Can I please get some help, no one in class knows how you do this. its impossibl

Question

Can I please get some help, no one in class knows how you do this. its impossible. Thabk You.

5. (20 points) An interplanetary satellite has been launched into a 400 km parking orbit about Earth. A Heliocentric Hohmann transfer from Earth to Saturn (a = 9.537 AU) is performed. Once at Saturn, the satellite is captured at the optimal planetary capture radius, rp using the minimum Av to be captured in that circular parking orbit. Use the following values: equatorial radius of 60,270 km, ?8aturn- km3 s2, and sun 132.7 x 10 km3/s2 (a) What is the total mission ??, starting from Earth parking orbit through plan- etary capture at Saturn'? (b) What is the final orbit radius at Saturn'? (c) Determine the amount of propellant required as a percentage of the spacecraft mass before the first ?? burn, assuming a specific impulse of 300 seconds.

Explanation / Answer

5. radius of earth, Re = 6,371,000 m

orbit altitude, h = 400,000 m

hence orbit radius = Re + h

speed in this orbit = v1

GMe/(Re + h) = v1^2

v1 = sqrt(GMe/(Re + h))

speed required for the first blast off from the heliocentric orbit from earths orbit = v2

GMs/(1 Au) = v2^2

v2 = sqrt(GMs/(1 Au))

speed at the orbit near saturn, but in heloicentric orbit = v3

GMs/(9.537 AU) = v3^2

v3 = sqrt(GMs/9.537 AU)

now, Ms is mass of sun, Me is mass of earth

let mass of saturn = M

then

final speed for orbit around saturn = v4

orbit radius around saturn = R

v4 = sqrt(GM/R)

a. total dv = v4 - v1

dv = sqrt(GM/R) - sqrt(GMe/(Re + h))

now v4 = v3 for hohmann transfer

hence

GMs = mus = 132.7*10^18 m^3/s^2

Me = 6*10^24 kg

hence

v4 = v3 = sqrt(mus/9.537*1.496*10^11) = 9644.1459319195 m/s

v2 = 29783.083882658 m/s

v1 = 7687.9780596714 m/s

hence dv = v4 - v1 = 1956.1678 m/s

b. v4 = sqrt(GM/R)

GM = 37,931,000*10^9

hence

R = 407818333.618690949 m

orbit radius = 407818.3336186 km

c. specific impulse dt = 300 s

then

dt = T/m'

m' = dm/dt

T = thrust

dv = ve*ln(mo/mf)

ve = dt*g

hence

we take g = 9.81 m/s/s

then

1956.1678 = 9.81*300*ln(mo/mf)

mf/mo = 0.51443571

hence propellant required = (mo - mf) = 0.4855642836*mo

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