HELP on a spacecraft mission to saturn for the injection velocity

[gikwiye]

HELP on a spacecraft mission to saturn for the injection velocity i beg you please

Homework Statement

on intial approach to Saturn, the spacecraft mass is 3500 kg. To enter orbit around Saturn , the vehicle has a 400N thrust primary propulsion engine, with a specific impulse of 310 sec. The orbit insertion burn takes place when the spacecraft is at 4R_s from Saturn centre.
R_s being the radius of Saturn.its speed at that point is 18.39 km/s.
4R_s is the perigee of Saturn's elliptical orbit.
find injection burn delta V

Homework Equations

V=sqrt(mu(2/r(apogee or perigee))-(1/a)) (for an ellipse)
a=(r_apogee+r_perigee)/2
mu=3.8*10⁷
delta V=Vex*ln(M_o/M_b)
Vex=3.041km/s
V=sqrt(mu/r(apogee or perigee)) (for a circle)

The Attempt at a Solution

writing my attempt would be too long but my values for
V_p=18.39 as given in the question
V₁=12.58 calculated from the last equation given
V_a=0.26calculated from first equation
V₂=1.53
delta V=(18.39-12.58)+(1.53-0.26)=7.076 km/s
this in the end will give me for the mass of fuel consumed =3158.39kg
which would mean our spacecraft is only 350 kg which is really low.
HELP PLEASE

 

[Filip Larsen]

If the spacecraft approaches Saturn on a hyperbolic orbit with a given radius and speed at its closest approach, the change in speed to a circular orbit at same distance is simply (using your variable names) v_p-v₁. Not sure what your other two speeds signify, but if you feel they are important I guess there is something missing in your problem description.

 

[gikwiye]

I used the hohmann orbit transfer that's why I have two other velocities

 

[Filip Larsen]

Where does the Hohmann come from? Does the problem text you quoted describe the full problem or have you left something out. If you have, then please include it or we will have no idea why you are including mystical terms.

As it stands, you seem to calculate with two maneuvers. First one breaks from hyperbolic speeds at 4R down to circular orbit and is the maneuver that the problem text seems to invite you to make. The second maneuver is rather mystical and involves some speeds and inferred radii that does not fit the rest of your description.

 

[gikwiyefriend]

deleted

 

[Filip Larsen]

Gikwiey, please keep the discussion in this thread. You PM'ed me the following:

hi this is the exact question

on intial approach to Saturn, the spacecraft mass is 3500 kg. To enter orbit around Saturn , the vehicle has a 400N thrust primary propulsion engine, with a specific impulse of 310 sec.
The burn starts when spacecraft ’s approach trajectory is a distance of 4 RS from Saturn’s centre, and its speed is 18.39 km/sec. Here RS is Saturn’s planetary radius. This is the point of closest approach, when the vehicle’s velocity vector is perpendicular to its Saturn-centred position vector.
The engine is fired to insert the spacecraft into its initial orbit around Saturn, which has a periapsis radius of 4 RS, and an apoapsis radius of 270 RS. the engine firing is impulsive, estimate:

• The injection burn ΔV
• The propellant mass required for the burn
• The duration of the engine firing (note that the acceleration is not constant!)

USE
Earth grav. acc., g=9.81 m/sec2
Grav. con. of Saturn, μS=3.8km3/sec2
Saturn planetary radius, RS=6×10^4km
Rocket equation ΔV=Vex*log(M0/Mb), where Vex=g*specific impulse

The text describes two orbits, the approach orbit and the initial orbit. The approach orbit is characterized by having periapsis at 4 R_s with the given speed. The spacecraft now performs a single breaking maneuver so that it enters an elliptical orbit with perikrone at 4 R_s and apokrone 270 R_s. To calculate the change in speed you need to calculate what speed the spacecraft must have at 4 R_s in order to be in an orbit with apoapsis of 270 R_s, and subtract that from the 18.39 km/s.

 

[gikwiye]

cool thanks got it

 

[mastervig]

hi,

i have to answer a similar question.

can you please post the appropriate equations for this one??

thank you