbit.
- \(\mathbf{\hat{x}}\): \(\mathbf{\hat{y}}\times\mathbf{\hat{z}}\).
We will use primed quantities to denote the updated value of a quantity after an infinitesimal times step, and will use x, y and z indices to indicate projections of quantities alonge \(\mathbf{\hat{x}}\), \(\mathbf{\hat{y}}\) and \(\mathbf{\hat{z}}\) respectively.
\[\begin{split}\mathbf{S} &=& S\mathbf{\hat{z}}\\
\mathbf{L} &=& L\sin\theta\mathbf{\hat{x}}+L\cos\theta\mathbf{\hat{z}}\\
\mathbf{\hat{y}} &=& \frac{\mathbf{S}\times\mathbf{L}}{LS\sin\theta}\\
\mathbf{\hat{p}} &=& - \sin\omega\cos\theta\mathbf{\hat{x}}
+ \cos\omega\mathbf{\hat{y}}
+ \sin\omega\sin\theta\mathbf{\hat{z}}\\
\mathbf{S}' &=& (T_x+\mathscr{T}_x)dt\mathbf{\hat{x}}
+ (T_y+\mathscr{T}_y)dt\mathbf{\hat{y}}
+ (S+T_zdt+\mathscr{T}_zdt)\mathbf{\hat{z}}\\
\mathbf{L}' &=& (L\sin\theta-T_xdt-\tilde{T}_xdt)\mathbf{\hat{x}}
- (T_y+\tilde{T}_y)dt\mathbf{\hat{y}}
+ (L\cos\theta-T_zdt-\tilde{T}_zdt)\mathbf{\hat{z}}\\
\frac{1}{S'} &=& \frac{1}{S} - \frac{T_z+\mathscr{T}_z}{S^2}dt\\
\frac{1}{L'} &=& \frac{1}{L}
+ \frac{(T_x+\tilde{T}_x)\sin\theta
+ (T_z+\tilde{T}_z)\cos\theta}{L^2}dt\\
\frac{1}{L'S'} &=& \frac{1}{LS}\left(1 - \frac{T_z+\mathscr{T}_z}{S}dt
+ \frac{(T_x+\tilde{T}_x)\sin\theta
+ (T_z+\tilde{T}_z)\cos\theta}{L}dt
\right)\\
\Rightarrow \frac{d}{dt}\left(\frac{1}{LS}\right) &=&
\frac{1}{LS}\left(\frac{(T_x+\tilde{T}_x)\sin\theta
+ (T_z+\tilde{T}_z)\cos\theta}{L}
- \frac{T_z+\mathscr{T}_z}{S}
\right)\end{split}\]
In order to derive the evolution rate for the inclination:
\[\begin{split}\cos\theta &=& \frac{\mathbf{S}\cdot\mathbf{L}}{LS}\\
\cos\theta' &=& \frac{\mathbf{S}'\cdot\mathbf{L}'}{L'S'}\\
&=& \frac{L\sin\theta(T_x+\mathscr{T}_x)dt + LS\cos\theta
+ L\cos\theta(T_z+\mathscr{T}_z)dt
- S(T_z+\tilde{T}_z)dt}{LS}
\left(1 - \frac{T_z+\mathscr{T}_z}{S}dt
+ \frac{(T_x+\tilde{T}_x)\sin\theta
+ (T_z+\tilde{T}_z)\cos\theta}{L}dt
\right)\\
&=& \cos\theta -\frac{\cos\theta(T_z+\mathscr{T}_z)}{S}dt
+ \frac{(T_x+\tilde{T}_x)\sin\theta\cos\theta
+ (T_z+\tilde{T}_z)\cos^2\theta}{L}dt
+\frac{\sin\theta(T_x+\mathscr{T}_x)}{S}dt
+ \frac{\cos\theta(T_z+\mathscr{T}_z)}{S}dt
- \frac{T_z+\tilde{T}_z}{L}dt\\
&=& \cos\theta + \frac{(T_x+\tilde{T}_x)\sin\theta\cos\theta
- (T_z+\tilde{T}_z)\sin^2\theta}{L}dt
+\frac{\sin\theta(T_x+\mathscr{T}_x)}{S}dt\\
\Rightarrow \dot{\theta} &=& \frac{(T_z+\tilde{T}_z)\sin\theta}{L}
- \frac{(T_x+\tilde{T}_x)\cos\theta}{L}
- \frac{T_x+\mathscr{T}_x}{S}\end{split}\]
Now to derive the evolution rate for the argument of periapsis:
\[\dot{\omega} = -\frac{1}{\sin\omega}\frac{d(\mathbf{\hat{p}}\cdot\mathbf{\hat{y}})}{dt}
= -\frac{1}{\sin\omega}\left(
\mathbf{\hat{y}}\cdot\frac{d\mathbf{\hat{p}}}{dt}
+ \mathbf{\hat{p}}\cdot\frac{d\mathbf{\hat{y}}}{dt}
\right)\]
\[\begin{split}\frac{d}{dt}\left(\mathbf{S}\times\mathbf{L}\right)
&=& - S(T_x+\tilde{T}_x)\mathbf{\hat{y}}
+ S(T_y+\tilde{T}_y)\mathbf{\hat{x}}
- L\cos\theta(T_x+\mathscr{T}_x)\mathbf{\hat{y}}
+ L\cos\theta(T_y+\mathscr{T}_y)\mathbf{\hat{x}}
- L\sin\theta(T_y+\mathscr{T}_y)\mathbf{\hat{z}}
+ L\sin\theta(T_z+\mathscr{T}_z)\mathbf{\hat{y}}\\
&=& \left[S(T_y+\tilde{T}_y)
+ L\cos\theta(T_y+\mathscr{T}_y)\right]\mathbf{\hat{x}}
+ \left[L\sin\theta(T_z+\mathscr{T}_z)
- L\cos\theta(T_x+\mathscr{T}_x) - S(T_x+\tilde{T}_x)
\right]\mathbf{\hat{y}}
- L\sin\theta(T_y+\mathscr{T}_y)\mathbf{\hat{z}}\end{split}\]
From \(\dot{\theta}\):
\[\frac{d}{dt}\left(\frac{1}{\sin\theta}\right)
=
\frac{(T_x+\tilde{T}_x)\cos^2\theta}{L\sin^2\theta} +
\frac{(T_x+\mathscr{T}_x)\cos\theta}{S\sin^2\theta} -
\frac{(T_z+\tilde{T}_z)\cos\theta}{L\sin\theta}\]
Combining \(\frac{d}{dt}\left(\frac{1}{LS}\right)\), \(\frac{d}{dt}\left(\mathbf{S}\times\mathbf{L}\right)\) and \(\frac{d}{dt}\left(\frac{1}{\sin\theta}\right)\):
\[\begin{split}\frac{d}{dt}\mathbf{\hat{y}}
&=& \left\{\frac{(T_x+\tilde{T}_x)\sin\theta
+ (T_z+\tilde{T}_z)\cos\theta}{L}
- \frac{T_z+\mathscr{T}_z}{S}
\right\}\mathbf{\hat{y}}\\
&& +
\frac{1}{LS\sin\theta}\left\{
\left[S(T_y+\tilde{T}_y)
+ L\cos\theta(T_y+\mathscr{T}_y)\right]\mathbf{\hat{x}}
+ \left[L\sin\theta(T_z+\mathscr{T}_z)
- L\cos\theta(T_x+\mathscr{T}_x)
- S(T_x+\tilde{T}_x)
\right]\mathbf{\hat{y}}
- L\sin\theta(T_y+\mathscr{T}_y)\mathbf{\hat{z}}
\right\}\\
&& +
\left\{\frac{(T_x+\tilde{T}_x)\cos^2\theta}{L\sin\theta}
+ \frac{(T_x+\mathscr{T}_x)\cos\theta}{S\sin\theta}
- \frac{(T_z+\tilde{T}_z)\cos\theta}{L}
\right\}\mathbf{\hat{y}}\\
&=& \left[\frac{T_y+\tilde{T}_y}{L\sin\theta} +
\frac{\cos\theta(T_y+\mathscr{T}_y)}{S\sin\theta}
\right]\mathbf{\hat{x}}\\
&& +
\left[\frac{(T_x+\tilde{T}_x)\sin\theta}{L}
+ \frac{(T_z+\tilde{T}_z)\cos\theta}{L}
- \frac{T_z+\mathscr{T}_z}{S}
- \frac{\cos\theta(T_x+\mathscr{T}_x)}{S\sin\theta}
+ \frac{T_z+\mathscr{T}_z}{S}
- \frac{T_x+\tilde{T}_x}{L\sin\theta}
+ \frac{(T_x+\tilde{T}_x)\cos^2\theta}{L\sin\theta}
+ \frac{(T_x+\mathscr{T}_x)\cos\theta}{S\sin\theta}
- \frac{(T_z+\tilde{T}_z)\cos\theta}{L}
\right]\mathbf{\hat{y}}\\
&& - \frac{T_y+\mathscr{T}_y}{S}\mathbf{\hat{z}}\\
&=& \left[
\frac{T_y+\tilde{T}_y}{L\sin\theta}
+
\frac{(T_y+\mathscr{T}_y)\cos\theta}{S\sin\theta}
\right]\mathbf{\hat{x}}
-
\frac{T_y+\mathscr{T}_y}{S}\mathbf{\hat{z}}\\
\Rightarrow
-\frac{\mathbf{\hat{p}}}{\sin\omega}\cdot\frac{d}{dt}\mathbf{\hat{y}}
&=& \frac{(T_y+\tilde{T}_y)\cos\theta}{L\sin\theta}
+ \frac{(T_y+\mathscr{T}_y)\cos^2\theta}{S\sin\theta}
+ \frac{(T_y+\mathscr{T}_y)\sin\theta}{S}\\
&=& \frac{(T_y+\tilde{T}_y)\cos\theta}{L\sin\theta}
+ \frac{T_y+\mathscr{T}_y}{S\sin\theta}\end{split}\]
The evolution of the direction of periapsis:
\[\begin{split} \frac{d}{dt}\mathbf{\hat{p}}
&=& -(\mathbf{T}+\mathbf{\tilde{T}})
\cdot\mathbf{\hat{p}}\frac{\mathbf{L}}{L^2}\\
&=& \left(\frac{(T_x+\tilde{T}_x)\sin\omega\cos\theta}{L}
- \frac{(T_y+\tilde{T}_y)\cos\omega}{L}
- \frac{(T_z+\tilde{T}_z)\sin\omega\sin\theta}{L}
\right)
\left(\sin\theta\mathbf{\hat{x}}+\cos\theta\mathbf{\hat{z}}\right)\\
\Rightarrow
-
\frac{\mathbf{\hat{y}}}{\sin\omega}\cdot\frac{d}{dt}\mathbf{\hat{p}}
&=&
0\end{split}\]
So we get:
\[\dot{\omega} = \frac{(T_y+\tilde{T}_y)\cos\theta}{L\sin\theta}
+ \frac{T_y+\mathscr{T}_y}{S\sin\theta}\]
Finally:
\[\begin{split}\dot{S} &=& T_z+\mathscr{T}_z\\
\dot{L} &=& -T_x\sin\theta - T_z\cos\theta\end{split}\]
What remanains is to find \(\tilde{T}_x\), \(\tilde{T}_y\) and \(\tilde{T}_z\). All that is necessary is to express the coornidate system unit vectors of all other zones in terms of the ones for this zone. We will use \(\mathbf{\hat{\tilde{x}}}\), \(\mathbf{\hat{\tilde{y}}}\) and \(\mathbf{\hat{\tilde{z}}}\) to refer to the unit vectors of another zone, and we will denote the difference between this zone’s argument of periapsis and the second zone by \(\Delta\omega\).
Clearly:
\[\mathbf{\hat{\tilde{y}}}=-\cos\theta\sin\Delta\omega\mathbf{\hat{x}}
+ \cos\Delta\omega\mathbf{\hat{y}}
+ \sin\theta\sin\Delta\omega\mathbf{\hat{z}}\]
Next:
\[\begin{split} \mathbf{\hat{\tilde{z}}}
&=&
\cos\tilde{\theta}\mathbf{\hat{L}}
+
\sin\tilde{\theta}\mathbf{\hat{L}}\times\mathbf{\hat{\tilde{y}}}\\
&=&
\sin\theta\cos\tilde{\theta}\mathbf{\hat{x}}
+
\cos\theta\cos\tilde{\theta}\mathbf{\hat{z}}
+
\sin\tilde{\theta}\left(
\sin\theta\cos\Delta\omega\mathbf{\hat{z}}
- \sin^2\theta\sin\Delta\omega\mathbf{\hat{y}}
- \cos^2\theta\sin\Delta\omega\mathbf{\hat{y}}
- \cos\theta\cos\Delta\omega\mathbf{\hat{x}}
\right)\\
&=&
\left(
\sin\theta\cos\tilde{\theta}
-
\cos\theta\sin\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{x}}
-
\sin\tilde{\theta}\sin\Delta\omega\mathbf{\hat{y}}
+
\left(
\cos\theta\cos\tilde{\theta}
+
\sin\theta\sin\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{z}}\end{split}\]
Finally:
\[\begin{split} \mathbf{\hat{\tilde{x}}}
&=&
\sin\tilde{\theta}\mathbf{\hat{L}}
-
\cos\tilde{\theta}\mathbf{\hat{L}}\times\mathbf{\hat{\tilde{y}}}\\
&=&
\left(
\sin\theta\sin\tilde{\theta}
+
\cos\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{x}}
+
\cos\tilde{\theta}\sin\Delta\omega\mathbf{\hat{y}}
+
\left(
\cos\theta\sin\tilde{\theta}
-
\sin\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{z}}\end{split}\]
Crosscheck that \(\mathbf{\hat{\tilde{x}}}\times\mathbf{\hat{\tilde{y}}} = \mathbf{\hat{\tilde{z}}}\):
\[\begin{split}\mathbf{\hat{\tilde{x}}}\times\mathbf{\hat{\tilde{y}}}
&=& \left(\sin\theta\sin\tilde{\theta}
+ \cos\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\cos\Delta\omega\mathbf{\hat{z}}
-
\left(\sin\theta\sin\tilde{\theta}
+ \cos\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\sin\theta\sin\Delta\omega\mathbf{\hat{y}}\\
&& +
\cos\tilde{\theta}\sin\Delta\omega \cos\theta\sin\Delta\omega
\mathbf{\hat{z}}
+
\cos\tilde{\theta}\sin\Delta\omega \sin\theta\sin\Delta\omega
\mathbf{\hat{x}}\\
&& -
\left(\cos\theta\sin\tilde{\theta}
- \sin\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\cos\theta\sin\Delta\omega\mathbf{\hat{y}}
-
\left(\cos\theta\sin\tilde{\theta}
- \sin\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\cos\Delta\omega\mathbf{\hat{x}}\\
&=& \left(
\sin\theta\cos\tilde{\theta}\sin^2\Delta\omega
-
\cos\theta\sin\tilde{\theta}\cos\Delta\omega
+
\sin\theta\cos\tilde{\theta}\cos^2\Delta\omega
\right)\mathbf{\hat{x}}
-\left(
\sin^2\theta\sin\tilde{\theta}\sin\Delta\omega
+
\sin\theta\cos\theta\cos\tilde{\theta}
\sin\Delta\omega\cos\Delta\omega
+
\cos^2\theta\sin\tilde{\theta}\sin\Delta\omega
-
\sin\theta\cos\theta\cos\tilde{\theta}
\sin\Delta\omega\cos\Delta\omega
\right)\mathbf{\hat{y}}
+\left(
\sin\theta\sin\tilde{\theta}\cos\Delta\omega
+
\cos\theta\cos\tilde{\theta}\cos^2\Delta\omega
+
\cos\theta\cos\tilde{\theta}\sin^2\Delta\omega
\right)\mathbf{\hat{z}}\\
&=& \left(
\sin\theta\cos\tilde{\theta}
-
\cos\theta\sin\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{x}}
-
\sin\tilde{\theta}\sin\Delta\omega\mathbf{\hat{y}}
+
\left(
\cos\theta\cos\tilde{\theta}
+
\sin\theta\sin\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{z}}\end{split}\]
Which is exactly \(\mathbf{\hat{\tilde{z}}}\).
Crosscheck that the direction of the orbital angular momentum matches, i.e. that
\(\sin\tilde{\theta}\mathbf{\hat{\tilde{x}}} + \cos\tilde{\theta}\mathbf{\hat{\tilde{z}}} = \sin\theta\mathbf{\hat{x}} + \cos\theta\mathbf{\hat{z}}\):
\[\begin{split} \sin\tilde{\theta}\mathbf{\hat{\tilde{x}}}
+
\cos\tilde{\theta}\mathbf{\hat{\tilde{z}}}
&=&
\left(
\sin\theta\sin^2\tilde{\theta}
+
\cos\theta\sin\tilde{\theta}\cos\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{x}}
+
\sin\tilde{\theta}\cos\tilde{\theta}\sin\Delta\omega\mathbf{\hat{y}}
+
\left(
\cos\theta\sin^2\tilde{\theta}
-
\sin\theta\sin\tilde{\theta}\cos\tilde{\theta}
\cos\Delta\omega
\right)\mathbf{\hat{z}}\\
&&
+
\left(
\sin\theta\cos^2\tilde{\theta}
-
\cos\theta\sin\tilde{\theta}\cos\tilde{\theta}\cos\Delta\omega
\right)\mathbf{\hat{x}}
-
\sin\tilde{\theta}\cos\tilde{\theta}\sin\Delta\omega\mathbf{\hat{y}}
+
\left(
\cos\theta\cos^2\tilde{\theta}
+
\sin\theta\sin\tilde{\theta}\cos\tilde{\theta}
\cos\Delta\omega
\right)\mathbf{\hat{z}}\\
&=&
\sin\theta\mathbf{\hat{x}} + \cos\theta\mathbf{\hat{z}}\end{split}\]
Finally, crosscheck that the direction of periapsis is consistent, i.e. that
\[ - \sin(\omega-\Delta\omega)\cos\tilde{\theta}\mathbf{\hat{\tilde{x}}}
+ \cos(\omega-\Delta\omega)\mathbf{\hat{\tilde{y}}}
+ \sin(\omega-\Delta\omega)\sin\tilde{\theta}\mathbf{\hat{\tilde{z}}}
=
- \sin\omega\cos\theta\mathbf{\hat{x}}
+ \cos\omega\mathbf{\hat{y}}
+ \sin\omega\sin\theta\mathbf{\hat{z}}\]
We will go component by component:
\[\begin{split} \mathbf{\hat{\tilde{p}}}\cdot\mathbf{\hat{x}}
&=&
-\sin(\omega-\Delta\omega)\cos\tilde{\theta}
\left(
\sin\theta\sin\tilde{\theta}
+
\cos\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\\
&&
-\cos(\omega-\Delta\omega)\cos\theta\sin\Delta\omega\\
&&
+\sin(\omega-\Delta\omega)\sin\tilde{\theta}
\left(
\sin\theta\cos\tilde{\theta}
-
\cos\theta\sin\tilde{\theta}\cos\Delta\omega
\right)\\
&=&
-
\cos\theta\cos^2\tilde{\theta}\sin(\omega-\Delta\omega)\cos\Delta\omega
-
\cos\theta\cos(\omega-\Delta\omega)\sin\Delta\omega
-
\cos\theta\sin^2\tilde{\theta} \sin(\omega-\Delta\omega)\cos\Delta\omega\\
&=&
-
\cos\theta\sin(\omega-\Delta\omega)\cos\Delta\omega
-
\cos\theta\cos(\omega-\Delta\omega)\sin\Delta\omega\\
&=&
- \sin\omega\cos\theta\\
&=&
\mathbf{\hat{p}}\cdot\mathbf{\hat{x}}\end{split}\]
Next:
\[\begin{split}\mathbf{\hat{\tilde{p}}}\cdot\mathbf{\hat{y}}
&=&
-
\sin(\omega-\Delta\omega)\cos\tilde{\theta}
\cos\tilde{\theta}\sin\Delta\omega
+
\cos(\omega-\Delta\omega)\cos\Delta\omega
-
\sin(\omega-\Delta\omega)\sin\tilde{\theta}
\sin\tilde{\theta}\sin\Delta\omega\\
&=&
-
\sin(\omega-\Delta\omega)\sin\Delta\omega
+
\cos(\omega-\Delta\omega)\cos\Delta\omega\\
&=&
\cos\omega\\
&=&
\mathbf{\hat{p}}\cdot\mathbf{\hat{y}}\end{split}\]
Finally:
\[\begin{split} \mathbf{\hat{\tilde{p}}}\cdot\mathbf{\hat{z}}
&=&
- \sin(\omega-\Delta\omega)\cos\tilde{\theta}
\left(
\cos\theta\sin\tilde{\theta}
-
\sin\theta\cos\tilde{\theta}\cos\Delta\omega
\right)\\
&&
+ \cos(\omega-\Delta\omega)\sin\theta\sin\Delta\omega\\
&&
+ \sin(\omega-\Delta\omega)\sin\tilde{\theta}
\left(
\cos\theta\cos\tilde{\theta}
+
\sin\theta\sin\tilde{\theta}\cos\Delta\omega
\right)\\
&=&
+
\sin\theta\cos^2\tilde{\theta}\sin(\omega-\Delta\omega)\cos\Delta\omega
+
\cos(\omega-\Delta\omega)\sin\theta\sin\Delta\omega
+
\sin\theta\sin^2\tilde{\theta}
\sin(\omega-\Delta\omega)\cos\Delta\omega\\
&=&
\sin\theta\sin(\omega-\Delta\omega)\cos\Delta\omega
+
\sin\theta\cos(\omega-\Delta\omega)\sin\Delta\omega\\
&=&
\sin\theta\sin\omega\\
&=&
\mathbf{\hat{p}}\cdot\mathbf{\hat{z}}\end{split}\]