Differentiation of conic section equation

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SUMMARY

The forum discussion centers on the derivation of the time derivative of a conic section equation, specifically transitioning from equation 1.5-4 to 2.5-2 in "Fundamentals of Astrodynamics" by Roger Bate, Donald Mueller, and Jerry White (1971 edition). The user initially struggled with the derivation but ultimately found a solution. The key equation discussed involves the time derivative expressed as \(\frac{d}{dt}\frac{p}{1+e \cos \sigma} = \frac{pe\, \sin \sigma \dot{\sigma}}{1+2e\, \cos \sigma + e^2 \cos^2 \sigma\).

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tauon
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Homework Statement



I do not understand how the authors got the time derivative of equation 1.5-4 in the form given at 2.5-2.

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Homework Equations





The Attempt at a Solution





\frac{d}{dt}\frac{p}{1+e cos\sigma}=-\frac{p}{(1+ecos\sigma)^2}(-esin\sigma\dot\sigma)=\frac{pe\, sin\sigma\dot\sigma}{1+2e\,cos\sigma+e^2cos^2\sigma}

?

I tried various rewrites using trigonometric identities, but the equation just got so complicated that it'd take me a long time to typeset it in this post.
 
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Never mind. I got the derivation. I also found out I am blind/can't read.
 
Hi tauon! I'm curious if you could share how you got the derivation, or which textbook your above excerpt is from. I'm trying to get to equation 2.5-2 from this excerpt, but seem to be getting stuck the same spot you were. Thanks!
 
Hi. This might be very late now, but I didn't check this thread since I got the solution. :p
In case someone runs into it in the future, this is the derivation I used
xAij0s7.png

Oh, and the excerpt is from "Fundamentals of Astrodynamics" (authors Roger Bate, Donald Mueller, and Jerry White) the 1971 edition.
 
Last edited:

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