# How many observations are needed to determine a comet's orbit?

Based on the following facts:

1. We have Kepler's laws of planetary motion.
2. We have a good knowledge of the positions and orbits of the gravitationally significant objects in the Solar System.
3. We can numerically calculate orbital variables quickly and accurately using computers.

What is the minimum number of accurate visual observations we need to make in order to calculate the orbital elements of a newly discovered comet or asteroid?

I do appreciate the more observations we get, the more accurate the orbit we can calculate, and we will look for more, especially for example, if a comet produces a tail(s), but theoretically would just two/three/four observations be enough to be, say 90% sure we know it's orbital elements.

This may be a question for AstronomySE, no problem placing it there if necessary.

• This question is appropriate for physics stacke exchange because both physics and astronomy concepts are allowed according to the help center. But according to meta posts I have read, astronomy questions are better and more quickly answered in the astronomoSE since the traffic their is higher than the traffic for the astronomy and astrophysics tags for physics SE. – Cicero May 25 '15 at 19:00
• The difficulty with determining the orbit comes from the observations of where the comet is in its orbit. Since the comet is recently discovered, the comet is most likely approaching the sun. Even if the comet has traveled 1 AU, it may barely show any movement against the background and the orbit would still be difficult to predict. – LDC3 May 25 '15 at 19:04

Martin Hoecker-Martinez's Answer is correct for perfectly noiseless observations of a two body Kepler system, i.e. the force between the bodies is directed along the vector linking them and the force magnitude follows and inverse square law with distance. An alternative to Martin's answer is that perfectly known position and velocity will determine all future motions for the two body problem.

However, these assumptions do not strictly hold: a comet not only interacts with the Sun but also the planets (especially Jupiter) and other gravitational sources and moreover there are other nonideal "noises" that one must account for (solar wind and so forth) as in CuriousOne's answer.

Practically, the way one handles this is to assume a Kepler model (or more elaborate model if you know where all the planets are and can thus account for their effects) and treat other perturbations as additive Gaussian noise. One then finds the current maximum likelihood estimate of the orbit model parameters (position, velocity) using a Kalman Filter Algorithm, which I describe here and here. So you will begin six observations and then make new ones regularly, using the Kalman filter to update your estimates. The Kalman filter will give you current variances on your estimates so you can always put fairly rigorous error bounds on theoretically calculated future positions. So there will be some finite number of observations needed for whatever accuracy you need in your calculations: the Kalman filter will let you know when you have enough.

Historically, exactly your problem motivated the invention of the Kalman filter. For, although we credit Rudolf Kalman as its inventor, it was in fact first published in 1809 by Carl Friedrich Gauss, where he documented his use of it for simplifying hand calculations made in estimating the orbital parameters of celestial bodies. See

"Recursive Estimation and the Kalman Filter" in D.G.S. Pollock's "Kalman Filters"

• Thanks for that answer and your time. I am amazed how astronomers managed without our current techniques, say the Mercury precession problem 100 plus years ago, getting worked up about 43'' per century. Amazing commitment in cold observatories. Also if the solar wind is strong enough to deflect X tons of comet, even slightly, there must be solar wind propulsion potential there somewhere for light spacecraft. – user81619 May 26 '15 at 1:14
• Sorry, I didn't take on board your comment about long history of Kalman filter. Still impressed by the work they did though. – user81619 May 26 '15 at 1:23
• AcidJazz, I would note that while CuriousOne's answer is nicely succinct and accurate in the sense that no finite number of predictions can predict infinitely into the future, I would note that I think you might want to reconsider your Answer vote given that @WetSavannaAnimal has given you a very specific answer in terms of how you do it in practice. But also: If you were thinking e.g. "How many photographic snapshots of a comet or asteroid or dwarf planet are needed to give you a fighting chance of knowing where to look for it next?", then you might want to look again at Martin HM's answer. – Terry Bollinger May 26 '15 at 4:04
• @TerryBollinger good advice and taken, didn't see the votes till now and didn't know you could switch answers. Still a newbie..... – user81619 May 26 '15 at 8:41

An ideal Kepplerian orbit is defined by six (6) parameters:

Therefore you need at least six (6) independent observations. Astronomical observations are direction (but not range usually) given by a pair (2) of angles therefore three (3) independent observations are the absolute minimum you need.

("ideal Kepplerian" meaning you ignore general relativity and all gravitational interactions except for the Sun)

• Actually, radar observations of many bodies provide range and range-rate to high accuracy, while the angular precision of such observations is very low. – 2012rcampion May 26 '15 at 1:35
• I retract my down vote and bow to the far greater understanding of this area of @WetSavannaAnimal! Alas, the stupid system won't let me, something about some kindo of one-hour limit expiring (I'm expecting flying monkeys at any moment...) My apologies, Martin, I'll reverse it when I can. – Terry Bollinger May 26 '15 at 3:58
• @2012rcampion actually, I had thought of radar but I thought the comet would just soak up the signal and have a low albedo. But I am probably wrong and further out from the sun it would be solid ice, not an absorbing material by any means. I will Google comet composition. – user81619 May 26 '15 at 8:35
• I will just repeat here what WetSavannaAnimal aka Rod Vance mentioned in his answer: Another way to realize that there are six (6) degrees of freedom, or parameters to determine, is to observe that the vectors of angular momentum (3 coordinates) and position (3) are necessary and sufficient to determine the motion of the comet in all eternity, in this simplified setup. – Jeppe Stig Nielsen May 26 '15 at 13:35

An infinite number of observations are needed because a comet does not have a well defined orbit. It is strongly deflected from ideal Newtonian orbits by outgassing, solar wind etc., so if you want to know where it is, and especially where it will be, at a given time in the future, you have to keep observing.