# How many measurements does it take to determine an asteroid's period?

#### Kim Lang

originally posted to the Minor Planet Mailing List, Aug 31, 2002

Regarding your question about the minimum number of observations needed to get a rough idear about the rotational period, I can see that the answers you have had until now are as vague as your question (just teasing a bit :-). If I may rephrase your question I can shed some lights on the problem, since I have spend a lot of time on the following question, in preparation of my thesis work, that sounds as follows: "What is the maximum number of rotational periods (of quality 3; "No ambiguity of period.") that I can bag, when I have 12 full nights on a 1.54 meter telescope." I have made a substantial number of simulations of observing schedules and have added field practice, and based on this, I pass on to you the following advice.

The basic knowledge is:

• periods are on average 8 hours, most between 4 and 20 hours.
• Most lightcurves are dubblepeaked, but there are some with 3 or 4 peaks and a very few are single peaked.
• Amplitudes depend on ecliptic longitude, and on occations amplitudes goes to zero.

I assume the lightcurves are differential. Make it a habbit to make a lightcurve of a neighbour star of approximately the same magnitude as the asteroid, thereby you can estimate the noiselevel of the asteroid lightcurve. If your asteroid lightcurve amplitude is 2 or 3 times that of the control star I will belive you have detected a signal. You can space your observations with 30 to 60 minutes apart. If the nightarc of the asteroid is 8 h, you will get 8 to 16 data points each night which is good. Do this each clear night until your composite lightcurve coverage passes 80% of your best period solution. It is not the number of data points each night that matters, it is the percent coverage of the true rotational period that is important. If you have more data points its fine, but it is of no use to have good looking data that is inconclusive !!

You need at least 80% coverage of the lightcurve to have confidence in the solution. If you have less, you havent convinced the sceptics that the lightcurve is that of a common two-peaked lightcurve. 80% coverage of a two-peaked lightcurve will have 3 welldefined extremas like two minimas and a maximum. If You have exactly 75% coverage you still have 3 extremas, you just dont now that for sure because you havent seen the curve starting rising again. When I analyse a lightcurve I use at least 3 harmonic terms in my Fourier fitting. This makes it possible to discriminate between solutions with one, two and three maximas per rotation. If I fit with only one harmonic term, I can detect a weaker amplitude (assuming a symmetric lightcurve) but at the expense of knowing if the period determined is half or the whole rotational period. Each night image the asteroid at regular intervals the whole night! (The time when it is above your minimum horizon elevation). Do not observe an "evening" asteroid for 4.5 h and then swich to a "morning" asteroid for another 4.5 h. Select asteroids with a longer night arc. Let me make an example: The asteroid you observe for 4.5 h each night, is "mean" and has a period of 16 h. Even if you observe each night for two weeks your lightcurve covers less than 60% of the lightcurve. You have then proven nothing except the lightcurve shows an amplitude and is periodic. Observing 6.5 h each night you quickly passes the 80% mark, and you have a solid case. If you cant stay up all night then use a robot or get yourself a playmate in Europe or Japan to continue your nightarc.

Harris & Lupishko (1989) demostrates that a Fourier series of 20 harmonics reproduce composite lightcurves to the 1 mmag level so 20 equally spaced observations over the period could extract the maximum amount of information aviable. Since equal spacing is impossible in practice they advice to use 50 evenly spread points. Karttunen (1990) made some theoretical studies too, and found that Fourier series of 2 harmonic terms with as little as 8 data points reproduces the correct period to an relative accuracy delta P / P of 0.1. The way I used Karttunen's finding was to set a lower limit around 20 points per composite lightcurve.

I put a lot of thinking into making as few measurements of each asteorid each night because I wanted to observe as many asteroids each night as possible and still get a unique period solution. I ended op making loops on the sky with on average 15 asteroids in the loop. This technique worked in practice and the outcome was pretty much as expected considering the number of nights with bad weather and the fact that you cannot beat the 24h commensurability problem when observing from a single site. The analysis of the data I brought home from La Silla can be summarized as follows:

• Most period solutions found unique had a coverage from 81% and up.
• Even the very finest composite lightcurve had only from 5 to 15 data points per night.
• Highly sampled lightcuves failed in yielding a unique solution because of periods too long or close to 12 and 24 hours.

Bill, I hope you (and everybody else who kept on reading!) can make some use of my findings, and I would be happy to continue the discussion with any one who takes interest in the subject. I have kept the arguments to a minimum as to keep the conclusions visible. 90% of my masters thesis work was about this very question, so I could keep on writing for hours!!!

```Yours sincerely
Kim Lang

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Cand. Scient. Kim Lang
Blegdalsparken 23, 1. tv
DK-9000 AAlborg
DENMARK

Office phone: +45 9932 2906
Email: mip@phys.au.dk
or     KimLang@aas.nja.dk
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