Homework answers / question archive / CHAPTER 1 ASSIGNMENT The motion of the planet takes place according to the law of universal gravitation

CHAPTER 1 ASSIGNMENT The motion of the planet takes place according to the law of universal gravitation

Math

Share With

---

CHAPTER 1

ASSIGNMENT

The motion of the planet takes place according to the law of universal gravitation. Accordingly, the Newtonian gravitational force between any two objects with masses m1 and m2 is

?

F12 = −Gm1m2|?/r1 − ?r2|2 ?*e12

where ?e12 is the unit vector in the direction ?r1 − ?r2 and the gravitational constant G = 6.67 ×

10−11 N m2 kg−2

.Note that all the planets in the solar system move (approximately) in the same a flat surface; so two-dimensional simulation will be temporary enough and save time.

1. Simulate the motion of a planet under the influence of the sun’s gravity (which you can

assume it doesn’t move) from time tmin to time tmax using time step ?t = τ. Use ode45 or

ode23.

It can be assumed that the force of gravity is inversely proportional to the exponential distance

β, it mean ?

F ∼ |?

r1 − ?

r2|−β

(where β = 2 is the normal force of gravity). Output to a file

the columns containing position, velocity, kinetic, potential and (total) energy data as time

dependent functions.

2) Determine the reasonable value for the time step deltat =. What is the initial τ prediction? Why? For a particular planet e.g. Earth and using β = 2, optimize τ using the requirement of total conservation of energy. To achieve this goal, calculate the energy change over a traverse and plot it as a function of τ. Choose an appropriate way to plot the results.

3) Check Keplers third law for all planets with near circular orbits (for planetary parameters see table below). Show how to choose the initial conditions to obtain circular orbits.

Request:

1. Write the theory to pdf file

2. Answer the questions in pdf file