How can I calculate the delta-v correctly, this way does not seem to be correct?

Question

I'm learning about Astrodynamics, and I would like to ask how can I calculate the $\Delta V$ for an interplanetary mission.

I used a website for the porkchop plots (http://sdg.aero.upm.es/index.php/online-apps/porkchop-plot), which gives me the $\Delta V$ of the mission, and I am comparing my calculations with the $\Delta V$ that the site gives me, but they aren't alike at all, so I was thinking that maybe you could help me to calculate it correctly.

The calculations I did are following the site http://www.braeunig.us/space/interpl.htm

Solving the Kepler equation, I get the position and velocity of each planet at departure and arrival ($\vec{V_{p1}}$ and $\vec{V_{p2}}$). In my case, the planets are Earth and Jupiter and the date of departure is 21/01/2030, with a time of flight of 869 days (I used the porkchop plots in the first site I linked for choosing a departure date). After that, I solve the Lambert problem given the two positions and the time of flight, getting two velocities ($\vec{V_1}$ and $\vec{V_2}$).

With those values, I obtain the difference between the spacecraft heliocentric velocity and the planet orbital velocity. That is, $\vec{V_1}-\vec{V_{p1}}$. Taking that value as $\vec{V_\infty}$, I calculate the injection velocity as $V_o=\sqrt{V_\infty^2+\frac{2\mu}{r_o}}$. Being $\mu$ and $r_o$ the values respect to the Earth. As I have a parking orbit with a radius of 200 km, $r_o$ is the sum of the Earth radius plus 200 km. At last, I calculate the $\Delta V$ as the difference between the injection velocity and the orbital velocity

$$ \Delta V=V_o-\sqrt{\mu/r_o}$$

Qeustion: With these calculations, I have a $\Delta V=19$ km/s approximately, but the $\Delta V$ I have according to the porkchop plot is about 8.8 km/s. I am new with Astrodynamics, so I am trying to learn how to calculate this properly, but I don't find the mistake and I don't know what's wrong.

Answer 1

note: This is a very helpful extended comment that may be of use to the OP but it can't currently be posted as a comment until this user reaches 50 reputation points.

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Oh this is a fantastic question. It is common to fall into the following trap when making these types of calculations.

• Check carefully your reference frames. Celta V numbers are all relative, and trying to calculate delta V between orbits in different frames can be very problematic.

• Figure out what reference frames these v-infinities are from in the pork-chop plot. Are they relative to the planet or are they inertial with respect to the sun?

Once you figure that out, how do you transition from one frame to another? What parameters do you need? Once you have those nailed down, then make sure that all of your vectors are all in the same frame when doing addition, subtraction, cross-products, etc.

Edit: I wanted to leave this as a comment, but I need the reputation in order to do that. When I hit 50 I will delete the answer and put it as a comment.

Answer 2

It would help if you showed your results at different steps. Then it would be less difficult to spot the error.

End of January 2030 is about the right time for a Hohmann launch window to Jupiter (according to my arithmetic). But an Earth to Jupiter Hohmann is just short of a 1000 days. An 869 day trip would boost delta V. Regardless, I did a quick diagram of an Earth to Jupiter to Jupiter Hohmann trip. This quick BOTE assumes circular, coplanar orbits for departure and destination planet.

Earth's heliocentric velocity is about 30 km/s. Hohmann transfer orbit's heliocentric velocity at perihelion is 38.8 km/s. So Vinf at earth departure is 8.8 km/s.

Do these numbers agree with your calculations?

If so, I will move on to injection velocity from a 200 km low earth orbit.