The transit of a planet across HD 209458 lasts for about 3 hours, dimming the star by 1.5%. In this lab we are interested in whether or not we could be able to detect this change. Error calculations were done to find what the minimum change that we could detect is. We were only interested in the change in magnitude, and not with an absolute scale, so we compared the brightness of the star to a bright object of constant magnitude. This method of finding the ratio had the added benefit of cancelling out errors due to changes with time in the CCD, telescope efficiency, or atmospheric transmission. HD 210074 is one such object that's brightness is very constant; this is known from previous work (Henry et al. 2000).The Leuschner Telescope was used, with a 512x512 CCD, to collect this data. Data was taken in sessions, with a group of students working for about an hour; it was shared after it was analyzed.

The ratio in brightness of HD 209458 to HD 210074 was measured over a period of one hour. We took 18 images of HD 209458 "bookended" by images of HD 210074, for 19 images of the reference star in all. The reported ratio of HD 209458 to HD 210074 is actually the ratio of the DN of HD 209458 to the average DN of HD 210074 for the DN values that bookend the HD 209458 image. In this way changes in the DN of the reference star due to changes in the CCD, telescope efficiency, or atmospheric transmission. We calculated the flux from the star using a program that summed the DN within a given radius.

This table is of data that was taken by our group, Emily Deb and I, on Oct 19, 2002. Included are the errors associated with the DN counts and the calculated ratio.

Times of star exposures (sec) Starting at UT 5:23:00	DN HD 209458	Poisson Error in DN	DN HD 210074	Poisson Error in DN	Ratio DN Star to Ref. Star	Propagated Uncertainty (dr/R)	Propagated Uncertainty (dr)
48	64421.2	253.813	328151.	572.856	0.196510	0.00412893	0.000811375
272	63189.5	251.375	327490.	572.267	0.190968	0.00416371	0.000795137
416	64317.8	253.610	334290.	578.178	0.194330	0.00413019	0.000802621
629	61560.0	248.113	327654.	572.411	0.186978	0.00421462	0.000788041
768	60635.3	246.242	330819.	575.168	0.187520	0.00424716	0.000796427
1199	63190.7	251.378	315889.	562.040	0.196290	0.00416873	0.000818280
1321	63764.9	252.517	327961.	572.679	0.194252	0.00414799	0.000805757
1507	62276.1	249.552	328555.	573.197	0.191189	0.00419435	0.000801912
1660	63871.3	252.728	322907.	568.249	0.195778	0.00414597	0.000811692
1815	62195.5	249.310	329579.	574.089	0.188433	0.00419577	0.000790623
2128	61320.0	247.629	330131.	574.570	0.187613	0.00422347	0.000792377
2291	62409.0	249.818	323570.	568.820	0.192232	0.00419088	0.000805622
2449	63162.9	251.341	325752.	570.747	0.191738	0.00416501	0.000798590
2594	59091.1	243.043	333206.	577.229	0.180020	0.00429567	0.000773308
3060	64607.0	254.179	323068.	568.391	0.198986	0.00412531	0.000820879
3323	62383.9	249.768	326276.	571.221	0.195948	0.00419527	0.000822056
3505	60765.1	246.506	310445.	557.176	0.194704	0.00424958	0.000827411
3640	63147.6	251.292	313734.	560.120	0.199573	0.00417326	0.000832873
-	-	-	319092.	564.882	-	-	-

Poisson Statistics:	Ratio	Star	Reference Star
mean	0.192393	62572.7	325188.
standard deviation	0.00492553	1478.86	6445.30
standard deviation of the mean	0.00116096	348.571	1478.65

Mean Ratio	Propagated fractional Error w/stdom	Propagated Error w/stdom	Propagated Fractional Error w/Poisson	Propagated Error w/Poisson
0.192420	0.00719083	0.00138366	0.00287565	0.00055333

Our Group's Ratio Plot

Ratio of HD 209458/HD 210074 vs. the time of image (s), starting from 05:23:00 UT.

Ratio Plots: The first plot shows data taken on Oct. 19, 2002 by Tuan Do and then by Maureen, Emily and Deb. The second shows data taken by Brandon Swift on Oct. 21, 2002.


Ratio Plot: The first cluster of data was taken by Sabrina, Phil, Garrick and Alicia, the second by Mohan and the last by Karl.

Statistics for Class Data

The weather was horribly cloudy on the evenings of the transit; so nature decided for us: we couldn't detect the transit. The question of whether we would have been able to can be answered by analyzing the errors in the collected data. The propagated errors were much smaller than the error found by doing Poisson calculations on our final data. Error was introduced by changing weather conditions, and by the way we analyzed the data with our circle programs. Also, the reference star was a certain distance, a certain sky angle, away from HD 209458, so atmospheric conditions in one area of the sky would not necessarily affect both stars. This would change the ratios. Our group data has a smaller error than the class data as a whole. This is probably because of the varying weather conditions throughout the week.

Calculating the error, and the subsequently the minimum size of the planet detectable, raised a lot of debate in our class. Everyone used the equation:
Error/R = (radius(pl)/radius(star))^2
We disagreed on what error should be defined as. Our group initially used the standard deviation. Using this, and taking HD 209458 to be a star like our sun, we found that we would be able to detect a planet of radius 1.1005x10^8. This would make the planet just barely too small for us to see. An argument was made for using the standard deviation of the mean for the Error term. Doing the calculation over, the smallest detectable radius of the planet is .60 Jupiter radius', which means that we would be able to detect the planet.

Below are several simulations that create an array of points with a dispersion defined by the error in the ratio. The simulations are set up so that they show what data gathered during a transit would look like. The ratios are normalized. During transit, the ratio drops by 1.5%. The dotted line shows the true ratio value.

This is a simulation with an error predicted by our propagated error. It was .00091. The transit is clearly visible.

This is a simulation with error from the standard deviation of the mean it appears to have a y-range similar to the y-range in our group plot, above.

This is a simulation with error corresponding to 2.5% from the standard deviation. The transit is just barely not visible.

The second plot is similar to our group data plot, while the third gives a much larger distribution.

I hoped to find an answer to whether the standard deviation, or the standard deviation of the mean should be used to find the minimum radius. I drew how the standard deviation and the standard deviation of the mean would change with more data: the mean value is known more and more precisely, but the standard deviation approaches a finite value. Using the standard deviation of the mean is correct.

Maintained by Maureen Teyssier.