Simple Orbital calculating spreadsheet

[drewmacrae]

https://docs.google.com/spreadsheet/ccc?key=0Au0rf83JePNndGRBUlBfam42TjBkUlpEM1BBNzZDYWc

I built this simple orbital calculator based on the wikipedia article on HoffmanHohmann transfer orbits. What I\'ve found is that an orbit of 42% of the radius of a target orbit allows you to fire prograde when the orbits are aligned, and then intercept by doing 2.5 ascending transfer orbits while the target does 1.5 circular orbits.

I then demonstrate how to use differentials to approximate the periapsis necessary to close on the target. If you make a copy of the spreadsheet\'s you\'ll find the flight plan for the interceptor is interactive. It took me on the order of 10 orbits to rendezvous, and in the end it required little more deltaV than the spreadsheet predicts.

Feel free to take a peek, comment or expand on it.

I\'m pretty sure I knew how to spell Hohmann, I blame the error on having just done a bunch of math, and spell-check.

[semininja]

Where do people keep coming up with \'Hoffman\' transfer orbits? Does a lowercase \'h\' look that much like two \'f\'s? Also, two \'n\'s on the end of \'Hohmann\'.

Sorry about the rant...

[togfox]

I\'d really like to know how to use this - or something similar if there are others about, but I\'m not a rocket scientist so I\'m not sure where to start.

[drewmacrae]

Ah yes, reading this over in retrospect there isn\'t much info on what any of it means.

OK, here I\'m going to try to actually explain it without drawing a wall of equations.

This strategy for orbital intercept allows you to perform the difficult computation at the time of writing your flight plan when intercepting objects in circular orbits. A target at a known circular orbit can be intercepted accurately and without any measurements that can\'t be made within either the map view or the flight view. The computations at the time of the flight are approximate, but may be performed by hand. All computations are assumed to occur in plane.

My inspiration for this was based in part on not understanding the info in rincomp (orbiter 2010 has since taught me what most of it meant.) It\'s difficult to measure the orbital phase between a craft and a target at points other than 0 and 180 degrees.

Approximate, ascending interception is performed as described for orbits higher than 158,000m:

Craft climbs to circular staging orbit at 1/1.42* the target orbital radius.

Craft circularizes in staging orbit

When the target passes over the craft (phase is 0) the craft executes a burn to raise apogee to target orbit

After the craft performs 2.5 transfer orbits, (and the target performs 1.5) the craft should achieve an approximate intercept and may either circularize or begin the next process immediately which in the event of a leading target will save fuel by not elevating the perigee to the target altitude until later.

Final phase adjustment is more interesting and involves some precomputation when producing the flight plan (based on the target altitude)

At apogee the craft should be at the target altitude. Measurements, computations and burns will be made only at this point in the orbit (very small burns at the opposite point may be necessary to adjust for drift in this point and inclination may be adjusted to bring orbit onto the target orbit plane as appropriate).

At apogee and after each successive orbit:

The error should be measured approximately in map view or flight view (I didn\'t have radar when I first developed this but that\'s probably a good way to measure now)

A precomputed differential for circular orbits, can be multiplied by the error to determine the target altitude for the opposing somethingee of the orbit. The precomputation is measuring how many meters of a lead or a lag will be produced by a small adjustment of the oposite somethingee of the orbit. (which may be an apogee or a perigee depending on the sign of the error).

Knowing this, and the number of meters of lead or lag for which we want to compensate, we\'re able to find an approximate altitude to place the opposite somethingee.

The craft orbit is adjusted accordingly. This pattern is repeated with each successive orbit until rendezvous is achieved. The approximation works for the same reason a first order approximation works for Newton\'s method: as the error is reduced so is the innacuracy in the approximation. You should be close enough to use RCS for stationkeeping after 2 or 3 orbits.

This method could be adapted for elliptical orbits as well timing burns at the apogee or perigee.

To perform the precomputation (approxmately) I used the equation for the period of an elliptical orbit: =PI()*SQRT((Rap+Rpe)^3/mu) to determine the period of the target orbit and one with a perigee one meter higher than the target. The difference can then be converted (approximately) to a distance using the circumference of the target orbit and its period. Later, knowing the error and how much error 1m of opposite somethingee altitude will correct, we can then produce an optimal gain for the error measurement to control the altitude of the opposite somethingee. Of course using half the value would use twice as much time but less fuel so the phrase optimal refers to single orbit interception targets and is optimal within the constraint that the system is linear (which it isn\'t)

Hope that helps. Let me know if you want me to clarify/correct/expand on any particular part.

[togfox]

Awesome. Lol @ somethingee.

I got about 3/4 of that and will work out the other 1/4 when i practice it. 8)

Thx.