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We know from Hohmann or bi-elliptic transfer maneuvers that a burn in the tangential direction of an orbit changes the radius of the orbit.

But what happens to an object in orbit, when there is a burn in radial direction (orthogonal to the Earth's surface directly below the object in space)?

costrom
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user36499
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1 Answers1

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From the diagram given in this answer

enter image description here

We see that radial "in" burns shift the perigee towards the burn point, and radial "out" burns shift the apogee towards the burn point.

Thrusting radially outwards (that is, away from the center of the Earth, or "up") creates an initial vertical motion in the desired direction, but the chaser then begins falling behind its original position while the upwards motion slows and stops. A quarter rev (22 minutes) after the impulse, it is about 900 feet higher (it would have been 1300 feet higher and still moving, if it hadn't been for orbital mechanics effects) and about 1700 feet behind its original position, with all motion in the horizontal direction. Drifting downwards as well as backwards, and 45 minutes (half a rev) after maneuver execute, the chaser drops through its original altitude at a range of about 3500 feet from where it started. It then continues in this "football" trajectory, dropping but moving forward, then rising and resuming its original position after a full orbit. The motion then repeats, subject to outside perturbations.

Thrusting radially inwards (downwards) creates the same-sized "football" orbit which first pulls ahead and then backwards in its 90-minute cycle.

Source Rendezvous and Proximity Operations Handbook Part 2 pp. 2-13, 2-14 (also contains an earlier version of the diagram)

For completeness, Part 1 of the document is located here.

Organic Marble
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    thanks a lot!!! from which book is this? i would like to read it! the reason i was initially asking is that i was wondering if a capsule approaches the ISS and it is like perfectly aligned to the docking adaptor but just radially 50m "under" the final position: how would the then increase the height/radius. radial burns or tangential burns? or a combination? – user36499 Jun 06 '20 at 07:10
  • @user36499 I managed to find the source of the quoted material online. Updated answer to include reference. – Organic Marble Jun 06 '20 at 12:58
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    @user36499 depending on the part of the orbit, such a position will result in the approaching craft being drawn towards, or drifting away from the target. The latter has been used for some docking approaches, where the approach vehicle must apply thrust to approach. If it fails to do so, its approach will slow and eventually it'll drift backwards, away, at least for a half orbit. – Innovine Jun 06 '20 at 13:46
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    Calculating the radial burn trajectories (in the limit of a small ∆v) is a fun exercise that I sometimes assign in my upper-division classical mechanics class. If I recall correctly I cribbed it from Arnol'd's Mathematical Methods of Classical Mechanics. – Michael Seifert Jun 06 '20 at 13:47
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    Up is back, back is down, down is forward, forward is up. – Bob Jarvis - Слава Україні Jun 06 '20 at 14:31
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    I have some info (including an interactive plotting script) on the effect of a radial burn on a circular orbit here: https://physics.stackexchange.com/a/677727/123208 Sorry, I didn't include the full derivation, but I didn't feel like typing half a page of MathJax on my phone. ;) – PM 2Ring Feb 22 '23 at 16:59
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    @PM2Ring Nice! Thanks for the link. – Organic Marble Feb 22 '23 at 17:35