(Solved): 5. Lagrange Points (10 points): Recently, a new space telescope (JWST) was launched. Its final des ...

5. Lagrange Points (10 points): Recently, a new space telescope (JWST) was launched. Its final destination was an orbit near the so-called L2 "Lagrange" point - a region where the gravitational forces of the Earth and the Sun partially cancel. Illustration the path the JWST took to reach its final orbit near the L2 point, along with (inset) the location of the five Lagrange points of the Earth-Sun system. We will assume the telescope is in a circular orbit with radius \( r \) with a period matching that of the Earth. If \( r>R_{e} \) the acceleration of the telescope due to the gravity of both the Sun and Earth is \[ a=\frac{G M_{s}}{r^{2}}+\frac{G M_{e}}{\left(R_{e}-r\right)^{2}} \] where \( G \) is the gravitational constant, \( M_{s} \), is the mass of the Sun, \( R_{e} \) is the radius of the Earth's orbit. and \( M_{e} \) is the mass of the Earth. (a) (2 point) The telescope is executing an orbit with angular frequency \( \omega=\sqrt{G M_{s} / R_{e}^{3}} \), matching that of the Earth. What is its acceleration, \( a \) ? Hint: Remember what you know about circular motion. (b) (3 points) Combining your result for (a) with the given gravitational acceleration, show that the position of the telescope must satisfy \[ x=\frac{1}{x^{2}}+\frac{M_{e}}{M_{s}} \frac{1}{(1-x)^{2}} \] where \( x \equiv r / R_{e} \) and \( M_{e} / M_{s} \approx 3 \cdot 10^{-6} \). (c) (4 points) Write a Python program to produce a plot of the both sides of the equation from part (b) and find the (physical) solution for \( x \) to an accuracy of three digits. Note: Use numpy arrays compute each side. You will have to use plt. \( \mathrm{xlim} \) and plt.ylim to zoom in and isolate the location of the crossing point. Submit your code and a plot showing the \( \mathrm{cr} \) point. Hint: It isn't very far from \( x=1 \). (d) (1 point) How far is the L2 point from the Earth? Use that \( R_{e} \approx 1.496 \cdot 10^{8} \mathrm{~km} \).