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I'm starting to study by myself orbital mechanics, but I don't understand well how can be calculated the transfer orbit. If I have for example a route Earth-Jupiter, how can I calculate it? I've read some books about it, but as I don't know when the spacecraft would reach Jupiter, I don't know the ∆t and therefore I cannot calculate anything of the Gauss problem. How can I approach this problem? I'm pretty new with this subject and I am not so familiar with it.

Thank you very much

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There are a multitude of possible maneuvers and trajectories possible, but the one most used historically is the Hohmann transfer orbit.

Where it falls short

  • The Hohmann transfer is only accurate for orbits that are both in the same plane (in our solar system there is always around 1º tilt between the orbits of each two planets). At this point you can still more or less do a Hohmann transfer, but do a plan change or broken plane maneuver at some point. The trajectory however stays more or less the same as in the coplanar system.
  • The orbits have to be circular, which no real orbit is. This can change the Hohmann transfer orbit either way, to be longer, faster, require a little bit more or a little bit less fuel. All Planets orbit are at least slightly elliptical, for Mars the eccentricity is about 0.0934.
  • It assumes a two body problem, so there are (in the mathematical model) no gravitational influences from any body to the spacecraft.
  • The Hohmann transfer might in a perfect planetary system still require course adjustments, due to incorrect measurements, engine shutoff not being precise to the millisecond, solar wind and other effect not leaving the orbit exactly as wanted.

All of these are effects that make the 'vanilla' Hohmann transfer utterly unsuitable for real application. However what is used in reality is usually still similar to a Hohmann transfer. Depending on what you want to do it is still good enough for understanding a basic transfer, estimating a rough travel time or even simulating it in a model solar system (it is good enough in KSP for the most parts for example). Also some homework problems might specifically ask for this one, as it is the most common one.

Math for the Hohmann transfer

wikipedia

Source: Wikipedia

When starting from Earth (green orbit) and going to Jupiter (red orbit) you'd go into an elliptical transfer orbit (yellow), that touches both of them. Its semi major axis is

$$a_{Transfer} = \frac{a_{Jupiter} + a_{Earth}}{2}$$

According to Kepler's third law the period for one entire orbit is thus:

$$T=2 \pi \sqrt{\frac{a_{Transfer}^3}{G \cdot M_{Sun}}}$$

Since you only go half the transfer:

$$\Delta t = \frac T 2 = \pi \sqrt{\frac{a_{Transfer}^3}{G \cdot M_{Sun}}}$$

Conclusion

Most transfers, especially to close bodies like Mars and Venus use a variation of the Hohmann transfer. When going further out (or in) you might use multiple gravity assists to save fuel, which however usually increase travel time. The MESSENGER probe to Mercury went through quite a lot

wikipedia Source: Wikipedia

When going even further out you probably want to use less efficient maneuvers, which require more fuel both for escape from Earth and breaking at the other body (if you want to do that), but are a lot faster. New Horizons did that when traveling to Pluto, because otherwise it would have taken about 45 years.

I suggest you also play around with that on Wolfram Alpha like this, it is a quite powerful and easy to use tool!

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  • $\begingroup$ This misses the mark for a number of reasons. One is that an optimal transfer orbit from Earth to Mars is not a Hohmann transfer. A Hohmann transfer transfers between coplanar circular orbits. Mars's orbit is noticeably elliptical, and its orbit is inclined with respect to Earth's orbit. Another is that a transfer orbit takes a spacecraft from object A at time initial time to object B at some later time. The transfer orbit needs to reach the target rather than some random point on the target's orbit. $\endgroup$ – David Hammen Apr 3 '18 at 9:34
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    $\begingroup$ Yes, forgot to mention the elliptical aspect, adding that back in. Also, of course you need to wait for the right transfer point, but (for coplanar circular orbits) the transfer time is always the same. Also with the still approximately circular orbit of Mars the transfer is still reasonably similar to a Hohmann one, just requires plane change maneuvers, but the overall time (which OP seems to have asked for) stays more or less the same. $\endgroup$ – Hans Apr 3 '18 at 10:40
  • $\begingroup$ The time does not stay more or less the same. The optimal route will either be a good bit less than or a good bit more than a 180° Hohmann transfer: either ~200 days for the short route, ~400 days for the long route, vs ~270 days for a Hohmann-style transfer. $\endgroup$ – David Hammen Apr 3 '18 at 21:52

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