CLASS: Technology, Astronomy and Physics
GRADE: Senior High School
1. Educational outcome:
 To learn how to study the properties of a cosmic physical phenomenon by making use of the previously acquired knowledge. To learn, by applying their knowledge of geometrical optics and using simple mathematics, how to estimate the Asteroid rotation.
2. Purpose:
The students should:
 be able to understand the origin of the constant rotation of an asteroid along its axis
 be able to understand the origin of the asteroid motion around the sun
(Newton's universal attraction law)
 be able to recognize a periodicity within a set of data
 be able to compare and contract the differences between orbital rotation and self rotation
 be able to make astronomical imaging observations and extract
Photometric data from their CCD frames
 be able to learn how to complete and apply the method of finding the period by plotting the asteroid light curve
 be able to understand the scientific methodology
3. Educational approach:
The students use the experimental method (observation-collection of experimental data and their analysis) to record and analyze data in order to study a cosmic physical phenomenon and estimate its parameters. The phenomenon under study is the presence of asteroid rotation. The lesson involves the observation of selected asteroids, with the Andreas Michalitsianos robotic telescope of the Eudoxos center for education and research, and the estimation of the rotation periods.
4. Equipment tools:
• Andreas Michalitsianos Telescope (TAM)
• A personal computer with internet connection
• A sheet of paper, a pencil
• A ruler

5. Short description:
The students organize and perform an experiment with the purpose to determine
(estimate) the asteroid rotation periods. The realization of the experiment is accomplished by observing the moon with the AM telescope and collecting images of asteroids for further analysis. The control of the telescope is done via a computer interface, which contains special software to determine the parameters of an astronomical observation and allows the student to request the specific observation from the Eudoxos site. It also helps with the acquisition and the analysis of the images. The interface is accessible via the internet as is located in the Eudoxos web site: http://www.ellinogermaniki.gr/ep/eudoxos .
6. The structure of the lesson:
 Detailed description of the teaching procedure
The students will access the web site of Eudoxos that will guide them to conduct the lesson-experiment. At start, they should study the theory involved. This is a necessary step to be taken in order to be able to follow and understand the instructions. This also involves the procedure of the determination of various parameters, which are needed in the estimation of the asteroid rotation period.
 Theory
Introduction to Asteroids
Asteroids are small solid bodies, typically about a few kms large, that orbit the
Sun between Mars and Jupiter (most of them occupy the region around 2,5AU), which is often called the asteroid belt. The name asteroid is of Greek origin and means ‘kind of a star’, or better put ‘looks like a star’. The reason for that is that since asteroids are such small bodies and reside far from the Earth, in the telescope they are not resolved into the heavily cratered worlds we are familiar with (from spacecraft images), but look just like ordinary stars: ie small dots of light. The only difference in the telescope is that since asteroids are orbiting around the Sun they appear to move in respect to the fixed stars on the background (and this is utilized for finding asteroids).
Most people assume that asteroids are rocky in composition, but as many studies have proved asteroids can have a wide range of mineral compositions. A large percentage of the asteroids are carbonaceous (which might be very dark), while others are made up of metallic minerals rich in iron and/or nickel. There is also the possibility that asteroids could contain quantities of water ice, especially those farther for the Sun towards the edges of the asteroid belt.
Asteroids when first discovered get a provisional designation, something like 1995
CB, which denote the year, month and order of discovery (eg. CB= second asteroid discovered in the first part of February). Once the orbit of the asteroid is accurate enough the asteroid gets numbered and named. The name is proposed by the discoverer, while the number denotes the order in the numbering process
(eg. asteroid 291 Alice is the 291 st asteroid to be numbered). The largest asteroids have small numbers, since being large obviously helped their early discovery. Therefore the numbering order can be taken as an indication of an asteroid’s size (but this is no absolute rule as the asteroid brightness is also an

issue and chance played an important role –however we can expect asteroid 2
Pallas to be larger than 219 Alice and that larger than 4.179 Toutatis).
Although the best way to study an asteroid is by in situ measurements –ie by a spacecraft, this is very expensive and therefore only a small number of asteroids can be studied this way. However there are quite a few studies that the Earthbound astronomer can perform by studying the asteroid’s light.
Firstly, the asteroid’s path can be determined across the sky by measuring with precision the position of the asteroid at different dates. This will allow the determination of the asteroid’s orbit around the Sun, and therefore its path on the sky in subsequent days, months or years.
Another very useful method is to study the photometric curve (also called light curve) of the asteroid, which can contain information on the asteroid’s rotation period, shape, and whether it’s binary. This is the method that this exercise focuses onto.
Of course there are more studies one can carry out such as spectroscopy which could possibly give information on the surface composition. Or if the asteroid gets close to Earth, radar can be utilized to determine the exact shape and size of the asteroid, and even produce low-resolution images of the surface, but of course these require more specialized equipment.
Early brightness measurements of asteroids revealed both periodic timedependent variations and a phenomenon called the opposition effect (this is the sudden rise of the asteroid’s brightness when it’s very close to opposition).
However, from photometric measurements in the visual wavelengths alone, it is not possible to conclude on what causes these variations as it could be either due to changes of the reflectivity on different parts of the asteroid, or due to a nonspherical shape (which would cause variation of the cross-section that is observed, and therefore of the light reflected toward Earth), or maybe both.
Whichever is the correct interpretation the periodic brightness variations betray the rotation period of the asteroid around its axis.
But which axis? Since each asteroid would have suffered several large impacts, one would expect asteroids to have in general a complex rotational behavior and display precessional motion (rotation around more than a single axis) as well to the normal single-axis rotation, but this is not the case. Instead the study of brightness variations shows that precessional motion is quite rare in our days. It has been eliminated by viscous damping over millions of years, and therefore like the planets- only a uniform rotation remains around the shortest axis. Notice that this is not the case for the Earth that has a precessional motion maintained by a torque that the forces of the Moon and Sun create.
From the asteroids studied so far, the average rotation period is of the order of
10h, while most asteroids have periods between 7h and 30h. Extreme cases of very slow rotators with periods a few weeks are known, and very fast rotators with periods around 2h have been found. The latter are quite important as the minimum density of the asteroid can be estimated, and a conclusion can be reached on whether the object is rigid or not. Since some density measurements from spacecrafts have revealed some low-density asteroids, these are most probably not consolidated bodies but a “rumble pile” held together by gravity.
Calculations show that the centrifugal is equal to the force of gravity on the equator of fast rotators with periods 1,5h to 3h (depending on the exact density) and therefore such asteroids are unstable and begin to fragment.
Also, the typical brightness variation is of the order of 0,2mag (which corresponds to 20% variation), but NEAs (Near Earth Asteroids) tend to show greater variation than typical. This could mean that these asteroids are more irregular than typical, but this is not a certain conclusion given the fact that most of them are measured

when close to Earth, far from opposition. With that been said it must be noted that the brightness variation amplitude is certainly not constant throughout the asteroid’s orbit, and the changing Asteroid-Earth-Sun geometry adds up to the complexity due to the change of the illumination angle of the asteroid.
Asteroid Rotation Period Determination
The aim here is to measure the rotation period of an asteroid by means of photometry. For most asteroids the brightness variations are due to their nonspherical, elongated shape that is similar to a three-dimensional ellipsoid –ie something like a rugby ball. In this case the variation is due to the fact that the asteroid cross-section as seen from Earth changes, so the light reflected towards us varies as well and therefore the asteroid’s brightness appears to vary.
For a rugby-ball shaped asteroid the photometry light curve has two minima and two maxima per revolution, as the shape is 180 degrees symmetrical around an axis perpendicular to it’s greatest dimension –ie it appears the same when rotated 180 degrees. So the same pattern of cross-section change will appear as the asteroid rotates from 0-180 degrees and 180-360 degrees. This concept is illustrated on fig.1 and fig.2. But notice that when the viewing direction and the rotation axis are the same there is no cross-section variation no matter the asteroid shape.
Figure 1: Photometry light curve of asteroid Geographos that displays the 2 minima and 2 maxima per revolution rule. Here the maxima have almost the same amplitude, but the minima display considerable amplitude difference. The numbers in the red dots on the curve denote the respective view on figure 2.

Figure 2: A sequence of views from Earth of asteroid Geographos that demonstrates how the crosssection varies with time. It starts from a maximum at (1) goes to a minimum at (3) and then again at a maximum at (5) when the other side of the asteroid is turned in view, and a second minimum at
(8), finally the revolution is completed when the same side of the asteroid is again in view. Notice that the asteroid brightness displayed at (3) is less than at (8) due to irregular shape and shadows.
The amplitude of the maxima however will be different for a number of reasons: there could be some reflectivity difference between the two sides of the asteroid, also the asteroid will have features such as craters on it, while of course it won’t be exactly an ellipsoid. Also shadows can play an important role when the asteroid is not close to opposition. Although, most asteroids are potato-shaped rather than rugby-ball shape, the ellipsoid approximation is reasonable, but other more irregular cases are possible that would produce a different number of maxima. For example a pyramid-shaped asteroid will produce four maxima, since this shape is 90 degrees symmetrical. Therefore, we cannot conclude on the rotation period unless we have covered more than a whole revolution of the asteroid, and can trace the light curve repeat itself –ie it’s not safe to assume a 2 maxima rule and rely on measuring the time between a minimum and a maximum.
Asteroid Photometry
At this point, before we get into any details concerning photometry, a word on the magnitude system is in order. The magnitude system is the scale that astronomers have traditionally been using to denote the brightness of objects.
This scale is logarithmic and not linear, because the eye perceives illumination in such a non-linear fashion. And since the first brightness measurements of stars were carried out with the eye, either unaided, or through the telescope, this scale was extended and is still in full use. So astronomers, when quoting the brightness of a star (or other object in the sky) are speaking in terms of magnitudes, and the greater the magnitude is said to be the fainter the object is. The brighter stars in the sky are zero or 1 are 5 th or 6 th st magnitude and the dimmest the unaided eye sees
(in dark skies). For example the nearby bright star Vega (near zenith on summer evenings) is exactly zero magnitude, while the brightest star in the sky, Sirius (seen in the winter) is -1.4mag. Notice on this example that the magnitude system is extended to negative values for objects brighter than zero magnitude, and the brighter the object the more negative it gets. For example
Venus is even brighter than Sirius and shines at its brightest at -4.6mag, while

the even brighter full moon is at -12.5mag. Also notice that silently the unit ‘mag’ has been used to denote the brightness in the magnitude system, and that tenths of a magnitude have been used. Even higher precision can be used, but this is not justified for measurements made by the eye alone, since the accuracy is about
½mag, and this is why the magnitude scale at first had only integer values.
However today with the use of electronic sensors it is possible to measure the intensity of light from an object -and therefore its brightness- to very high precision. With such measurements the magnitude system once based on the eye has been linked to the real intensity of light. The basic relation is that the intensity becomes 10 times larger for every 2.5 magnitudes we subtract (or 100 times for each 5mag). Therefore a star of 6 th magnitude is 100 times fainter than a 1 st mag star, and a 2 nd mag star is 1000 times brighter than a 9.5mag star, or
10000 times brighter than a 12 mathematical formula of the form: th mag star. This can be expressed in a m
1
 m
2
 
2 .
5 log(
I
1
I
2
) (1)
Where m
1
, m
2
are the magnitude of two objects and I
1
, I
2
are the
corresponding light intensities for each of them. Eq.1 is the most basic photometry equation and is going to be utilized to convert measurements to the magnitude system.
Now since we have discussed the magnitude system, let’s examine the process of asteroid photometry. As you can imagine to get a photometric light curve of the asteroid we need to make measurements of its light intensity at certain intervals to get the data points we need to make a plot. But the question is how many points do we need and what should the time interval between the
measurements be? The answer to the first part of the question is of course as much as we can get, more data points are always better, but typically 50 points seems to be a good minimum number that gives a good enough plot to find the period. For the second part of the question again the answer is as fast as possible, but the matter is more complex since large intervals on a fast rotator would give too few points per revolution, while too short intervals would limit the exposure time of the sensor and therefore would reduce the signal (and produce more data to be reduced by the observer).
Asteroid Photometry Process
The process of making photometric measurements is based on eq.1, which states that we can calculate the magnitude of one object if we measure its intensity and then compare it with another of known magnitude and intensity (this technique is called differential photometry). Furthermore, if we are only interested on brightness variations we can just compare with an object of constant brightness, but we don’t need to really know what its actual brightness is (in the magnitude system). This simplifies this exercise since we only need to determine the periodic brightness variation and not the actual brightness of the asteroid. Of course finding the actual brightness would be good for comparing results with other astronomers, or in cases that data from different observers are needed to compile a light curve.

Anyway, since the aim of the exercise is period determination alone, we will not need to measure a standard star (these are stars of known brightness in the magnitude system). Therefore the measurement process will be too simply to measure the signal from the asteroid and at least one comparison star of similar
color that we can find in the same field as the asteroid. But because we can not be certain if the comparison star is variable, it’s better to pick two or more comparison stars and use the average intensity. In this manner if one is slightly variable the variation will be reduced as we average, and moreover the comparison intensity will be of better precision, as noise in the intensity measurements will be reduced as well. Further, this way (by using 3 comparison stars or more) we can check if one of the comparison stars is variable by checking its magnitude difference compared to the average of the rest. If the star is OK this should be constant. From eq.1 the formula for the calculations is:
 m
 
2 .
5 log
1
N
( I
1

I
2
I

...

I
N
)
(2)
One more thing we must pay attention to is that the comparison stars should be of similar color to the asteroid, as briefly noted above. This is necessary because the atmosphere absorbs and scatters light differently, depending on the wavelength of light. Smaller wavelengths –ie blue light, are attenuated more than the longer wavelengths of red or near-infrared, this is why the Sun becomes red at low altitudes. And since asteroids are usually reddish, if we use a blue comparison star then as we get to lower altitudes the blue star will be dimmed more than the asteroid and we will end up in our light curve with an erroneous brightness increase (a negative Δm).
To find suitable stars, if we don’t have any filters we’d need to use a catalogue such as USNO A2.0 or SA2.0 which contain millions of stars with photometry accurate to about ¼mag, and use it to pick stars with B-R>1.0mag.
Else we can use filters at the start of the observation by taking first one image with a B or V filter and a second with a V or R respectively (we can say roughly that B=blue, V=visible, R=red). Then we can check the B-V or V-R of the asteroid itself and pick comparison stars that have values as close to that as possible. An even better solution is to make a filtered observation and totally eliminate the possible problem. The only disadvantage with filtered observations is that most of the light of the object is blocked and we end up receiving something like 20% of it. Making the observation with an R filter is advantageous since the asteroid itself is reddish so less light is lost. There is also less extinction due to the atmosphere, and moreover the sky is bluish, so we eliminate some of the background and therefore the noise is reduced as well.
Some sensors are much more sensitive to red light than blue light, and therefore no R filter is needed! Another solution would be to limit our observation to high altitudes in the atmosphere, such as 30 degrees and above. This way the dimming of the blue light won’t matter as much, especially when averaging lots of comparison stars. Of course the main problem with this method is that we limit our observation in respect to the time period available, which may not be enough to get a good light curve.

Student activities
A.
Picking an Asteroid
For this exercise it is important to be able to determine the period with a single photometric run (this means with one night’s data alone). So we need to pick an asteroid that has a short enough period for us to see a good part of the curve repeat itself, and therefore a period of less than 4h is a quite reasonable criterion.
Of course we need the asteroid to be bright enough to give a good signal for photometry and to be favorably placed in the sky so that it’s observable most of the night (more than 6 hours). With these criteria the 60 asteroids currently know to have periods <4h where checked to see which are bright enough and well placed. Of those only 7 are suitable, and the details are listed below:
522 Helga: At opposition on mid-May that reaches 14.4mag. The best period to observe it is on the first week of May when it displays its greatest elevation ~40 degrees. At that time it is possible to observe it over 30 degrees of elevation for
5½ hours.
699 Hela: At opposition on the first week of June, that it reaches 13.5mag.
Better to observe from the end of April and on that it will be brighter than
15.0mag and can be viewed more than 6hours. Maximum elevation ~30 degrees in mid-May.
841 Arabella: At opposition ~10 th of May, that reaches 15.5mag. Better observed from end of April to mid-May that will be visible all night (about
9hours). Maximum elevation ~30 degrees throughout most of spring.
1250 Galanthus: At opposition on the first week on June, that reaches
15.6mag. This is an unfavorable opposition as it will be in Scorpio, very low in the sky and within reach star-fields (difficult to detect). Maximum elevation only 12 degrees at the beginning of June.
1368 Numidia: At opposition around 20 th of April, that it reaches 14.0mag. Will be brighter than 15mag from mid-March to start of June, but its better placed for observation at April to mid-May that will be visible all night (about 10 hours).
Maximum elevation ~35 degrees at mid-April, when we can observe it for 4 hours over 30 degrees of elevation.
1727 Mette: Although opposition was in the winter, this asteroid is brighter than 15mag in mid-March and will be brighter than 16.0mag till the last week of
April. Due to its high declination (meaning it’s near the pole) it will be visible almost all night (about 10 hours) at the end of March- beginning of April. This is a fast rotator with period around 2½hours. Maximum elevation 87 degrees (almost zenith) at the end of March.
4979 Otawara: At opposition on the second week on June, when it reaches
15.4mag. However it will be fainter than 16.0mag before June 1 st . This is also a fast rotator with about 2¾ hour’s period. Maximum elevation 29 degrees at the beginning of June.
As we can see from the above list what mostly determines the target we choose is the desired observation period. Arguably, in the spring, the best placed object by date of opposition alone is 1368 Numidia, while it will be as bright as
14.0mag. However the low maximum elevation (=altitude) of 35 degrees means that if we set a limit for the object to be over 30 degrees of altitude to observe it, we end up with just 4 hours of observing time.

The other good spring opposition is that of asteroid 522 Helga that is at opposition at mid-May, when it also displays a good maximum elevation of slightly over 40 degrees. Therefore it could be observed about 6 hours with good elevation. The timing is also good for Helga, since there is new moon at the beginning of May.
But, in respect to sky position alone the best-placed object is 1727 Mette, which almost gets to zenith! This is also a fast rotator, which means that ~5hours of data would enough. However the object was in opposition in the winter and it will be getting fainter as time goes by. It’s better to observe it early March to mid-
April at most.
B.
Planning the observation
Once we have decided on the asteroid (eg Numidia) we need to decide on the date we want to observe it, so we can find its sky coordinates to point the telescope to it. We also need to see when the object will be high enough in the sky to carry out the observation, and to check if the starfield that the asteroid will be in is suitable. To do all these we need to use a sky-charting program. The one proposed here is an excellent program called “Cartes du Ciel”, which is free to download from Eudoxos web site: http://www3.ellinogermaniki.gr/ep/eudoxos/htm/private/private.asp
.
To decide the date the criteria are: the asteroid should be as close to opposition as possible and the moon should not be in the sky for most of
the night. Concerning the first criterion the best period to observe the asteroids was given in the list above. Concerning the moon presence we can eliminate any problems by working near new moon. Here is a list of the periods in spring which is free of moonlight interference –calculated for moon less than 25% illuminated
(in parenthesis is the date of new moon):
 27-2-2003 to 8-3-2003 (3-3-2003)
 28-3-2003 to 7-4-2003 (2-4-2003)
 26-4-2003 to 7-5-2003 (1-5-2003)
 26-5-2003 to 5-6-2003 (31-5-2003)
Now let’s assume that we decided to observe 1727 Mette on the 1 st of April, and we want to see what the circumstances are:
1) Before we run Cartes du Ciel copy the Asteroides.dat
file which can be found on the Eudoxos web site at: http://www3.ellinogermaniki.gr/ep/eudoxos/htm/private/private.asp
, to the hard disk on the C:\Program Files\Ciel\cat\planet
directory (assuming the program is installed on C:\Program Files\Ciel which is the default location). This file has the orbital elements for our asteroids added since they did not exist on the original CBAT file.
2) Select Preferences  Observatory.
(or click the observatory icon on the left side toolbar). And on the new window enter the telescope coordinates: Latitude: 38 degrees 10 min 17 sec North.
Longitude: +20 degrees 37 min 11 sec East. Altitude: 1050 meters. And set Time= UTC +2. Then click OK.
3) Then select Preferences  Date/Time.
(or click the clock icon on the left side toolbar over the observatory icon). And on the new window uncheck the ‘Use system time’ option and the ‘Auto refresh’ –if

enabled. Then set the date to 2003-4-1 (notice the time there is in year-month-day format) and the time to 17:00:00 which is afternoon with the Sun still up. Click OK to accept the settings.
4) Now select Preferences  Catalog and Object parameters . (or click the icon above the date/time icon on the left side toolbar). Then click on the ‘Asteroids’ tab and the list of asteroids should appear, with the asteroids for this exercise on top. Select ‘1727 Mette’ and click OK.
5) Then select Search  Find.
(or click the binocular icon on the topcenter toolbar). Click the Solar System tab and select Asteroids.
1727
Mette should be listed below. Click OK and asteroid will be presented on the screen as a white dot with a pink arrow pointing to it!
6) Move the mouse to center the asteroid and click on it.
The asteroid name ‘1727 Mette’ should appear next to it. If we click on the name
(notice that the pointer becomes a hand) an ‘Identification’ window appears which gives us useful information on the object such as distance from Earth or the Sun, its coordinates on the sky, the time it rises and set, etc. Close that window for now.
7) Point again on the asteroid and right-click on it.
A menu appears, select Track 1727 Mette. A new window will pop up that will enable us to see how the object moves in the sky. It should be tilted ‘2003-4-1
17h00m00.0s’ which tells us the date and time that we view the sky chart.
8) Now set in the window 0 days, 0 hours, 15 minutes and 0 seconds.
This selects a 15min interval on the time lapse action between each screen redraw. We can make the time advance by clicking the right arrow or we can go backwards by click the left arrow, or we can pause by clicking the square in the middle.
9) Click the right arrow and notice how the asteroid gains altitude in daytime, and then sky brightness is reduced until the Sun goes down and stars begin to appear. At this point click the pause button to stop the time lapse as soon as the sky becomes totally dark which happens at 20:45 local time. Use the left arrow to go back if you pressed the pause too late.
10) Click on the asteroid to get the identification window as in (6), and write down the sky coordinates given at J2000 as RA=9h20m20.25s,
DE=+35 o 05’00.4”. Also make a note of the local time (20:45).
11) Then click the right arrow so that time can lapse once more, and notice how the asteroid will cross zenith and start to get lower in altitude. On
the chart gray altitude and azimuth lines are displayed. Notice when the +30 degree line appears and click the pause when the asteroid is crossing it , which happens at 2:30. Make a note of this time, since this is the time we want the observation to end (we limit to above 30 degrees elevation).
12) Click again on the asteroid and write down the new coordinates (the asteroid has moved in these 4½ hours) RA=9h20m28.42s,
DE=+35 o 05’32.0”. The difference from the previous position therefore is 8.17seconds in RA (Right Ascension) and 31.6arcsec in DE
(Declination). The 8.17sec correspond to a motion of
8.17x15=122.6arcsec –since the Earth rotates 15arcsec per second of time. This motion is not too much compared to the field of view of the
CCD camera on the telescope which is 405arcsec.

13) Therefore we can calculate the middle of the asteroid’s track, and point the telescope there the whole night without worry that the asteroid might get outside the field of view. This can be easily done by averaging the two positions in RA and DE, and we get:
RA=20h20m24.34s and DE=+35 o 05’16.2”.
14) But before we decide on the exact position lets have a better view of the starfield the asteroid will be in. Use the left arrow to return to the starting time (20:45).
First select Move the 1 o resources  USNO-A 
 Field Width
icon on the right side toolbar). Then
 1 select File o (or click
 Online
and select either of the two A2 download sites.
Then click Connect (for this you must already be connected to the
Internet) and the specific area of the sky is downloaded from the
USNO-A2 catalogue (this may take a minute or so). Then click the OK button and the stars will appear on the screen! Then turn off the labels that will be flooding the screen by unchecking Preferences  Labels 
Display (select to it turn off since it will be selected).
15) Now to make the view exactly what in the CCD image will be select
Search  Locate new position. Then change on the new window the
Field Width to 6’ 45” and click OK.
From this screen we can understand that there are not many stars on this frame to use for comparison…
16) So click again the 1 o icon to enlarge the view. Then right-click on the asteroid and select End tracking.
Now the display will not follow the asteroid but will be locked on the stars like the telescope.
17) Then click the clock icon (date/time) and make the date 2003-3-31.
The asteroid will move towards the bottom-right, where there are more stars. If we make the date 2003-3-30 then the asteroid is within a good starfield with several stars 14-15mag. So it’s better to move the date to 30 th of March!
18) Now we have to determine the coordinates again. But since it’s only 2 days difference we can use the same start and end times, and to make things a bit easier lets make the start time 21:00. And since the end time is 2:30 the middle of the observation period will be 23:45. So by clicking the clock icon we set the time to 23:45:00 (date 30/3).
Now by clicking on the asteroid we get the coordinates we need to point the telescope to: RA=9h19m15.91s, DE=+34 o 59’43.5”.
19) Finally to view what the selected field will look like right-click on the asteroid and select Center. Then Search  Locate new position and set the view to 6’ 45”.
At this point save the chart for later use by selecting
File  Save chart  enter a name eg Mette, and click Save.
It would be a good idea to also print this chart (click on the printer icon).
20) Moreover, we can download an image of the selected field by selecting
File  Online resources  Images  ESO Skycat DSS (Digital Sky
Survey). Then we can save this image by selecting Images
Background image and then click the Save as BMP button.

C.
Submitting Request to the Telescope
Using the “Eudoxos” Robotic telescope user interface, we can request observations of any object. All we need to do is fill in the appropriate request form at: http://www.ellinogermaniki.gr/ep/eudoxos/etool

There we need to select the AM Telescope , and enter the observer code and information below. Then we can point the telescope to the desired object (the asteroid) by entering its sky coordinates. According to the above example for
1727 Mette these are RA=9h19m15.91s and DE=+34 o 59’43.5”. So enter
09:19:16 into the RA box and 34:59:44 into the Dec box. Don’t forget to select
‘Manual’.
To set the correct integration time we select the C (clear –ie no filter) setting and we type in 60 next to it, below the column ‘Filter(s) and Duration(s)’. This makes the exposure time 60sec without any filter. If we want to try another asteroid that gets lower in the sky select the R setting (red filter).
Then we need to type in the desired date and time for the observation, below the column ‘Start Time’. Keep the default selection of LST which selects local time and type in the time and date in the respective boxes (date: 30-3-2003 and time: 21:00:00). Finally we need to enter how many images we want to take and the intervals between them. Since we want the observation to end at 2:30am and start at 21:00 this means that the total duration is 330min. A reasonable interval between the exposures is 3minutes given the fact that 1727 is a fast rotator. Therefore we need to take 330/3=110 exposures total. So below the column tilted ‘Repeat Count and Delay’ we enter on the top box 110 and below
00:03:00, which sets for 110 exposures (of 60sec duration each as selected below ‘Filter(s) and Duration’) spaced by 3 minutes.
D.
Reducing the Data
Once the observation is completed by the telescope the images can be downloaded from the appropriate web page. The images are going to be in the
FITS format, and an appropriate viewer is needed to open them. Such a viewer that can be found on the for free http://www3.ellinogermaniki.gr/ep/eudoxos/htm/private/private.asp
is AVIS:
Although for photometry there are several software packages in use, we could even exploit AVIS to do the work. A professional astronomer doing photometry all the time would of course prefer a more specialized software package that does batch processing of all the images on its own but this is not so easy concerning asteroid photometry since the asteroid moves with regard to the background stars. Anyway here we’ll have to the process manually but given the small number of images (~100) and the fact that the work could be splited between students this is no obstacle.
But first we need to find the asteroid in the images!
This is not difficult if we had printed the chart and/or saved the DSS image that we generated while planning the observation. Then we could just compare an image with the chart or the saved DSS image and notice the star patterns. Since we even know the asteroid’s path it should be quite easy to find it. Once we locate where we think the asteroid is we can confirm it by loading another image on the viewer maybe with half an hour distance. The asteroid should have moved and it would be very easy to track its new position.
Once we are familiar with the position of the asteroid and the stars in the image we need to pick a few comparison stars (ie the stars labeled U1200_06280778 and U1200_06280144 which are 13.0mag and 13.1mag, or the stars
U1200_06280595, U1200_06279835 and U1200_06279735 which are 14.7mag,
15.1mag and 14.7mag respectively are good choices). We can see the chart again in order to pick comparisons by simply loading it from file ( select File
Load chart  select the file eg Mette.cdc
and click Load

). Then we can inspect the

chart by clicking on stars to see their magnitude . We need to be careful not to use too bright stars (eg having more than 2.5mag difference from the target) as the sensor may be near saturation, and definitely not fainter! Also we need to be careful of the color not being bluish. On the identification window there will be a B
(blue) magnitude next to the label ‘Description 1’, while the magnitude quoted next to the ‘Magnitude’ label is an R (red) magnitude. Suitable comparison stars should have B-R>0.5mag.
Finally the data reduction procedure is:
1) We open the image on AVIS ( File  Open FITS , or click the folder icon).
2) Then select View new window.
 Stars , or click the stars icon. This will open up a
3) Place the pointer over the asteroid and notice that the cross turns into a reticule. This means that the star has been detected and its parameters determined. Double click on it and some values will appear on the Stars window. These give the exact position and total brightness in arbitrary units (ADU).
4) Next place the pointer over each of the comparison stars and measure them. Their values will appear in the Stars window below the asteroids.
Figure 3: Using AVIS we can view FITS images and measure the properties of stars in them.
5) Write down the values for the asteroid and comparisons in a table.
6) Close the image.
7) Open the next image (it is important to maintain the images order in time) and then repeat steps 2 to 6.
By doing that for all images we end up with all photometry data on a table.
8) Then we use equation 2 to calculate Δm for all images –ie times.

9) Finally we plot these values against time (we know that images are
3min apart), and we determine the rotation period by measuring the time between maxima and minima of the same amplitude!
As an example of period determination lets observe the two light curves of asteroid 4183 Cuno that are presented on fig.4 and have taken about a week apart. The asteroid gives 2 maxima and 2 minima per revolution, with the minima having similar amplitude. However the maxima have amplitudes that differ by a factor of 2!
Figure 4: Photometric light curves from asteroid 4183 Cuno taken a week apart. On the top curve the observing period was roughly equal to the asteroid’s rotational period, but unfortunately this does not provide enough data to decide what the period is. This is not the situation on the bottom curve however, where we observe the curve repeat
On the first observation we can determine the smaller (secondary) maximum and the two minima but the observation was started and ended at the times of the larger (primary) maximum. Therefore we cannot be sure that we have observed the primary maximum at all. However the time between the two minima is indicative of what the period most likely is!
In the second light curve of fig.4 the observation was extended in time and therefore the light curve is doing more than a full cycle and we can definitely see a pattern repeat itself. We can see two primary and secondary maxima and four

minima, and we can definitely tell where the two primary maxima are. Here period determination is easy, we just need to measure the distance in time between two consecutive minima or maxima of the same amplitude. This is better done with the minima since they are sharper (ie better defined in time) than the maxima. And since we can see both primary and secondary minima it’s even better to measure the period with both and take the average for better accuracy.