1) (50pts) A radar station at Georgia tech observes a...

1) (50pts) A radar station at Georgia tech observes a spacecraft with the range and range rate given below at 8:00pm EST on September 22$^{nd}$, 2023: $\vec{\rho} = -0.31902\hat{X}_h + 0.07651\hat{Y}_h + 0.45728\hat{Z}_h$ DU $\dot{\vec{\rho}} = -0.31041\hat{X}_h + 0.60848\hat{Y}_h - 0.03624\hat{Z}_h$ DU/TU a. (10 pts) What was the local sidereal time (in degrees) for Georgia Tech (33.7718°N, 84.395°W). b. (15 pts) Find the radius and velocity vectors in the planet centered, equatorial coordinate frame (X,Y,Z) for the spacecraft. c. (15pts) Find the orbital elements of the spacecraft. d. (10pts) Find the time rate of change of the RAAN and argument of periapsis, then recompute RAAN and argument of periapsis after 20 orbits.

Transcript

00:01 So we have given this that satellite a is at a distance of r0 from earth and the satellite b is at a distance of 1 .7 r0 from the earth.

00:10 So in the first part, we have to calculate the time period of satellite b.

00:14 So according to the kepler's law, so by using kapler's third law, so by using kapler's third low, we can say that the time of satellite b will be equal to r.

00:33 X0 cube divided by 1 .7 r not cube because this is the radius of satellite b and we will multiply this with the time period of satellite a so according to this if we solve this equation then we will get the final answer of the time period of satellite b it will be equal to 2 .27 multiply by time period of satellite a so this is the answer to the answer answer of first part the time period of satellite b is equal to 2 .27 multiplied by time of satellite a and then we have to calculate the orbital acceleration so how can we calculate the orbital acceleration we know that acceleration is equal to omega square r and from here i can say my acceleration will be equal to v square divided by the radius that is equal to r 0 square multiply by r not from this equation i will get my orbital acceleration that i'm denoting with a note that will be equal to v0 square divided by r not so this is the acceleration or i should say the orbital acceleration in the b part they are saying to find orbital velocity so the b part is saying to find the orbital velocity so the v will be equal to 2 pi r divided by time period so from here that will be equal to 2 pi multiply by the radius will be equal to radius of satellite b will be equal to 1 .7 multiply by r not and we will divide this with the time period that we have calculated above that is equal to 2 .27 multiplied by time of satellite e so if we will calculate this then we will find that the velocity or i should say the orbital velocity will be equal to 0 .76 v .0...

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