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Question

a weather satellite with an orbit about the earth reaches a minimum altitude of 1000 miles and a maximum altitude of 1600 miles. the path of its orbit is approximately an ellipse with the center of the earth at one focus. find the equation of this curve. assume the radius of the earth is 4000 miles and the x-axis is the direction of the major axis.

 

 

Technical calculus with analytical geometry class

Answer 1

(note: I had to edit several times for the diagram further below to line up right.

Spacings in the editor do no equal spacings in the display) 

 the equation of the ellipse is:     x/a² + y/b² = 1

 

so we need to find the values of:    a and b

 

a = semi-major axis (half the long axis), b = semi-minor axis (half short axis)

 

2a = perigee alt + dia of the earth + apogee alt          

                   note: (for orbits,  perigee alt = min alt, apogee alt = max alt) 

 

2a = min alt + 2*radius of earth + max alt

 

we also know a*ecc = center of ellipse to focus     (earth center is focus pt)

 

b = a*sqrt(1 - ecc²)

 

 

|<-- min alt --|<--earth radius--|--earth radius--->|------------- max altitude -------->|

                                                       +

                                                 focus (at center of earth)

                                                      

                                                            center of ellipse 

|<--------------------- a----------------------------- | ----------------------- a -------------------------->|

                                                   |-- a*ecc --|

 

the length of the major axis of the elliptical orbit is the distance shown above, or = 2a

a = semi-major axis of the orbit (sma) 

 

2a = min alt + 2*radius of earth + max alt

 

so the sum above divided by 2 = a  = ( 1,000 + 4,000+4,000 + 1,600 ) / 2

                                                       a = 10,600/2 = 5,300

so a = 5,300

 

we now need to find eccentricity to find b (semi-minor axis)

 

we know that  a*ecc = the dist from the center of the ellipse to the focus (at earth center).

 

so,  a*ecc = a - (earth radius + min altitude) 

 

a*ecc = 5,300 - (4,000 + 1,000) = 300   

 

or ecc = 300/5,300  

    ecc = 0.0566   

 

now, the semi-minor axis b = a*sqrt(1 - ecc²)

so b = 5,300*sqrt(1 - 0.0566²)

 

b = 5,283.02

 

so now just substitute in the ellipse equation (top of page)

 

x/a² + y/b² = 1

 

or   x / (5300)²  + y/(5283.02)² = 1