Partnership for Integration of Computation into Undergraduate Physics

### Exercise 1 First, let's decide what parameters we will use for our binary star system. Assume Star 1 is one solar mass and Star 2 is twice the mass of Star 1. Place the center of mass of the system at the origin. Use the initial position and velocity of Star 1 given in the table below. | Parameter | Value | | -- | -- | | $m_1$ | $2\times 10^{30}$ kg | | $m_2$ | $2m_1$ | | $\vec{r}_1$ | $<1.5\times 10^{11},0,0>$ m | | $\vec{v}_1$ | $<0,3.3 \times 10^4,0>$ m/s | Compute the initial position and initial velocity of Star 2 given that the center of mass of the system should be at the origin and the center-of-mass velocity should be zero. ### Exercise 2 Model the motion of the stars using Newton's Second Law and Newton's Law of Gravitation. Plot their orbits. Create graphs of the total energy $E$, kinetic energy $K$, and potential energy $U$ as a function of separation distance $r$ for the system. ### Exercise 3 Represent the system as a single particle and model its motion by numerically solving: $$\mu\ddot{r}=-\frac{Gm_1m_2}{r^2}+\frac{L^2}{\mu r^3}$$ and $$\mu \dot{\phi}=\frac{L}{r^2}$$ where $L=|\vec{L}|$ is the magnitude of the angular momentum of the system. Create graphs of the total energy $E$, kinetic energy $K$, and potential energy $U$ of the system as a function of $r$. ### Exercise 4 **Now here is the big deal! This single star and its motion can be used to tell you what the two-star system is doing without using Newton's Second Law.** (Admittedly, this is a bigger deal from a theoretical perspective than a computational perspective, perhaps.) Starting with your program in Exercise 3, during each time step, calculate the position of Stars 1 and 2 in Cartesian Coordinates based on the polar coordinates $r$ and $\phi$. Show the orbits of both stars along with the single particle model. Numerically computing the orbits of stars from the single particle model is more difficult perhaps than using Newton's Second Law as you did in Exercise 2. In some ways, this exercise demonstrates that the single particle model is far more useful theoretically than computationally. The potential energy graph for the single particle model can give useful insights. Furthermore, treating a two-body problem as a one-body problem has applications for analyzing diatomic molecules and applying the Bohr model to determine energy states of positronium.