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ASP: Star Science in the Autumn Sky

The Universe in the Classroom

Star Science in the Autumn Sky

3. Measure Delta Cephei for a month

Once students get the hang of it, they can find and measure the brightness of Delta Cephei in a few minutes. They should do this as many nights as possible over the course of a month. If a student forgets to observe one night, or the sky is cloudy, that's not a problem; just try to observe as many nights as possible for a whole month. If two or more students independently measure the brightness at roughly the same time, they may decide to take the average.

4. Plot the magnitude of Delta Cephei

Such a graph is called a light curve. On the horizontal axis, put the day of the month. If observations spilled into the next month, pretend that the first month got longer; for instance, Oct. 10 would be Sep. 40 for purposes of plotting. On the vertical axis, put the magnitude. Flip the vertical axis upside- down so that magnitude gets smaller as you go up. That way, brightness increases as you go up. Many teachers find that this is a good time to review graphing.

5. Draw a smooth curve through the points

This is often a new experience for students, who are used to drawing a curve connecting each point. Since each magnitude measurement has an uncertainty of 0.2, it's sufficient that the curve passes within 0.2 of each point, on average.

6. Estimate the date of maximum and minimum brightness

The smooth curve should tell you the answer.

7. Estimate the period

Many variables brighten and fade regularly. The number of days it takes to go from brightest to dimmest to brightest is the period. Does Delta Cephei seem to behave in this way? How accurate do you think your estimate is? [The period of Delta Cephei is actually 5.366269 days, or 5 days 8 hours 47 minutes.]

8. Plot a phase diagram

If a star (or any other phenomenon) is periodic, its variation depends only on where it is within its cycle, and not on which particular cycle it's in. The location within the cycle is called the phase. A calculator makes it easy to calculate the phase: Simply divide the date by the period and note the decimal part. For instance, if the period is 5.4 days, the phase of the 12th day of observations is 0.22; the phase of the 40th day is 0.41.

Using a period of 5.4 days, calculate the phases of the Delta Cephei measurements. Then you can plot magnitude versus phase. You should find that the magnitudes at the same phase will be about the same -- not exactly the same, though, because of the measurement errors that inevitably creep in. The accuracy of the plots is higher when the dates are expressed in terms of fractional days.

Try recomputing phases using different periods, say, 4.0 days or whatever you want. Prepare new phase diagrams. How does they compare to the first one?

9. Compare your estimate of the date of maximum brightness with astronomers' predictions

Based on the past behavior of the Delta Cephei, the maximum brightness occurs on the following dates for the upcoming months. These dates are for the Pacific time zone; the date in other locations occasionally differs by one day.

Dates of Maximum Brightness
of Delta Cephei
Sep. 29 Oct. 31 Dec. 2 Jan. 4, 1996
Oct. 4 Nov. 6 Dec. 8 Jan. 9
Oct. 10 Nov. 11 Dec. 13 Jan. 14
Oct. 15 Nov. 16 Dec. 19 Jan. 20
Oct. 21 Nov. 22 Dec. 24 Jan. 25
Oct. 26 Nov. 27 Dec. 29 Jan. 30

Extra credit

If you're looking for a way to challenge advanced students, have them express their data in terms of "Julian date,'' an astronomical calendar system described in almanacs (such as The World Almanac) and books for advanced amateurs (such as The Observer's Handbook). The integer part of a Julian date represents the number of days elapsed since noon on Jan. 1, 4713 B.C. The fractional part represents the Universal Time. One quirk of the system is that a Julian day begins at noon, rather than midnight. The Julian date for midnight Jan. 1, 1995 is 244 9718.5; for midnight Jan. 1, 1996 the Julian date is 244 0083.5.

If students express their data in terms of Julian date, they can send them to the American Association of Variable Star Observers in Cambridge, Mass. (email: aavso@aavso.org). The maximum brightness of Delta Cephei occurs at the Julian dates 2449722.05 + 5.366269 N, where N is an integer.

The following table lists the measurements of Delta Cephei made by one amateur astronomer in autumn 1990. Your students can use these data for practice.

Measurements of Delta Cephei, Autumn 1990
Calendar Date (Pacific time zone) Julian Date Magnitude
Oct. 5 2448170.63 4.0
Oct. 8 2448173.55 4.4
Oct. 10 2448175.58 4.0
Oct. 11 2448176.58 3.5
Oct. 12 2448177.55 3.8
Oct. 13 2448178.57 4.0
Oct. 15 2448179.59 4.4
Oct. 16 2448181.54 3.7
Oct. 17 2448182.55 3.8
Oct. 19 2448184.49 4.0
Oct. 20 2448185.51 4.2
Oct. 21 2448186.55 4.0
Oct. 23 2448188.55 4.0
Oct. 27 2448192.59 3.8
Oct. 28 2448193.50 4.0
Oct. 29 2448194.49 4.1
Oct. 30 2448195.57 4.3
Nov. 1 (Oct. 32) 2448197.51 4.0 
Nov.  5 (Oct. 36)
 2448201.50 4.5
Nov. 6 (Oct. 37) 2448202.51 4.1

Source: American Association of Variable Star Observers

This activity is a simplified version of one that appeared in Hands-On Astrophysics, a series of activities prepared by the American Association of Variable Star Observers with support from the National Science Foundation.

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