Measuring Asteroid Rotational Periods (Observational)

Transcription

1 Student Name Peter Caspari HET606 Student ID Project Supervisor Eduardo Alvarez SAO Project Cover Page Project 141 Measuring Asteroid Rotational Periods (Observational) All of the work contained in this project is my own original work, unless otherwise clearly stated and referenced. I have read and understood the SAO Plagiarism Page What is Plagiarism and How to Avoid It at

2 Table of Contents Introduction... 1 Aim... 1 Equipment... 2 Photometry... 3 Collecting Lightcurve Data and Data Reduction... 4 Producing Lightcurve Plots... 5 Shape Modeling... 6 Convex Hulls... 6 Light Curve and Shape Analysis... 7 Procedure... 7 Asteroid 77 Frigga... 8 Asteroid 347 Pariana Conclusion References: Appendix A - SAC 8II Camera Specifications Appendix B - Shape Model for Asteroid 77 Frigga Appendix C - Shape Model for Asteroid 347 Pariana Appendix D - Mission Control Software... 20

3 Introduction The opportunity to image an asteroid directly does not often present itself. Even the most powerful Earth based or Earth orbiting telescopes cannot resolve the image of an asteroid as the apparent angular size of these objects is smaller than one pixel in the detector. The rare opportunities to image asteroids directly come about from radar observations of close approaches and space probes. As a result astronomers are typically limited to measuring the amount of light produced by the asteroid as a function of time. This is what is called a light curve. Most asteroids have a non spherical appearance, often resembling a potato. During each rotation the asteroid presents two elongated sides and two shortened ends. This results in cyclic changes in brightness. The light curve will typically show two bright peaks and two dim troughs. These light curves help us to understand aspects of the asteroid such as its rotational period and sometimes the asteroid s spin axis and shape. The well respected astronomer, Henry Norris Russell, who in part developed the Hertzsprung-Russell diagram, declared, in relation to asteroid light curves, It is quite impossible to determine the shape of the asteroid (Russell 1906). His basic argument assumed that albedo variations would prevent the determination of the shape of the asteroid. His theory held for many years. We now know however, that asteroid albedo variations are only a small fraction of the total amplitude change as an asteroid rotates. Now that this has been accepted, the possibility of shape modeling can be realized (LCInvertweb). Aim Any study involving observational astronomy is limited by the equipment employed. However it is important that the equipment is used to its full potential. To achieve the full potential of the equipment available specifics such as image exposure times and data reduction methodology will be considered. This project will endeavor to maximize the potential of the modest equipment available and the sub-optimal environmental conditions at the observing site. Two asteroids will be studied initially to confirm the published period of rotation. With that established a further analysis of the light curve data will be conducted with the aim of determining the sidereal period, spin axis and shape model. The number of asteroids with sufficient light curve data for shape modeling and spin axis possibilities is limited. Of these, many have already been modeled by the developers of the technique (DAMITweb). The opportunity exists however, to observationally collect light curves for asteroids with substantial but still insufficient light curve data and produce a shape model. These shape models of asteroids can be used by those studying the evolution of our solar system and the evolution of asteroids. 1

4 Equipment The telescope currently in use at BDI Observatory (BDIweb) was initially a budget Chinese 8" f/4. The optics were discarded in favour of better quality hand finished mirrors. The telescope was extended to an f/6 focal ratio. The standard spider was replaced with a custom made spider that used 3 arcs made from high tensile steel. This was to minimize spider diffraction lines. The mount is a Konus EM10 which is similar to an EQ5. Meade DS motors and an Autostar were adapted to make the mount fully GOTO capable. The weight of the telescope is at the upper limit of the mount. This plus the physical size of the telescope make it susceptible to tracking problems even with relatively minor winds. This is one reason exposure times are kept to a minimum. The main camera is the SAC 8-II. While very sensitive it is only an 8 bit camera. With the gain at 100%, sky glow becomes a problem when exposure time is greater than 5 seconds. The location of BDI Observatory suffers from moderate light pollution from the city of Sydney. The solution to the problems of an 8 bit camera, poor tracking and sky glow is to stack short exposure images, usually 1 to 5 seconds depending on task and wind. The observatory control software, BDI Mission Control (see Appendix D) can automatically stack literally thousands of images. Using stacked 5 second exposures a magnitude 20+ star (USNO B1 Catalog ID: ) has been identified. Figure 1: BDI Observatory Interior Another less significant problem with the SAC8-II camera is a shadowing artifact. It arises from some of the inner workings of the camera particularly when a signal comes in that is bright enough to drive the analog-to-digital (A/D) converter to its full level. It takes some time for the A/D to recover to normal operation so saturated pixels will peg the A/D enough that the next few background sky pixels end up black. This artifact can be removed by a Van Goph filter however this filter has a broader effect and could reduce photometric accuracy. Figure 2: BDI Observatory Exterior The observatory is remotely controlled over a wireless network using Remote Desktop and custom software. The custom written observatory control program, BDI Mission Control, sends instructions to the custom written remote control program, BDI Remote, running on a notebook computer in the observatory. This notebook is connected to the camera and the Meade Autostar controller. A typical observing session of 5 hours, weather permitting, can produce 5000 images. 2

5 Photometry Photometry is a technique of measuring the intensity of an astronomical object's electromagnetic radiation in this case, visible light. All photometric observation made from BDI Observatory are unfiltered and thus only differential photometry is used. To measure the photometric accuracy of the optical train at BDI Observatory multiple measurements will be taken of non variable stars. The variations will be measured for three stars at varying magnitudes to determine the error or scatter. Due to the primary camera having a high gain and sensitivity with only an 8 bit dynamic range many short exposures need to be captured and stacked. Magnitude 12 stars begin to saturate with 1 second exposures. Typical targets for photometric study at BDI Observatory range in magnitude from 12.5 to Photometric targets are usually asteroids which move in relation to the background star field which limits exposure time. The methodology considered here is for a total exposure time of 1 minute which is typically a worst case situation. Observations of slower moving asteroids or variable stars allow for longer exposure times which would improve the SNR and reduce the scatter. For a 1 minute total exposure approximately 60 1 second individual exposures are captured and stacked. Over a typical photometric observing session literally thousands of images are captured and stacked in this way. The resulting stacked images are then analyzed by MPO Canopus (Warner, B.D. (2009a) ultimately producing a lightcurve. The following table shows magnitude compared to an average SNR for 1 and 4 minute total exposures. The SNR values were determined by Astrometrica (Astroweb). Magnitude Average SNR 1 Min Average SNR 4 Min Table 1: Magnitude vs SNR Summary The differential magnitudes from three non variable stars from a recent photometric asteroid study have had their magnitudes plotted against time. Approximately 60 images from this study were analyzed. The magnitudes of the selected stars are typical of objects studied at BDI Observatory. Based on the following formula a measure of accuracy based on SNRs can also be calculated (AAVSOweb). Standard Error = (2.5 * log (1 + 1/SNR) * log (1-1/SNR))/2 3

6 The following table summarizes the amount of scatter for observations of stars with different magnitudes. All measurements are based on 1 minute total exposure time which is the worst case scenario. 1 Minute Test Magnitude / Star Visual Scatter Estimate Standard Deviation from data Phototest / UCAC Phototest / UCAC Phototest / UCAC Table 2: Photometric accuracy summary Std Error based on SNR To evaluate a better case scenario the same images were stacked into sets of 4 images. This equates to an exposure time of approximately 4 minutes. The additional exposure time provided a significant reduction in the scatter. 4 Minute Test Magnitude / Star Visual Scatter Estimate Standard Deviation from data Phototest / UCAC Phototest / UCAC Phototest / UCAC Table 3: Photometric accuracy summary Std Error based on SNR Collecting Lightcurve Data and Data Reduction A typical lightcurve data collection observing session can involve capturing up to second exposures. The images are captured with Astrovideo (AVweb) which also does dark frame subtraction. During these exposures the BDI Mission control software loads and plate solves the images using the Charon software (Charonweb). Based on the plate solving solution tracking instructions are sent back to the BDI Remote software and then onto the Meade Autostar controller. This setup allows for an unattended automated process. Once all the images are collected for the observing session synthetic flats are generated and applied to the images. The SAC8-II camera is peltier cooled however ambient temperature changes do alter the flat field during the course of an imaging session. As a result multiple synthetic flats may need to be generated. The application used to generate synthetic flats is called iprep (iprepweb) The resulting images are then automatically stacked into batches of 60 producing approximately 1 minute exposures which are usually short enough to prevent trailing for most asteroids studied. The automated stacking process is run from the BDI Mission control software which shells out to the command line version of Deep Sky Stacker for individual stacking events (DSSweb). The BDI Mission control runs each stacking event twice. This first run is used to evaluate the quality of the resulting image allowing the second run to drop low quality images. 4

7 Producing Lightcurve Plots The resulting stacked images are then loaded into MPO Canopus for analysis and to produce the Lightcurve plot. The first step is to blink a sample of the images including the first and last to identify the asteroid. Once identified the process of differential photometry can begin using the powerful Lightcurve Wizard feature of MPO Canopus (Warner, B.D. 2009a). The following are the steps involved in the lightcurve wizard:- 1. Select if you require the star subtraction feature. This feature virtually removes any stars directly in the path of the asteroid as it transits the field. While this feature can be useful, on most occasions the data points collected during the occultation are subtracted from the resulting data due to questionable results. 2. Load the first image in the series and select the asteroid and at least 2 reference stars. These reference stars are used for calculating the position of the asteroid in the field not for differential photometry. 3. The same as step 2 except you select the last image and the same reference stars. 4. If the star subtraction feature is used the software allows you to select the stars you wish to subtract. 5. The wizard then goes through the series of images identifying the reference stars and the asteroid. If it is unable to make a match it will allow the used to manually identify the asteroid. It also gives you the option to override the automatically selected asteroid however this is very rarely required. 6. The software then does the differential photometry automatically selecting reference stars using star catalogues, USNO A2 and UCAC3. Once these steps have been completed the raw plot of the data can be viewed. At this stage obviously faulty data points can be removed. The most likely scenarios in which this is required is as a failure in the star subtraction feature failed or a passing cloud. The typical session is usually no more that 5 hours which is rarely enough to provide sufficient coverage for a useful lightcurve thus multiple sessions need to be captured using the above procedure. Once sufficient data has been collected multiple sessions can be merged into one lightcurve. Each session will need to be adjusted as the reference stars used will differ between sessions. As the asteroids used in this study have published periods the period determination process can be skipped at this stage. However when the period is unknown, the process requires finding the period and the relative differences between the sessions simultaneously. Juggling these two unknowns to find the period can be a challenging exercise. MPO Canopus also supports searching for binary systems which can appear as a secondary sinusoidal effect on the light curve. No signs of a binary system were evident during the analysis of the light curves as part of this project. 5

8 Shape Modeling Early this century Mikko Kaasalainen, Josef Durech (Kaasalainen, J., Torppa, J. 2001a) (Kaasalainen, J., Torppa, J. 2001b) and others developed algorithms and code to determine the shape and spin axis models for asteroids. This code has been further developed by Brian D. Warner (Warner, B.D. 2009b) with a software application called LCInvert. The shape and spin axis modeling process involves many complexities and is far from simply inputting light curve data and outputting the shape model and spin axis. The modeling process produces numerous results with each solution having a value called chi-square (ChiSq) as an indication of accuracy (ChiSqweb). ChiSq is widely used as a theoretical probability distributions in probability theory and statistics. In this scenario it is calculated by the software to estimate the accuracy of a specific solution or goodness of fit. For example if all the ChiSq results produced by the shape modeling process are very similar it indicates that the available data is insufficient to generate a reliable model. Also any model in which the longest axis is the axis of rotation is likely to be a false result. A broad range of observing geometries with sufficient data points are most likely to produce a reliable result. Lightcurves that exhibit non sinusoidal features are usually caused by the shape and it is this type of detail that is important to the process (LCInvertweb). The following are required for a reliable shape modeling solution and spin axis solution: 1. Lightcurves of different geometries are required. 2. Lightcurves covering a range of phase angles including some at phase angles of >10 and ideally >20 if possible. 3. The range of geometries and phase angle covered is of higher priority than the density of the data however excessively sparse data is counter productive. 4. Removing redundant data can aid in finding a reliable solution. 5. The period must be known to a high degree of precision. 6. The ChiSq for best solution from the initial pole search must stand out from the rest of the solutions by at least 10% 7. The weighting factor applied to the "dark facet" must ensure that the dark facet remains below 1%. When the dark facet is large, it suggests there is an albedo variation over the surface. This needs to be compensated for by adjusting the weighting factor. If the dark facet remains large with increased weighting the albedo variation is likely to be near the equator. In this case, increasing the weighting only reduces the credibility of the solution. 8. It is not unusual to have 2 solutions with similar ecliptic latitudes and differ by about 180 in ecliptic longitude. In this scenario more data may help identify which of these 2 solutions is correct. Convex Hulls The modeling code assumes only convex shapes and it creates a shape as if wrapped in wrapping paper. Concavities will form a flat area which will most likely represent a large crater. 6

9 Light Curve and Shape Analysis Procedure The following is the procedure developed to determine the spin axis and modeling solution. It is reiterative in that it evaluates the solution based on: 1. Available data (such as from APC) 2. Available data + existing sparse data (such as from U.S. Naval Observatory (Flagstaff, AZ)). 3. Proven data from the above steps above plus data collected as part of this project. If the quality of the solution is reduced by the addition of data at any of the above stages that additional data is discarded. Stage 1: APC and other Import Stage 2: Period Search Stage 3: Initial Spin Axis Search Stage 4: Minkowski Triangles Search Stage 5: Generate Preliminary Model Stage 6: Import USNO data Stage 7: Initial Spin Axis Search Stage 8: Minkowski Triangles Search Stage 9: Generate Preliminary Model Stage 10: Import Canopus Data Stage 11: Initial Spin Axis Search Stage 12: Minkowski Triangles Search Stage 13: Generate Final Model Import and load all light curves available from the Standard Asteroid Photometric Catalog (APCweb) and other sources. This stage is used to refine the rotational period as the accuracy of the period is critical to a reliable model. ChiSq is used to define accuracy. The dark facet value also needs to be less than 1%. This stage searches for an initial spin axis. A credible solution is defined as a ChiSq that stands out from the other solution by being at least 10% lower. The search iterates through the light curve data initially trying 30 different initial spin axis pairs. This step converts the initial areas file data found in Stage 3 into a closed convex hull using the "Minkowski" method. This method creates convex polyhedron which is solved iteratively so approximately 20 minutes of computing time is required. This stage loads the resulting model data including the spin axis solution as triangular facets or mesh data for rendering. The U.S. Naval Observatory (Flagstaff, AZ) (USNO) data is available from the AstDys website (AstDysweb) and is in the form of so called sparse data. Sparse light curve data contains observations that may span months or years in a single dataset. This data can be used in conjunction with the data captured as part of this project and the APC data to help provide a more complete model. Same as Stage 3 with additional data. Same as Stage 4 with additional data. Same as Stage 5 with additional data. This stage introduces the data captured as part of this project. It is converted and appended to the existing data. This data typically contains more data points than the light curve data available from APC. Same as Stage 3 with additional data. Same as Stage 4 with additional data. Same as Stage 5 with additional data. Ideally this stage will generate the final and reliable solution. 7

10 Asteroid 77 Frigga Lagerkvist and Rickman (1982a) Figure 3: 2 Light curve for Asteroid 77 Asteroid 77 was imaged at approximately Magnitude 12.8 with an approximate phase angle of 20 (Grey 2008). The resolved period was 9.0 ±0.03 hours which matches the published period of hours (Callweb). The error figure produced by MPO Canopus is unreliable. The recommended method for determining the error is by shifting the period until such time as the light curve is no longer folding correctly. This amount of shift becomes the amount of error. The sessions were all sub optimal due whether or proximity to a near full moon. The resulting light curve is presented in Figure 3 which also includes a light curve from Lagerkvist and Rickman (1982) as a comparison. Both light curves are typically bimodal with the data collected as part of this project showing differing peak magnitudes. This difference in peak magnitudes is a strong indication of a non spherical shape. Date Time Coordinates Phase Angle Airmass Imaging data Sky Conditions 17/10/ 21:02-> RA:22h11m49s > 33 sets of approx 60 Seeing 2/ :20 Dec: -11:56: stacked 1 second exp. Transp. 4/10 18/10/ 20:30-> RA:22h11m48s > 44 sets of approx 60 Seeing 2/ :24 Dec: -11:54: stacked 1 second exp. Transp. 5/10 22/10/ 20:19-> RA:22h12m00s > 26 sets of approx 60 Seeing 3/ :00 Dec: -11:49: stacked 1 second exp. Transp. 5/10 25/10/ 20:16-> RA:22h12m26s > 5 sets of approx 60 Seeing 6/ :35 Dec: -11:56: stacked 1 second exp. Transp. 6/10 26/10/ 20:38-> RA:22h11m49s > 35 sets of approx 60 Seeing 3/ :56 Dec: -11:43: stacked 1 second exp. Transp. 3/10 Table 4: Imaging Data for asteroid 77 Frigga Note Approx 10 from near full moon Approx 10 from near full moon Scattered cloud Technical Malfunction Hazy 8

11 Shape Modeling Procedure Stage 1: APC Import Stage 2: Period Search Stage 3: Initial Spin Axis Search Stage 4: Minkowski Triangles Search Stage 5: Generate Preliminary Model The APC data contained 28 light curves with 19 being utilised due to having >=8 data points. Phase angles ranges from 0.27 to st run did not produce a Dark Facet of <1% - Increased weight to 0.5. Through a process of elimination the data from 1980 was discarded as it only reduced the quality of the resulting ChiSq value. 2 nd run produced a period of which is only has a 2% lower ChiSq value. This is a questionable result. A single initial Pole of (60.0, -30.0) was found however it does not stand out by the recommended 10% lower value of ChiSq. Clearly more data required. Completed successfully. While its appearance seems credible the ChiSq value does not stand out from the other solutions. This raises doubt over its accuracy. Hopefully more data will improve the ChiSq results. Stage 6: Import USNO Data Stage 7: Spin Axis Search Stage 8: Minkowski Triangles Search Stage 9: Generate Preliminary Model Completed successfully with 171 data points. A single initial Pole of (270.0, 0.0) was found again however it does not stand out by the recommended 10% lower value of ChiSq. More data required. Completed successfully. The addition of the UNSO data has reduced the credibility of the solution and produced an unlikely shape model. The USNO data will be discarded. Stage 10: Import Canopus Data Stage 11: Initial Spin Axis Search Stage 12: Minkowski Triangles Search Stage 13: Generate Final Model Light curve with 5 sessions loaded successfully. As with stage 3 a Pole of (60.0, -30.0) was found however again it does not stand out by the recommended 10% lower value of ChiSq. These results are typical for an asteroid with insufficient light curve data. The addition of the data obtained as part of this project was insufficient to produce a reliable shape model. The final shape model is presented is Appendix B. Incomplete. Incomplete. 9

12 Asteroid 347 Pariana Figure 4: Four light curves for Asteroid 347. Asteroid 347 was imaged at approximately Magnitude 13.4 (Figure 4, Phased Plot 347x3) with an approximate phase angle of 11 (Grey 2008). The resolved period was ±0.03 hours which matches the published period of hours (Callweb). The sessions were all sub optimal due to being relatively low in the sky and towards the north which is affected by light pollution from the city of Sydney. The low amplitude was also a challenge for the limited equipment used. This light curve has the lowest amplitude change of any light curve yet collected with the equipment at BDI Observatory. This light curve provides a useful reference when considering attempting similar low amplitude targets. 10

13 Also included in Figure 4 for comparison are 2 light curves collected for 347 by BDI Observatory from 2009 (347x1 and 347x2) and 1 collected by Luis Martinez in November Luis Martinez kindly provided this data to aid with the shape modeling effort. His location, Arizona USA, has 347 much higher in the sky and has generally better conditions and has better equipment resulting in a much better quality data set. Both light curves from 2009 indicate a much larger amplitude change then the 2 from 2010 supporting a significantly non spherical shape. Date Time Coordinates Phase Angle Airmass Imaging data Sky Conditions 11/11/ 22:50-> RA:02h18m09s > 35 sets of approx 60 Seeing 4/ :02 Dec: 00:00: stacked 1 second exp. Transp. 2/10 24/11/ 22:21-> RA:02h07m32s > 28 sets of approx 60 Seeing 5/ :01 Dec: 00:04: stacked 1 second exp. Transp. 6/10 25/11/ 21:35-> RA:02h06m50s > 37 sets of approx 60 Seeing 6/ :52 Dec: 00:06: stacked 1 second exp. Transp. 7/10 Table 5: Imaging Data for asteroid 347 Pariana Note Hazy Good Night Good Night Shape Modeling Procedure Stage 1: APC Import Stage 2: Period Search Stage 3: Initial Spin Axis Search Stage 4: Minkowski Triangles Search Stage 5: Generate Preliminary Model The APC data contained 5 Light curves with 5 being utilised due to having >=8 data points. Phase angles ranges from 2.8 to st run did not produce a Dark Facet of <1% - Increased weight to nd run produced a period of , the published value is A credible single initial Pole of 0.0, was found with a stand out lower value of ChiSq. Completed successfully. The hamburger shape indicates that the light curve data does not contain enough detail to reliably produce a model. Hopefully more data will improve the result. Stage 6: Import USNO Data Stage 7: Initial Spin Axis Search Stage 8: Minkowski Triangles Search Completed successfully with 223 data points. A pole of (180.0, 60.0) was found with a stand out lower value of ChiSq however the dark facet was higher than in stage 3. The difference in pole values as compared with Stage 3 introduces significant doubt in the value of the USNO data. Completed successfully. 11

14 Stage 9: Generate Preliminary Model While the addition of the UNSO data has provided more detail to the model the dramatic difference in pole solutions render this solution as most likely incorrect. Stage 10: Import Canopus Data Stage 11: Initial Spin Axis Search Stage 12: Minkowski Triangles Search Stage 13: Generate Final Model 4 light curves with 11 sessions loaded successfully. 2 light curves that I collected from 2009 plus 1 that I collected from 2010 and 1 curtesy of Luis All solutions within 10% of the lowest chisq values were clustered around 230 ±10 and 30 ±30 suggesting a prograde rotation. While ideally only 1 solution should be within 10% of the lowest chisq valuie this clustering is compelling. A Pole Search Plot for 347 Pariana is presented in Appendix C. Done Plausible result. The 2 light curves from 2010 were also processed individually to investigate the difference to the solution these data sets offered. The results in both cases were remarkably similar to the combined solution. This was also done to check that the comparatively noisy data collected as part of this project did not deter from the quality of the combined solution. Early in 2010 the author paper published in the Minor Planet Bulletin (Caspari 2010) that included a preliminary spin axis and shape model for 347 Pariana. The methodology used is identical to what has been discussed above except for the addition of the 2 light curves from The additional data resulted in very similar ChiSq results and similar appearance of the shape model. While the additional data unfortunately did not improve the ChiSq values the fact that the results are so similar to the original results increases the credibility of this solution. A comparison of the preliminary shape model already published with the results of this effort can be seen in Appendix C. 12

15 Conclusion The performance potential and limitations of the equipment at BDI Observatory as it relates to photometry was thoroughly explored and considered in obtaining the light curves covered by this project. What was learnt from the analysis of the optical train performance will benefit future observations. Even with the modest equipment available informative light curves can be produced. The weather was uncharacteristically inclement during semester 2 of 2010 with only 8 observing nights available. By comparison the same period in 2008 offered 17 observing nights. While generating the light curve for 77 Frigga was successful, the attempt to produce a shape model was not. The term insufficient data summarizes the shape modeling effort for 77 Frigga. The sky location, light pollution and low amplitude change in magnitude limited the quality of the data collected for 347 Pariana however a useable light curve was still produced. The most substantial result from this project is the additional data for 347 Pariana that served to support the preliminary shape model (Caspari 2010). One of the reasons that the published shape model for 347 Pariana was categorized as preliminary was due to the sparse nature of the data. The solution is now based on a more substantial data set. While this new data might be worth publishing it does not warrant a paper in its own right. This revised and further substantiated shape model will be submitted as part of a future paper to the Minor Planet Bulletin which will hopefully also contain additional shape models solutions and / or other related research. The methodology developed as part of this project will continue to be utilized by BDI Observatory with the eventual goal of finding further reliable asteroid modeling solutions. References: AAVSOweb:CCD Observing Manual, (Accessed 20 October 2010) ASTROweb: Astrometrica - Shareware for research grade CCD Astrometry, (2010) AVweb: Astrovideo imaging software, (accessed October ) BDIweb: BDI Observatory, (accessed October ) Callweb: Collaborative Asteroid Lightcurve Link, Lightcurve Parameters Page (accessed 2 December 2010 Caspari, P 2010: Minor Planet Bulletin, Edition 37-3, 107 Charonweb: Charon Plate soiling software: (Accessed 22 November 2010) ChiSqweb: Chi-square distribution Wikipedia: (2010) 13

16 DAMITweb: Database of Asteroid Models from Inversion Techniques, (2010) DSSweb: Deep Sky Stacker software, (accessed October ) Gray (2008). Guide software v8. Project Pluto, Bowdoinham,Maine iprepweb: iprep Interactive Image Preprocessor : (Accessed 31 October 2010) Kaasalainen, J., Torppa, J. (2001a) "Optimization Methods for Asteroid Lightcurve Inversion: I. Shape Determination" Icarus 153, Kaasalainen, J., Torppa, J., Muinonen, K., (2001b) "Optimization Methods for Asteroid Lightcurve Inversion:II. The Complete Inverse Problem" Icarus 153, LCInvertweb :LCInvert Installation and Operating Instructions, (2007) Russell H.N. (1906) On the light variations of asteroids and satellites. The Astrophysical Journal. XXIV APCweb: The standard asteroid photometric catalog. (accessed December ) Warner, B.D. (2009a). MPO Canopus v Bdw Publishing, Colorado Springs, CO. Warner, B.D. (2009b). LC Invert. Bdw Publishing, Colorado Springs, CO. 14

17 Appendix A - SAC 8II Camera Specifications CCD Camera - SAC 8 Technical Specifications Pixel Size: 9.6µm x 7.5µm CCD Chip Type: Sony 1/3" EX-VIEW HAD Interline Transfer CCD Pixel Layout: 640 X 480 displayed Exposure Length: 1/1,000 Seconds to infinity Image Formats: JPG, BMP, FITS Required Computer Interface: USB and standard Parallel (needs both) Cooling: Peltier thermo-electric cooled Cooling Power: 3.5 amps Figure 5: Spectral Sensitivity Characteristics for the SAC8II CCD camera 15

18 Appendix B - Shape Model for Asteroid 77 Frigga Figure 6: This shape model is presented for educational purposes only as the results of the modelling process suggest that this is an unreliable solution. 16

19 Appendix C - Shape Model for Asteroid 347 Pariana Figure 7: This is the preliminary shape model was published early in

20 Figure 8: This is the shape model produced as part of this project. 18

21 Figure 9: Pole Search Plot for 347 Pariana. Darker blue represents lower chi-squared results (more probable solutions). Red represents less probable solutions. 19

22 Appendix D - Mission Control Software The Following is a screen shot of the main custom application that controls the telescope and processes the imaging at BDI Observatory. In addition it runs the automated surveys and stacking processes. Figure 10: A screen shot of the BDI Mission Control Software. 20