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Old Aug 6th 2017, 06:26 PM   #1
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Explorer 1 Orbit

Exercise taken from Caiken, pg 23



For part (a) I get (350+2549+2*6378)/2 = 7287.5 km

for part (b) ...

Used formula for conic sections $\displaystyle \frac{p}{r} = 1 + e \cos \theta$

assumed perigee when $\displaystyle \cos \theta = 0$ and apogee when $\displaystyle cos \theta = 1$.

This gives two linear equations:

p/(360+6378) - e = 1
p/(2549+6378) + e = 1

I get p = 7679.55646345356 and e = 0.139738270028726

for part (c) I get a bit stuck.

I can't think of any formula that will help without knowing the mass of the satellite.

Edit: Looks like Keplers 3rd law is a possible candidate for (c)

$\displaystyle T^2 = \frac{4 \pi^2}{G M}a^3$ where a is the semi major axis and T is the period. That's a step closer anyway.

Last edited by kiwiheretic; Aug 7th 2017 at 01:18 AM.
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Old Aug 7th 2017, 07:27 AM   #2
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part (a) ... if the Earth's equatorial radius is 6378 km, the the semi-major axis is

a = [360+2549+2(6378)]/2 = 7832.5 km


part (b) ... focal length, c = 7832.5 - (360+6378) = 1094.5 km

e = c/a = 1094.5/7832.5 = 0.1397...


part (c) ...

at the Earth's surface, $g = 9.81 \, m/s^2$

$g = \dfrac{GM}{R_e^2} \implies GM = g \cdot R_e^2$

$T^2 = \dfrac{4\pi^2}{GM} \cdot a^3 = \dfrac{4\pi^2}{g \cdot R_e^2} \cdot a^3$
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Old Aug 7th 2017, 07:18 PM   #3
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Looks like you did something different for eccentricity that didn't involve a system of linear equations?

you got a different answer for (a) but used same numbers? I just double checked with IPython as a calculator.

Last edited by kiwiheretic; Aug 7th 2017 at 07:20 PM.
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Old Aug 8th 2017, 05:03 PM   #4
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Originally Posted by kiwiheretic View Post
Looks like you did something different for eccentricity that didn't involve a system of linear equations?




you got a different answer for (a) but used same numbers? I just double checked with IPython as a calculator.
get a new calculator ...

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Old Aug 8th 2017, 05:22 PM   #5
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Originally Posted by kiwiheretic View Post

For part (a) I get (350+2549+2*6378)/2 = 7287.5 km
Oh, was a typo, I used a 350 instead of 360
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Old Aug 8th 2017, 05:56 PM   #6
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Originally Posted by kiwiheretic View Post
Oh, was a typo, I used a 350 instead of 360
For part (a) I get (350+2549+2*6378)/2 = 7287.5 km
you also transposed the digits 2 and 8
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