Impulsive delta v hyperbolic and elliptical orbit

[orbitsnerd]

Homework Statement

On July 1, 2004, the Cassini spacecraft approached
Saturn with hyperbolic excess velocity 5.5 km/s to
swing by the planet at the closest approach distance
rp = 80,680 km. Compute the impulsive ΔV
required for a maneuver performed at the closest
approach to Saturn to transfer the Cassini spacecraft
into a 116-day elliptical orbit having the same
periapsis point as the approach (hyperbolic)
trajectory.
rp=80,680 km
vinf=5.5 km/s
T=116 days
GMsaturn=3.7931187e7 km^3/s^2

Homework Equations

vinf=sqrt(GMsaturn/a)

The Attempt at a Solution

I know it is an in-plane maneuver going from a hyperbolic approach to an ecliptic approach. If the vinf=sqrt(GM/a) is correct, I get a=1,253,923 km.
I am pretty much lost on how to complete this problem. I have done previous examples of Hohmann transfers and heliocentric orbits including the various velocities involved. Any help would be appreciated.
Thanks!

 

[gabbagabbahey]

Hi orbitsnerd, welcome to PF!

What does vinf physically represent? Does that really correspond to the value of 5.5km/s excess velocity you are given?

What must be true of the total energy (kinetic plus gravitational potential) of Cassini for it to be captured by Saturn's gravity? If Cassini is traveling at 5.5km/s in excess velocity, doesn't that mean it is traveling 5.5km/s too fast to be captured? What does that tell you it's total energy is? What would its total energy be if it were traveling in a 116 Day elliptical orbit having the same
periapsis point as the approach (hyperbolic)? What is the difference in these two energies? What chance in speed would account for this difference?

 

[orbitsnerd]

I was able to begin the problem (so I hope) this way:
T=2Pi sqrt(a^3/mu) where a=299290 km
e=1-(rp/a) so e=.7304
a=ra+rp/2 where ra=517900 km
vp=sqrt((2mu/rp)-(2mu/rp+ra))=28.53 km/s
I am using mu=musaturn=3,7931187e7 km^3/s^2.

Is this this correct initial approach? I am still a bit lost by what you meant by 5.5 km/s not being vinf.

For the energy equation are you referring to the E=(v^2/2)-(mu/r)?
Thanks.

 

[gabbagabbahey]

I was able to begin the problem (so I hope) this way:
T=2Pi sqrt(a^3/mu) where a=299290 km
e=1-(rp/a) so e=.7304
a=ra+rp/2 where ra=517900 km
vp=sqrt((2mu/rp)-(2mu/rp+ra))=28.53 km/s
I am using mu=musaturn=3,7931187e7 km^3/s^2.

Is this this correct initial approach?

I haven't checked your numbers, but your approach looks good.

This tells you what final speed you want the satellite to have at the periapsis point. Now you need to find its initial speed at that point...

I am still a bit lost by what you meant by 5.5 km/s not being vinf.

If I told you that the velocity of the spacecraft was 5.5km/s in excess of 100km/s, would you tell me its velocity was 5.5kms or 105.5km/s?

In this case, the "hyperbolic excess velocity of 5.5km/s" tells you that the spacecraft is traveling 5.5km/s faster than the maximum velocity it could have if it were in a bounded orbit. You'll need to figure out what that maximum velocity is if you want to find the velocity of the spacecraft .

For the energy equation are you referring to the E=(v^2/2)-(mu/r)?
Thanks.

Close, shouldn't the mass of the satellite be in that equation as well?