AsteroidRotationPeriods小行星自转周期.doc

Asteroid Rotation PeriodsCLASS:Technology, Astronomy and Physics GRADE:Senior High School1. 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 (Newtons 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 ruler5. 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 procedureThe 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. TheoryIntroduction to AsteroidsAsteroids 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 291st 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 asteroids 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 Earth-bound astronomer can perform by studying the asteroids light.Firstly, the asteroids 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 asteroids 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 asteroids rotation period, shape, and whether its 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 time-dependent variations and a phenomenon called the opposition effect (this is the sudden rise of the asteroids brightness when its 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 non-spherical 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 asteroids 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 DeterminationThe 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 non-spherical, 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 asteroids 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 its 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 cross-section 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 wont 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 its not safe to assume a 2 maxima rule and rely on measuring the time between a minimum and a maximum. Asteroid PhotometryAt 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 1st magnitude and the dimmest the unaided eye sees are 5th or 6th (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 mag, 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 6th magnitude is 100 times fainter than a 1stmag star, and a 2ndmag star is 1000 times brighter than a 9.5mag star, or 10000 times brighter than a 12thmag star. This can be expressed in a mathematical formula of the form:(1)Where m1, m2 are the magnitude of two objects and I1, I2 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, lets 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 ProcessThe 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 dont 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, its 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:(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 l