For central force motions with an inverse square law force of attraction the relation between energy \(E\) and actions \(I_{1}, I_{2}\) is (14.17), demonstrated in Problem 9 . If the strength of the force \((\) i.e. \(|k|)\) decreases slowly ('tired sun'), use the principle of adiabatic invariance to show that the period \(\tau\) of a bounded orbit (an ellipse, see Problem 5) varies so that \(\tau \propto k^{-2}\) (i.e. \(\tau\) increases). Find how the semi-major axis of the ellipse varies with \(k\) and show that the eccentricity of the ellipse remains constant.

The relation between energy E and actions I_{1}, I_{2} is given by: E = \frac{k^{2}}{2(I_{1} - I_{2})^{2}} - \frac{k}{I_{1} - I_{2}} where E is the energy, k is the strength of the force, and I_{1} and I_{2} are the actions.

The principle of adiabatic invariance states that the action variables I_{1} and I_{2} are constant when there is a slow change in the force (tired sun). In this case, the change in k is slow, so the action variables remain constant.

We know that the period τ of a bounded orbit is given by: τ = 2 \pi (\frac{a^3}{G(M + m)})^\frac{1}{2} where a is the semi-major axis of the ellipse, and G, M, and m are the gravitational constant, the mass of the central body, and the mass of the orbiting body, respectively. Since we're only dealing with central forces, we can substitute F = k/r^2 for G(M + m). Therefore, τ = 2 \pi (\frac{a^3}{\frac{k}{r^2}})^\frac{1}{2} Now, we can square both sides and isolate k in terms of τ: k = r^2 \frac{a^3}{(τ^2/4 \pi^{2})} As we've shown earlier, the action variables I_{1} and I_{2} are constant due to the adiabatic invariance, which means the sum of constants squared involving semi-major axis a is also constant. Therefore, τ^2 is proportional to k^{-2} since k needs to decrease to maintain constant action variables.

From the previous step, we can write the proportionality between τ^2 and k^{-2} as: τ^2 \propto k^{-2} It can be rewritten as: τ^2 = C k^{-2} Now, we have: k = r^2 \frac{a^3}{(C k^{-2})} where C is a constant of proportionality. Simplifying the equation, we get: k^3 = r^2 a^3 C This shows that the semi-major axis a varies with k^{\frac{1}{3}}: a \propto k^{\frac{1}{3}}

The eccentricity e of an ellipse can be defined as: e = \sqrt{1 - \frac{I_{2}^2}{I_{1}^2}} Since I_{1} and I_{2} are constant due to adiabatic invariance, the eccentricity e remains constant, proving that it doesn't change with the variations in k. In conclusion, when the strength of the force k of a central force motion with an inverse square law force of attraction decreases slowly, the period τ of a bounded orbit (an ellipse) varies as τ ∝ k^{-2} (i.e., τ increases). The semi-major axis of the ellipse varies with k^{\frac{1}{3}}, and the eccentricity of the ellipse remains constant.