A spacecraft is initially in a 300 km altitude circular orbit about the Earth in the ecliptic plane. It is to be sent on a Hohmann transfer to Saturn, also in the ecliptic plane. Assume that Saturn is in the correct position in its orbit for a flyby to occur when the spacecraft gets there.
Calculate the initial \(\Delta V_{1}\) required to start the trip to Saturn.
\[ r_{b0}=r_{E}+alt \]
Where \(r_{E}\) is radius of earth and \(alt\) is spacecraft altitude. Hence
\[ r_{b0}=6378+300=6678\text{ km}\]
The distance from earth to sun is \(R_{E}=1.496\times 10^{8}\) km and the distance from saturn to sun is \(R_{s}=9.536\times 1.496\times 10^{8}=1.4266\times 10^{9}\) km therefore \(a=\frac{R_{E}+R_{s}}{2}=\frac{1.496\times 10^{8}+1.426\,6\times 10^{9}}{2}=7.8815\times 10^{8}\) km.
The earth speed around the sun is \(V_{e}=\sqrt{\frac{\mu _{s}}{r_{e}}}=\sqrt{\frac{1.327\times 10^{11}}{1.496\times 10^{8}}}= 29.783\) km/sec. When the spacecraft escape the earth it has to be at speed \[ V_{perigee}=\sqrt{\mu _{s}\left ( \frac{2}{R_{E}}-\frac{1}{a}\right ) }=\sqrt{1.327\times 10^{11}\left ( \frac{2}{1.496\times 10^{8}}-\frac{1}{7.8815\times 10^{8}}\right ) }=40.07\text{ km/sec}\]
Therefore, \(V_{\infty }\) is the escape speed found from
\begin{align*} V_{\infty } & =V_{perigee}-V_{e}\\ & =40.07-29.783\\ & =10.287\text{ km/sec} \end{align*}
Now the burn out speed is found
\[ \frac{V_{bo}^{2}}{2}-\frac{\mu _{E}}{r_{b0}}=\frac{V_{\infty }^{2}}{2}-\frac{\mu _{E}}{r_{SOI}}\]
Where \(r_{SOI}\) is the earth sphere of influense given by \(9.24\times 10^{5}\) km. Solving for \(V_{bo}\)
\begin{align*} \frac{V_{bo}^{2}}{2}-\frac{3.986\times 10^{5}}{6678} & =\frac{10.287^{2}}{2}-\frac{3.986\times 10^{5}}{9.24\times 10^{5}}\\ V_{bo} & =14.978\text{ km/sec} \end{align*}
Hence
\begin{align*} \Delta V_{1} & =V_{bo}-\sqrt{\frac{\mu _{E}}{r_{bo}}}\\ & =14.97-\sqrt{\frac{3.986\times 10^{5}}{6678}}\\ & =7.244\,2 \end{align*}
Calculate the angle past the Earth’s dawn-dusk line where the \(\Delta V\) should be applied.
\begin{align*} e & =\sqrt{1+\frac{V_{\infty }^{2}V_{bo}^{2}r_{bo}^{2}}{\mu _{E}^{2}}}\\ & =\sqrt{1+\frac{\left ( 10.287^{2}\right ) \left ( 14.978^{2}\right ) \left ( 6678^{2}\right ) }{\left ( 3.986\times 10^{5}\right ) ^{2}}}\\ & =2.768\,3 \end{align*}
Hence
\begin{align*} \eta & =\arccos \left ( \frac{-1}{e}\right ) =\arccos \left ( \frac{-1}{2.768\,3}\right ) =1.9404\text{ radian}\\ & =111.18^{0} \end{align*}
Hence \(\theta =180-111.18=68.82^{0}\)
For how long is the spacecraft on the heliocentric Hohmann transfer between Earth and Saturn? (Note: you do not need to calculate the time within either planet’s sphere of influence, as that will be small relative to the Hohmann transfer time, but you are welcome to do so and compare those values for yourself.)
The time is half the period of the elliptical orbit. Hence
\begin{align*} T & =\pi \sqrt{\frac{a^{3}}{u_{s}}}=\pi \sqrt{\frac{\left ( 7.8815\times 10^{8}\right ) ^{3}}{1.327\times 10^{11}}}=1.9082\times 10^{8}\text{ sec}\\ & =\frac{1.908\,2\times 10^{8}}{60\times 60\times 24\times 365}=6.051\text{ year} \end{align*}
After crossing into the sphere of influence of Saturn, the spacecraft is to be placed in a circular orbit about Saturn with an orbital radius of 150,000 km. Calculate the \(\Delta V_{2}\) required to place the spacecraft on this orbit.
Solution completed in the Mathematica solution. See above for links.
A spacecraft on an interplanetary mission in the same plane as Jupiter’s orbit about the Sun enters Jupiter’s sphere of influence. The spacecraft has a speed of 10 km/s relative to the Sun at this point, which you can estimate as the Jupiter’s average orbital radius about the Sun. (See the Planetary Constants sheet in your notes for values.) Assume that Jupiter is in a circular orbit about the Sun.
The largest possible value for the impact parameter, \(b\), that will still result in a hyperbolic orbit about Jupiter in the patched conic method is Jupiter’s SOI radius. Find that value on the Planetary Constants sheet in the course notes and enter it here for reference.
\(b_{\max }=R_{SOI,Jupitor}\) = Answer km
For parts (b) through (g), assume that, relative to the Sun, the spacecraft is moving in the same direction as Jupiter when it enters Jupiter’s SOI.
What is the speed of the satellite relative to Jupiter when it enters Jupiter’s SOI?
\(V_{\infty }\) = Answer km/s
What is the smallest possible value for the impact parameter b? This value of impact parameter will result in a burnout radius that just grazes the surface of Jupiter, \(r_{bo}=r_{Jupiter}\)
\(b_{min}=km\)
Select as your impact parameter the value halfway between \(b_{min}\) and \(b_{max}\). Note that value here for reference and use it as your impact parameter for the rest of the problem.
\(b\) = Answer km
Given the impact parameter from part (d), calculate the turning angle of the spacecraft relative to Jupiter during the flyby.
\(\theta \) = Answer degrees
What is the spacecraft’s heliocentric speed following the flyby?
\(V_{D}=km/s\)
What is the spacecraft’s heliocentric flight path angle following the flyby?
\(\gamma _{D}=deg\)
For the remaining parts, assume that, relative to the Sun, the spacecraft DOES NOT arrive at Jupiter’s SOI moving in the same direction at Jupiter. The spacecraft still has a heliocentric speed of \(10\) km/s at the distance of Jupiter’s orbit from the Sun. But now it has a heliocentric eccentricity of \(0.5\). (What was the heliocentric eccentricity when the spacecraft arrived in the same direction as Jupiter, assuming that point was aphelion?)
What is the spacecraft’s heliocentric flight path angle when it arrives at Jupiter’s SOI?
\(\gamma _{A}=deg\)
What is the spacecraft’s speed relative to Jupiter?
\(V_{\infty }=\) km/s
part(j)
Using the same impact parameter as in part (d), calculate the turning angle of the spacecraft relative to Jupiter.
\(\theta =deg\)
part(k)
Assuming that the spacecraft flies behind Jupiter, what is the spacecraft’s heliocentric speed following the flyby?
\(V_{D}=\)km/s
Assuming that the spacecraft flies behind Jupiter, what is the spacecraft’s heliocentric flight path angle following the flyby?
\(\gamma _{D}=deg\)