The Mean Light Curve of a Abstract The abstract A variable star in the Large Magellanic Cloud is studied using sho u ld su m m a- data from the MACHO Collaboration[1]. The star is found to rise the m ain featu res o f the have a period of 14.18975±0.00025 days and a sawtooth light rep o rt, in clu d in g curve typical of a classical . Its position on the period-luminosity plot is consistent with this interpretation. The in tro d u ctio n the m ain resu lts. sho u ld m o tiv ate D o n ‘t co n fu se the the an aly sis: abstract w ith the ex p lain w hy it is in tro d u ctio n . 1. Introduction im p o rtan t, w hy y o u hav e cho sen Although the Sun’s light output is nearly constant, there this p articu lar are many stars whose luminosity varies with time. Some vary o bject to stu d y , over a regular and predictable cycle, while others have irregular an d w hy y o u are outbursts. In some cases the variation is intrinsic to the star u sin g this p arti- itself, while in other cases it is caused by a binary companion. cu lar m etho d . Variable stars have many uses in : for example, eclipsing binary systems can be used to calculate stellar masses, while some types of regular variable stars can be used as distance indicators. The period of the variation, and the shape and regularity of the light curve, are of critical importance in understanding variable stars. Therefore it is necessary to be able to determine these from observation. Because of weather and other constraints, variable star observations are usually made on a rather irregular schedule, so methods that assume regular sampling of the curve (e.g. many applications of Fourier series) will not work. In this report, a method of successive approximation is used to N o te the determine the period and extract the light curve for a variable referen ce here. star in the Large Magellanic Cloud, based on data collected by If y o u u se d ata the MACHO gravitational lensing experiment[1]. The MACHO fro m a p u blished experiment collected a great deal of variable-star data as a by- so u rce, y o u product of its search for gravitational lensing events, and all of sho u ld tell the this information is available online, making this a very suitable There‘s n o —theo ry “ sectio n read er w here it test of the method. cam e fro m . in this rep o rt, D o n ‘t p lag iarise! 2. The Light Curve becau se w e d o n ‘t n eed o n e. This The data consist of a time series of observations through sectio n is a blue and red filters. They cover the date range from Julian Date co m bin atio n o f 2448917 (October 1992) to 2451431 (September 1999), and the —d ata co llectio n “ magnitude range −9 to −9.5 for the red filter and −8.4 to −9.1 for an d —an aly sis“. the blue filter (these are instrumental values and not true apparent magnitudes). Successive observations are typically about a day apart. The time series is plotted in Figure 1. It is clear that the star is variable, but not at all clear what the period is. However, on selecting points near maximum light and plotting the time between successive points (Figure 2), a periodicity of about 14 days was seen. Averaging 16 pairs of maxima separated by ~14 N o te that y o u d o n ‘t days (one period) gave 14.05±0.07 days (standard error). u su ally g iv e d etailed calcu latio n s in a This period was refined by considering pairs of maxima with fo rm al rep o rt, bu t larger separations, obtaining 14.20±0.02 days from 6 pairs of y o u d o ex p lain the maxima each separated by 9 periods. A folded light curve was M o re d etail co u ld so u rces o f then produced by calculating the location of each point within hav e been g iv en u n certain ty an d ho w here. It w o u ld be its cycle, i.e. all points were referred to a single period, and the they are estim ated . reaso n able to period was further adjusted until the points formed as narrow a in clu d e G rap h 3 band as possible. Finally, the zero time was adjusted so that the fro m the lab d iary . period started at the minimum. B u t o n e sho u ld n o t o v erem p hasise the -9.6 techn ical d etails o f -9.5 calcu latio n . -9.4

F ig u res, tables, etc. -9.3 g a

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they can be referred R -9.2 to in the tex t. G rap hs an d d iag ram s are -9.1 —fig u res“; tables are -9 n u m bered sep arately

(Table 1 , 2 , etc.). -8.9 E q u atio n s are also 48800 49300 49800 50300 50800 51300 51800 date (MJD) n u m bered sep arately . Figure 1: The red magnitude of MACHO 82.8408.22 as a function of time.

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Figure 2: The time between successive maxima (defined as red magnitude <−9.48). A periodicity of about 14 days is clearly visible. The final period was 14.18975±0.00025 days, where the best period and error were derived from visual inspection of the folded light curve for the red filter. The zero point was chosen as JD 2448928.5. This light curve is shown in Figure 3. The same parameters were used to analyse the data from the blue filter, and a good light curve was produced (also shown in Figure 3).

-10 -9.8 -9.6 red -9.4 -9.2 E rro r bars hav e

g been ad d ed a -9 m fo llo w in g co m m en ts -8.8 blue o n lab d iary . -8.6 -8.4 -8.2 -8 0 5 10 phase (days) Figure 3: The final light curve, for a period of 14.18975 days. Each of the reduced light curves contains a small number of points that do not fit (most but not all are fainter than expected). The quoted errors on these points are very small and do not account for their positions. It is possible that these are poor quality images or poorly calibrated; the MACHO data base does not give access to the original images, so this cannot be verified. The amplitude of the variation was measured to be 0.468±0.007 magnitudes in red and 0.720±0.007 in blue (errors estimated from scatter). As the blue amplitude is significantly larger, the star must change colour, and therefore surface temperature, through its cycle (see Figure 4). The star appears to be hottest fractionally before maximum light.

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o 0.4 c 0.3 0.2 0.1 0 0 5 10 phase (days) Figure 4: Colour index B – R through the cycle. 3. Comparison with previous results R ep o rts sho u ld The MACHO variable star database[2] gives for this star: alw ay s in clu d e a A lw ay s g iv e a co m p ariso n o f • period 14.1895 days referen ce fo r the resu lts w ith • amplitude in red 0.463 magnitudes stan d ard v alu es accep ted v alu es that y o u co m p are o r theo ry . It • amplitude in blue 0.711 magnitudes w ith. J u st say in g w o u ld hav e been No errors are given. The values are in good agreement with —tex tbo o k v alu e“ u sefu l to in clu d e is n o t en o u g h. the stu d en t‘s o w n those calculated above. v alu es in this The MACHO database also specifies the true apparent sectio n , tho u g h magnitude of the star in the Kron-Cousins system as V = 15.545, as the rep o rt is N o t all rep o rts R = 14.78. q u ite sho rt it is n eed a —d iscu s- n o t essen tial. 4. Interpretation sio n “ sectio n , bu t this o n e d o es. This star has a “sawtooth” light curve and a period of The lig ht cu rv e about 14 days. The variation is clearly very regular, since points an d p erio d are from many different cycles all fall on exactly the same folded n o t m u ch u se curve. These properties are characteristic of Cepheid variables. u n less w e id en tify The location of this star on the MACHO period-luminosity plot the ty p e o f star. is appropriate for a classical Cepheid (see Figure 5).

Figure 5: Period-luminosity plot for classical Cepheid variables, from MACHO data[3]. The position of the star studied in this report is shown by the large dot. The light curve of 82.8408.22 as shown in Figure 3 is very comparable to light curves of Cepheids of similar period as obtained from the MACHO website (see Figure 6). All four examples show the same secondary bump in the rising edge of the light curve. Examination of a diagram in Payne- Gaposchkin[4] shows that the position of this bump depends on the period of the Cepheid.

Figure 6: Example Cepheid light curves with similar periods to the star studied in this report, from [3]. The diagrams are presented for comparison of shape only. A ll rep o rts sho u ld en d w ith a co n clu sio n . 5. Conclusion This sho u ld refer back to the in tro - Analysing the light curve data by studying the separation d u ctio n : to w hat of maxima and then refining a trial period based on the ex ten t hav e the appearance of the folded light curve proved to be an effective aim s o f the stu d y way of determining the period. The accuracy of the method been achiev ed ? seems to be very good, although unfortunately the MACHO W hat d o the database does not give the uncertainty of its quoted period. resu lts m ean ? The star studied appears to be a classical Cepheid: its light curve is comparable to identified Cepheids of similar period and it lies on the appropriate period-luminosity relation. The identification of this star as an eclipsing binary in the MACHO Variable Star Catalog[2] is presumably an error by the automatic classification N o te that the program. referen ces are F o r o n lin e lin k ed by n u m ber References referen ces, g iv e to the tex t œ w e title, au tho r if 1. MACHO home page, http://wwwmacho.mcmaster.ca/ d o n ‘t ju st g iv e a sp ecified , an d biblio g rap hy 2. MACHO Variable Star Catalog, fu ll UR L w itho u t say in g http://store.anu.edu.au:3001//cgi-bin/varstar.pl; w hich so u rce this star can be found by doing a search based on the w as u sed fo r period. w hat p u rp o se. 3. MACHO Interactive PL plot, http://www.macho.mcmaster.ca/Demos/Cepheids/WebPL. html F o r bo o k s, g iv e 4. Cecilia Payne-Gaposchkin, Stars and Clusters, Harvard au tho r(s), title, University Press, 1979, page 121. ed itio n if sp eci- fied , p u blisher, d ate an d p ag e o r chap ter.