This is a problem using MATLAB and I\'m not sure how to write the code for it Pl

Question

This is a problem using MATLAB and I'm not sure how to write the code for it

Plotting Orbits When a satellite orbits the Earth, the satellite's orbit will form an ellipse with the Earth located at one of the focal points of the ellipse. The satellite's orbit can he expressed in polar coordinates as where rand B are the distance and angle of the satellite from the center of the Earth, p is a parameter specifying the size of the size of the orbit, and £ is a parameter representing tile eccentricity of the orbit. A circular orbit has an eccentricity £ of 0. An elliptical orbit has an eccentricity of the satellite follows a hyperbolic path and escapes from live Earth's gravitational field. C onsider a satellite with a size parameter p = 10(H) km. Plot the orbit of this satellite if How close does each orbit come to live Earth? How far away does each orbit get from the Earth? Compare the three plots you created. Can you determine w hat the parameter p means from looking at the plots?

Explanation / Answer

The orbit of every planet is an ellipse with the Sun at one of the two foci.

Mathematically, an ellipse can be represented as:

r = p/(1 - e cos)

where p is the semi-latus rectum, and is the eccentricity of the ellipse, and r is the distance from the Sun to the planet, and is the angle to the planet's current position from its closest approach, as seen from the Sun. So (r, ) are polar coordinates.

For an ellipse 0 < < 1 ;

In the limiting case = 0, the orbit is a circle with the sun at the centre

At = 0°, perihelion, the distance is minimum

At = 90° and at = 270° the distance is equal to .

At = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

At any rate, you can see that because "a" and "p" in that equation are constant for a given orbit, "r" will be greatest when the denominator is smallest; and smallest when the denominator is greatest.
And these conditions are controlled by cos, which goes between -1 and +1.

apogee = max(r) = p/(1 - e) = a(1 + e), with cos = 1; = 0º
perigee = min(r) = p/(1 + e) = a(1 - e), with cos =-1; = 180º The semi-major axis a is the arithmetic mean between rmin and rmax:

The special case of a circle is = 0, resulting in r = p = rmin = rmax = a = b and A = r2.

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