Measuring the Age of the Beehive Cluster (M44)

Transcription

1 UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 136 PROFESSOR: PROCHASKA Measuring the Age of the Beehive Cluster (M44) Benjamin Stahl (PI) Team: A. Callahan & J. Gillette May 23, 2014 Abstract The open cluster M44 was observed through Blue, Visual, and Red filters using the 40" Nickel telescope at Lick Observatory in mid The CCD detector used to collect the data was determined to have a bias level of counts with a read noise of 16.4 counts. All images were corrected by removing the bias level and then applying flat field corrections. By observing the Landolt standard star SA and conducting a standard analysis the zero point magnitudes of the apparatus specific to the B, V, and R filters were determined to be ± 0.007, ± 0.008, and ± 0.006, respectively. A catalog of stars and their AB magnitudes in each filter and associated errors was constructed using specialized software. From this catalog, HR diagrams in observer space (magnitude in a given filter vs. color index) were plotted. Additionally a single HR diagram in physical quantities of specific luminosity vs. temperature was plotted following a series of calculations conducted on the data from the catalog. Theoretical isochrone tracks corresponding to clusters of various ages were over-plotted on an observer space HR diagram. The best fitting of these isochrone tracks was used to determine the age of M44 to be roughly 550 My or younger.

2 CONTENTS 1 Introduction 5 2 Observations Experimental Apparatus Calibration & Set Up Science Exposures Post Science Exposures & Shutdown Observing Conditions Sources Observed Timeline of Observations Data General Information on.fits Files & Their Attributes Bad Column on CCD & Overscan Region Bias Level & Read Noise Flat Fielding Science Ready Frames Zero Point Magnitude Analysis & Results Catalog of Objects Color Indexes HR Diagrams in Observer Space Discussion Pre-Lab Exercises Random Walk Luminosity of the Sun Pressure Luminosity vs. Mass and Radius MS Lifetimes Age of Cluster Temperature Luminosity HR Diagram in Physical Quantities Closing Remarks LIST OF FIGURES 1.1 A sample Hertzprung Russel (HR) diagram given in absolute visual magnitude as well as luminosity for the vertical axis, the horizontal axis is given in temperature (increasing to the left), color index, and spectral class. A typical HR diagram will utilize only one of each of the presented quantities for each axis. Labeled on the diagram are the typical locations of stars in various stages of evolution [11] The grid of exposures that were taken of M44. The very center of the grid is at the center of M44 (J RA: 08h40m06s, DEC: ) and each square on the grid is the size of the CCD (6.3 x 6.3 arc minutes). The number in the center of each square corresponds to the order in which the telescope was pointed A raw image from the first pointing on M44 taken through the B filter. To the eye there are several questionable columns that stick out, but analyzing the counts shows that only the most prominent column is saturated and the others should be corrected by the flat fielding process HR diagrams in observer space corresponding to specific color indexes. The vertical axes are of AB magnitudes through a given filter and the horizontal axes are of color indexes

3 5.1 The Main Sequence lifetimes plotted as function of mass. Note that the behavior at either extreme of the mass spectrum will vary from this plot because a medium to high mass star was assumed when making the calculations Plots of the same isochrone (750 My) overlayed on the same observer space HR diagram. Figure 5.2a clearly shows a same that agrees wit the shape of the experimental diagram, only shifted away. Figure 5.2b shows the same plot with the isochrone shifted to fit best Plot of a 550 My isochrone overlayed on the observer space diagram. The turn off point on the isochrone is just at the end of the observed stars on the Main Sequence and thus it can be concluded that M44 should be 550 My old or younger HR diagram in physical units. The vertical axis of specific luminosity scaled in terms of solar luminosities per angstrom and the horizontal axis is of temperature in kelvins LIST OF TABLES 2.1 Timeline of observations containing the time, exposure time, filter utilized, target, declination, right ascension, and any necessary comments. The listing has been abbreviated where possible by indicating when multiple identical frames were taken in the comments as well as compressing all entries not corresponding to M44 or the necessary calibration exposures into as few lines as possible Calculated zero-point magnitudes and the associated uncertainties for each filter A sample of the first 20 catalog entries from the SExtractor analysis. Note that the quantities x & y discussed in the list of outputs are not displayed because they were note used in the analysis. The full catalog can be obtained by contacting the authors

4 1 INTRODUCTION All stars share fundamental similarities with one another: they form from collapsing clouds of gas, they fuse hydrogen into helium, they are the central components of solar systems of planets, and so on. Despite these similarities, stars actually vary in many ways: they have different masses, sizes, brightness, temperatures, and lifetimes. It turns out that the mass of a star is essentially the parameter that will dictate almost all of its physical characteristics, as well as how long it will live and how it will die [3]. Thus, knowing the mass of a distant star will tell you quite a bit about it and its evolution. Unfortunately, the mass is hardly an observable quantity for distant stars, instead astronomers can directly measure the number of photons hitting a detector. These photon counts N can be converted to magnitudes m I, giving the relative fluxes of objects in a given exposure according to the following equation: ( ) N m I = 2.5log 10 (1.1) t Where t is the exposure time. If exposures are also taken of a calibration star with known flux, these instrumental magnitudes m I can be converted to absolute AB magnitudes m AB (a magnitude system based on flux in physical units) according to the following equation: m AB = m I + m Z P (1.2) Where m Z P is the zero point magnitude determined from the calibration star. The zero point magnitude is defined as the AB magnitude at which the experimental apparatus would register one count per second for a given source. Following this process, the AB magnitudes of objects in a given frame can be determined. Typically, exposures on the same pointing (where the telescope is pointed on the sky) will be taken with several filters. By subtracting the magnitude in one filter from that in another (often the visual filter magnitude is subtracted from the blue filter magnitude), the color index can be found. The color index is representative of the ratio of the fluxes from the filters and can be used to infer the spectral shape of the source. Plotting the magnitude in one filter against the color index yields one version of what is known as a Hertzprung-Russell (HR) Diagram. Other versions of the diagram exist, which have related but different quantities on each axis. Figure 1.1 presents a sample HR diagram with the various quantities that are typically used. Figure 1.1: A sample Hertzprung Russel (HR) diagram given in absolute visual magnitude as well as luminosity for the vertical axis, the horizontal axis is given in temperature (increasing to the left), color index, and spectral class. A typical HR diagram will utilize only one of each of the presented quantities for each axis. Labeled on the diagram are the typical locations of stars in various stages of evolution [11]. 4

5 As shown in Figure 1.1 there are several distinct, yet related quantities that are typically used on either axis of the HR diagram. The vertical axis is either scaled in terms of magnitude, which can be be found according to the methodology expressed in Equations 1.1 & 1.2 or it it can be scaled in terms of luminosity. Converting between magnitudes and luminosity is relatively straightforward but discussion of this will be deferred to Section 5.4. The horizontal axis is scaled as color index, spectral class, or temperature normally. The spectral class is normally classified under the Morgan-Keenan (MKK) system, which was developed in the late 1800 s and is largely based on the strengths of the Balmer lines of Hydrogen [14]. In this system, stars are categorized as O,B, A,F,G,K, or M and each letter is also assigned a whole number from 0 to 9 to add further resolution. The classes decrease in temperature from O to M as well as from 0 to 9. Determining the temperature of a distant star is not always a trivial task, but can in principle be accomplished if the magnitude in two filters is known. An in depth discussion of the determination of temperature from filter magnitudes and the color index will be deferred to Section 5.3. Also shown in Figure 1.1 are the various stages of stellar evolution. Stars form when giant clouds of gas collapse under gravity to the point that hydrogen begins to undergo fusion into helium and an equilibrium (hydrostatic and thermal) is reached [12]. These newly born stars begin their lives on the Main Sequence of the HR diagram. Eventually, these stars will burn through their supply of hydrogen and evolve off of the Main Sequence, though the amount of time that this takes to occur as well as the outcome are sensitive to the mass. Without the pressure from the nuclear fusion, gravity causes the star to begin collapsing again. For low mass stars (under roughly half a solar mass) this collapse will occur after a great amount of time and it will not generate enough heat to fuse helium, and thus these stars will eventually collapse into white dwarfs [4]. Mid sized stars (ranging in mass from roughly 0.5 to 10 solar masses) will eventually evolve off of the Main Sequence and become red giants burning their shell hydrogen. They will burn helium before quietly dying and becoming a planetary nebula and then a white dwarf. More massive stars will burn through their hydrogen relatively quickly and evolve off of the Main Sequence as super giant stars. Most massive stars will end their lives in a colossal explosion known as a supernova. If the mass of the core of a star is below the Chandrasekhar limit (around 1.4 solar masses) the stellar remnant will be white dwarf, if it is below a certain other limit (which is a topic of current research, but perhaps around 2 or 3 solar masses) but still above the Chandrasekhar limit, the stellar remnant will be a neutron star, and if the mass is above it will become a black hole. From this rudimentary discussion of stellar evolution, there are a few factors that should be emphasized as they can be used to infer the age of star clusters. Knowledge of how stars evolve according to their mass and the time scales on which they do this is very valuable, along with how this behavior would be represented on an HR diagram. As discussed above, the higher the mass of the star, the faster in general that it will burn through its supply of hydrogen and evolve off of the Main Sequence. Thus, the point at which the Main Sequence curves towards the giants describes what is known as the turn off mass. The turn off mass is essentially the stellar mass that corresponds to a star having burned all of its hydrogen and moved off of the Main Sequence, which is related to how old that star is. Assuming that all of the stars in a given cluster are the same age, this can tell the age of the cluster. Using this idea, theoretical HR diagrams can be generated for populations of stars with a given age, which can then be compared to HR diagrams made from real observations. By adjusting the age of the stellar population in the theoretical model to match the observed HR diagram, the age of a given cluster can be inferred. The ultimate goal in this experiment was to construct an HR diagram of a star cluster and from that diagram make an estimate of the age of the cluster. There are two main types of star clusters: globular and open. The distinctions between these two types of clusters are numerous, however for the purposes of selecting a cluster to observe, the main consideration was that globular clusters tend to be more densely filled with stars as compared to open clusters. Open clusters also tend to be younger than globular clusters. For the purposes of constructing an HR diagram, more data points (ie stars in the cluster) were desired suggesting that a globular cluster be observed. Unfortunately, most globular star clusters are in the galactic arm, which made them inaccessible during the window of observation. Along with these considerations of making sure that there were enough stars to observe in the cluster and that the cluster was in thy sky at the right time, another consideration was how bright the cluster was. The cluster clearly must be bright enough to be observed with the experimental apparatus using reasonable exposure times. Weighing in on all of these considerations as well as the numerous clusters that they ruled out, it was decided that the open cluster NGC 2632, also known as M44 and named The Beehive Cluster would be observed. Though not a globular cluster with high stellar density, M44 is massive, with a dimension of approximately 95 on a side. By 5

6 taking exposures of multiple regions of this cluster, it was possible to observe a sufficient number of stars with which the scientific goals of this experiment could be accomplished. Additionally, M44 was easily visible in the sky during the observing window and it is quite bright (apparent magnitude of 3.7) and thus was easily observed using the 40" Nickel telescope at Lick Observatory. The observational goal of this experiment was to collect images of the cluster M44 through several distinct filters. From these images, a detailed scientific analysis was performed to determine the specific luminosities and temperatures of the stars in each frame. HR diagrams of the stars in the M44 cluster were constructed from the magnitudes and color indexes as well as from the specific luminosities and temperatures, thus allowing for the age of the cluster to be estimated. A detailed reporting of the measurements, calibrations, and analysis utilized to accomplish the scientific goals will be provided in the following sections of this report. 2 OBSERVATIONS 2.1 EXPERIMENTAL APPARATUS Experimental data was taken using the following primary instruments and devices: Remote Control Location The experimental apparatus was operated via remote connection from Room 170 of the Natural Sciences II Building at the University of California, Santa Cruz. Telescope Nickel 40-inch reflecting telescope located at Lick Observatory, Mt. Hamilton, California. Detector Nickel Direct Imaging Camera: CCD-C2 (Dewar #2) [8] Field of view: approximately 6.3 x 6.3 arc minutes. Linear response range: up to around 50k counts. Binning: 2x2 Fast Overscan region: 32 columns Filters Blue (B), Visual (V), and Red (R) filters were used with the CCD. 2.2 CALIBRATION & SET UP The program of configuration and calibration consisted of multiple phases. Start up was commenced at approximately 18:45 PDT on April 29, 2014 from the Remote Observations room at UCSC. The procedures provided in the Remote Operations manual were followed but will briefly be outlined now [9]. The workstation was connected to the observatory through the remote observing software. The telescope and CCD were enabled and then several test exposures were taken. Upon these test exposures being successfully completed, 9 bias exposures were collected. Then 8 (3 at 10s exposure and 5 at 30s exposure) dome flat field frames were taken with the B filter in place. Dark frames were not collected because of the extremely low (to the point of being negligible) dark current of the CCD, likely due to the fact that the CCD was kept at around -116 C. Next the telescope was prepared for observing by opening the dome, turning on the fans, lowering the windscreen, enabling telescope motion, opening the mirror cover, enabling the dome s auto feature, and turning on telescope tracking. With the telescope now in an operational and move-able state, flat field frames were collected of the twilight sky with each of the filters with exposure times ranging between 1s to 20s. Next the telescope pointing was calibrated and then the telescope was focused. At this point, the apparatus was considered to be ready for scientific data collection. 6

7 2.3 SCIENCE EXPOSURES After configuring the telescope and taking the necessary exposures to make corrections for instrumental bias, scientific data collection was commenced. The telescope was pointed to the desired cluster, and exposures were taken in a coordinated grid on the sky that will be detailed in Section 2.6. For each position on this grid, one exposure was taken with each filter (B, V, & R). Initially it was planned that each exposure would be 3s regardless of the filter, which was a decision made by considering the cluster s apparent magnitude of 3.7 and the need not to underexpose nor exceed the linear range the CCD [5]. This prescription was modified once a series of exposures were taken indicating that the CCD was registering an insignificant number of counts on sources compared to the background with a 3s exposure when using the R filter. It was thus decided that exposures taken with the R filter in place would be 10s, while the other two filters would remain at 3s exposures. With the exposure times decided and the coordinates layed in to point, the desired science frames were collected. Given the short exposure times, guiding was not utilized for the observations. In total 48 science frames were collected, from 16 different pointings with 3 filters per pointing. 2.4 POST SCIENCE EXPOSURES & SHUTDOWN The telescope time was shared between a total of three research teams. Upon the completing the collection of the data used for this experiment, another team took over and collected data on M67. After this team completed their collection run, a series of frames were taken (2 per filter) of the standard star SA to allow for the zero point magnitude of the apparatus with each filter to be determined and thus the AB magnitude (in each filter) of each star in the science frames to be extracted from the data. Upon completing the collection of these calibration frames, a series of fun frames were taken of characteristic night sky objects known for their aesthetic appeal. These frames were not collected with the hope of doing science, but merely for fun at the discretion of the investigators as the final team prepared for their data run. After collecting these images, the final team commenced their data collection run of the cluster M3. They also collected frames for another cluster with the designation of NGC5466. Note that only the data collected on M44 will be considered in this paper. Upon completing all data collection, the telescope was shut down following the prescription provided in the manual that was previously cited. Essentially the set up process was performed in reverse order and with the opposite action being taken for each step. The data was then made available and the observing session was considered complete. 2.5 OBSERVING CONDITIONS Observations were conducted using the Nickel telescope at Lick Observatory, Mt. Hamilton, California on the evening of Tuesday, April 29, 2014 controlled remotely from the University of California, Santa Cruz. The observers arrived to begin set up and calibration protocols at 18:45 PDT and science image collection of M44 commenced at UT 03:58:09.15 (April 30 on UT time). During the period of observations, the conditions were clear with very slight cirrus clouds. Temperatures ranged in Fahrenheit from the 50 s to low 60 s with calm to gentle (under 10 mph) winds coming from WNW [10, 13]. The moon was waning crescent, with the new moon coming the following day. Thus, it is not expected that moonlight contaminated the observations. 2.6 SOURCES OBSERVED Given the 6.3 x 6.3 arc minute field of view of the CCD and M44 s angular size of 95 arc minutes as well as the fact that it is an open star cluster (relatively low density of stars), it was determined that one pointing of the telescope 7

8 simply would not be enough to conduct a broad investigation into the age of the cluster [5]. Thus it was decided to take exposures in a 4 x 4 grid, allowing for a region of 25.2 x 25.2 arc minutes (4 times both dimensions of the detector) to be investigated. This grid was centered at the central point of the cluster (J RA: 08h40m06s, DEC: ). This grid is presented in Figure 2.1: Figure 2.1: The grid of exposures that were taken of M44. The very center of the grid is at the center of M44 (J RA: 08h40m06s, DEC: ) and each square on the grid is the size of the CCD (6.3 x 6.3 arc minutes). The number in the center of each square corresponds to the order in which the telescope was pointed. The offset feature of the telescope interface was utilized to point the telescope with respect to the center of the cluster. To center on the first pointing, the telescope was offset by 1.5 times the width of a square on the grid (567 ) to the West and also to the North starting from the very center. Each subsequent pointing was achieved by offsetting by the width of a square on the grid (378 ) in the appropriate direction as is apparent in Figure 2.1. As noted previously, two other research teams collected data of other clusters and fun images were taken with excess time. These objects will be be briefly presented in Table 2.1 to account for the use of the telescope time but will not discussed further in this report. 2.7 TIMELINE OF OBSERVATIONS A timeline of the observational tasks conducted for this experiment is presented in Table 2.1. All observations were conducted on UT 2014 April 10. Table 2.1: Timeline of observations containing the time, exposure time, filter utilized, target, declination, right ascension, and any necessary comments. The listing has been abbreviated where possible by indicating when multiple identical frames were taken in the comments as well as compressing all entries not corresponding to M44 or the necessary calibration exposures into as few lines as possible. UT t exp (s) Filter Target DEC RA Comments 02:23: B Bias frames 39:56: :50: exposures 02:32: B Dome flat B 39:56: :59: exposures 02:39: B Dome flat B -05:02: :06: exposures 02:53: B Twilight Flat 29:56: :19: :54: R Twilight Flat 29:56: :19:25.8 Continued on next page... 8

9 Table continued from previous page UT t exp (s) Filter Target DEC RA Comments 02:56: B Twilight Flat 29:56: :19: :57: B Twilight Flat 29:56: :19: exposures 03:03: B Twilight Flat 29:56: :19: exposures 03:10: B Twilight Flat 29:56: :19: exposures 03:13: V Twilight Flat 29:56: :19: :14: V Twilight Flat 29:56: :19: exposures 03:16: V Twilight Flat 30:02: :18: :17: R Twilight Flat 30:02: :18: :18: R Twilight Flat 30:04: :18: exposures 03:20: R Twilight Flat 30:10: :18: exposures 03:23: R Twilight Flat 30:14: :17: exposures 03:26: R Pointing 29:21: :16: exposures 03:44: R Focus 27:54: :29: :46: R Focus 27:56: :29: :58: B M :53: :41:43.6 Data collection commences 03:58: V M44 1 V 19:53: :41: :00: R M44 1 R 19:53: :41: :01: B M44 1 B 19:53: :41: :02: B M44 2 B 19:53: :41: :03: V M44 2 V 19:53: :41: :04: R M44 2 R 19:53: :41: :05: R M44 3 R 19:53: :40: :05: V M44 3 R 19:53: :40: :06: B M44 3 B 19:53: :40: :07: B M44 4 B 19:53: :40: :08: V M44 4 V 19:53: :40: :09: R M44 4 R 19:53: :40: :10: R M44 5 B 19:47: :40:23.5 Exposure error 1 04:11: V M44 5 V 19:47: :40: :12: R M44 5 R 19:47: :40: :14: B M44 5 B 19:47: :40:50.3 Log error 2 04:15: V M44 5 V 19:47: :40: :16: R M44 5 R 19:47: :40: :18: B M44 5 V 19:47: :41: :20: V M44 7 V 19:47: :41: :21: R M44 7 R 19:47: :41: :22: B M44 8 B 19:47: :41: :23: V M44 8 V 19:47: :41: :24: R M44 8 R 19:47: :41: :26: B M44 9 B 19:41: :41: :26: V M44 9 V 19:41: :41: :27: R M44 9 R 19:41: :41: :29: B M44 10 B 19:41: :41: :30: V M44 10 V 19:41: :41: :31: R M44 10 R 19:41: :41: :33: B M44 11 B 19:41: :40: :34: V M44 11 V 19:41: :40:50.7 Continued on next page... 1 The equipment operator did not change to the V filter for this exposure. As a consequence, data from the 5th pointing of the telescope was not used in the analysis. 2 The equipment operator did not change the target names here for several exposures, but the coordinates are correct. It is completely a book keeping mistake and does not affect the results. 9

10 Table continued from previous page UT t exp (s) Filter Target DEC RA Comments 04:35: R M44 11 R 19:41: :40: :37: B M44 12 B 19:41: :40: :38: V M44 12 V 19:41: :40: :38: R M44 12 R 19:41: :40: :40: B M44 13 B 19:35: :40: :41: V M44 13 V 19:35: :40: :42: R M44 13 R 19:35: :40: :43: B M44 14 B 19:35: :40: :44: V M44 14 V 19:35: :40: :45: R M44 14 R 19:35: :40: :48: B M44 15 B 19:35: :41: :49: V M44 15 V 19:35: :41: :50: R M44 15 R 19:35: :41: :51: B M44 16 B 19:35: :41: :52: V M44 16 V 19:35: :41: :53: R M44 16 R 19:35: :41: :58: B M67 C0 B 11:52: :52:02.2 Other team takes data 05:51: B SA 101 B -00:25: :57:10.7 Calibration image collection commences 05:53: B SA 104 B -00:29: :43: :54: B SA 104 B -00:29: :43: :55: V SA 104 V -00:29: :43: :56: V SA 104 V -00:29: :43: :57: R SA 104 R -00:29: :43: :57: R SA 104 R -00:29: :43: :01: B M58 B 11:51: :38:16.1 Fun image collection collected 06:36: B M3 1 B 28:34: :33:39.2 Other team takes data 07:37: V NGC5466 Center V 28:35: :05:53.8 Other team takes data 3 DATA The images collected with the experimental apparatus were delivered in the.fits file format. These raw images were processed into a science ready format by performing a host of processes using a Python script developed by the investigators. The methodologies employed in making these corrections to the data will be outlined in the following sections. 3.1 GENERAL INFORMATION ON.FITS FILES & THEIR ATTRIBUTES The.fits file type contains two distinct attributes: the header and the data. The header of a.fits file contains useful information about the image such as the time it was taken, the exposure time, the filter being used, where the telescope was pointed, and so on. The data of a.fits file contains a two dimensional array where each element corresponds to the counts registered by a pixel in the CCD. The Python script was written to pull useful information about a given frame from the header and import the data into a two dimensional array allowing for calculations and corrections to be performed. 3.2 BAD COLUMN ON CCD & OVERSCAN REGION A brief view of the collected data showed that column 257 on the CCD was a bad column in the sense that each pixel along this column was saturated. Thus, it was desired that the column be removed from the data to avoid 10

11 skewing or biasing the results. It was removed by cutting each frame directly on either side of this bad column, and then putting the two pieces back together without the bad column. This process was systematically performed on all the collected frames. Figure 3.1 presents a sample image from showing the bad column. Figure 3.1: A raw image from the first pointing on M44 taken through the B filter. To the eye there are several questionable columns that stick out, but analyzing the counts shows that only the most prominent column is saturated and the others should be corrected by the flat fielding process. The CCD also contained an overscan region on the last 32 columns to assess the bias level of the detector an an image to image basis. However, the investigators opted to determine the bias level in a broader and more constant fashion that will be discussed in Section 3.3. Because the overscan region wasn t utilized for determining the bias level, it was removed systematically from each from frame. This was accomplished by cutting the last 32 columns out from each frame that was collected. 3.3 BIAS LEVEL & READ NOISE The bias level of the CCD is responsible for raising the zero point of the detector to some nominal value, typically around 1000 counts. The CCD will not register negative counts, and thus it is beneficial to implement a bias level to prevent the data from being biased to one side. Though implementing a bias level is critical in preventing the data from being skewed, it must be accounted for and subtracted for the data to be useful for scientific analysis. The fluctuation about the bias level, which is caused by the electronics is defined as the read noise. The bias level can be determined and accounted for by taking bias frames with the experimental apparatus. These frames are taken as 0s exposures with the shutter closed, thus the only counts that should be registered will be due to the bias level. This isn t always the case however, because cosmic rays can be striking the detector simultaneously. To prevent outliers such as cosmic rays from contaminating the determination of the bias level, multiple bias frames are taken and then the median value of counts for each pixel is taken across all of the frames. The resulting frame of median bias level counts for each pixel gives a good representation of the bias level of the detector. The mean value can be taken of this frame to represent the bias level with one number. The read noise of the detector can be found by taking the standard deviation of the counts from each pixel in the median bias frame. Note that the bias level and read noise are functions of the CCD and in no way are affected by the filter being utilized. This process was implemented in a Python script to determine the bias level of the CCD from the 9 bias frames that were collected. From this analysis, the bias level was determined to be counts and the read noise was determined to be 16.4 counts. 3.4 FLAT FIELDING Most if not all raw data collected with a telescope and associated equipment (ie CCD, etc) will have unwanted signatures present. While in principle it may be conceivable that these individual signatures could be modeled, 11

12 predicted, and thereby removed, it is much easier and economical to measure them and then divide them out. This is typically accomplished by flat fielding. There are various methods for obtaining flat field methods, each with their own benefits and drawbacks. Every method will share the similarity however, that the exposure should be as uniformly illuminated as possible. Note that flat field exposures will be affected by which filter is in place and thus flat fields must be taken for each filter that is to be used when collecting science images. Two methods were utilized during the night of observation to obtain flat field images. The first method utilized was to take dome flats, where the light is illuminated on a screen and an exposure is taken with the dome closed. While this is an acceptable method for taking flat field exposures, the investigators ultimately decided not to use the dome flats for correcting the science images. Instead, flat field frames taken of the twilight sky were utilized because they generally yield better results. Each flat field frame that was collected then had the bias level removed by subtracting the median bias frame. Next, the bias corrected flat field frames were each normalized by dividing each frame by the median of its counts. Then the bias corrected and normalized flat field frames were stacked according to which filter they were taken with. Then the median value of counts for each pixel in these stacks was taken, resulting a single, median flat field frame for each filter. Lastly, these median frames for each filter were normalized by dividing by the median of the counts. Thus, a single, normalized flat field frame was obtained for each filter that was used, allowing for unwanted signatures of the apparatus to be divided out of raw images. 3.5 SCIENCE READY FRAMES Having determined the bias level of the detector as well as found normalized flat field images for each filter, the raw data was converted into images that were ready for scientific analysis. This conversion was accomplished using the following algorithm and repeating over all of the science images that were collected: RI (f ) B I SI (f ) = F F (f ) (3.1) Where the science ready image SI, the raw image RI, and the normalized flat field image F F are all functions of the filter type f, and where the median bias image is denoted by B I. 3.6 ZERO POINT MAGNITUDE As discussed in the Introduction, the instrumental magnitudes (found using Equation 1.1) of the objects that were observed were desired to be converted into AB magnitudes, which are based on physical units. This conversion is detailed in Equation 1.2, which requires the zero-point magnitude m Z P as an input. The zero-point magnitude of the apparatus, specific to each filter, was found using a methodology that will now be outlined. First, the apparatus was pointed to a calibration star and exposures were taken with each filter that was used to collect science images. These exposures were reduced according the the methods outlined in the sections preceding this one. Next, a software package known as DS9 was used to perform photometry on each calibration exposure in order to determine the number of counts caused by just the calibration star. Specifically, the the total counts of a region (using a 7 radius circular aperture 3 ) closed around the star, as well as the total counts of a similar region closed around a patch of sky with no stars were found using this software package. The counts of the region containing no stars were subtracted from the region containing the star to determine the counts caused by the star only. The counts were then used in conjunction with the exposure time of the frame from which they were determined in order to calculate the instrumental magnitude of the calibration star according to Equation This is equivalent to a pixel radius aperture as was used in DS9. This was determined using the plate scale of the CCD (0.184 arc seconds per pixel) [8]. 12

13 The calibration star selected was SA , which is a Landolt standard star with published AB magnitudes in each filter that was utilized in this experiment [2]. The zero point magnitude was then calculated using the instrumental magnitudes m I and the published AB magnitudes in each filter m AB according to the following equation: m Z P = m AB m I (3.2) Two exposures were taken in each filter of the calibration star. Thus the zero-point magnitude used for the rest of this experiment for each filter was taken as the average of the two individual zero-point magnitudes (from the 2 exposures for each filter). The uncertainty in the zero-point magnitude for each filter was found by taking the standard deviation of the two individual zero-point magnitudes. The results of this analysis are presented in Table 3.1: Filter m Z P σ Z P B V R Table 3.1: Calculated zero-point magnitudes and the associated uncertainties for each filter Thus, the zero-point magnitude and its associated uncertainty were determined for each filter, and were then used to convert science objects to AB magnitudes in a later step. 4 ANALYSIS & RESULTS After all of the images were corrected according to the prescription outlined in Section 3 and the zero-point magnitudes were determined, scientific analysis was conducted on the data. The details of this analysis will now be presented in the subsequent sections. 4.1 CATALOG OF OBJECTS In order to generate an HR diagram for the cluster, measurements of individual objects within each frame were required. A software package known as SExtractor (v ) was utilized to catalog and organize this data. The following quantities and parameters were used as inputs in this software: Counts per second frames 4 MAG_ ZEROPOINT: zero-point magnitude for each filter GAIN = 1.0 DETECT_ THRESH = 3.0: the number of standard deviations above the background an object must have to be detected ANALYSIS_ THRESH = 3.0: the number of standard deviations above the background an object must have in order for the specified analysis to be conducted PIXEL_ SCALE = : pixel scale in arc seconds [8] From these inputs SExtractor was utilized to get a number of outputs which will be summarized here: index: used to catalog the stars. Each star was assigned a unique index number with which to identify it. The convention that the first number (or two) in the index was in reference to which pointing it was from according to Figure 2.1 was adopted. 4 Counts per second frames were found by dividing each frame by its exposure time. 13

14 x & y: the pixel coordinates of the center of the object (filter)_ mag: the AB magnitude of the object in B,V, or R filter as specified in (filter) (filter)_ rms: the standard deviation of the magnitude of the object in a given filter following the same convention as above. After obtaining the raw outputs, further refinement was required to obtain the final catalog. Only objects that show up in all filters for a given pointing were included in the final catalog. Additionally any stars on the edge of the detector, or that touched the bad column were identified and excluded from the final catalog. After making these revisions the final catalog of stars was obtained, allowing for further analysis and investigation to be conducted into M44. A sample of the first 20 outputs resulting from the analysis conducted with SExtractor is presented in Table 4.1. Index V σ V R σ R B σ B Table 4.1: A sample of the first 20 catalog entries from the SExtractor analysis. Note that the quantities x & y discussed in the list of outputs are not displayed because they were note used in the analysis. The full catalog can be obtained by contacting the authors. 4.2 COLOR INDEXES The color index is simply the difference of two magnitudes through different filters. This difference is analogous to the ratio of the the fluxes from the filters, which also describes the energy output at the two peak frequencies of the filters relative to one another. This information can be used to characterize the shape of the source, which is approximately a blackbody in the case of stars. Supposing the magnitudes (m 1 & m 2 ) in two filters (1 & 2) are known, the color index C I is found according to Equation 4.1 as follows: C I = m 2 m 1 (4.1) Where the convention in this case is that filter 1 is a redder band than filter 2. Additionally, if the uncertainties (σ 1 & σ 2 ) in each of these filter magnitudes are known, then following the rules of differential calculus the uncertainty σ C I in the color index is given in Equation 4.2: σ C I = σ σ2 2 (4.2) 14

15 Using Equations 4.1 & 4.2 the V - R, B - R, and B - V color indexes as well as their associated uncertainties were found for each star in the catalog. 4.3 HR DIAGRAMS IN OBSERVER SPACE Having constructed a catalog of the magnitudes and associated uncertainties of stars in M44 for each filter used, and having also found color indexes and associated uncertainties for each of these stars as detailed in Section 4.2, HR diagrams in observer space were plotted from this data. A total of three plots were generated based on the color indexes, and they are presented in Figure 4.1: 9 Magnitude vs Color Index 9 Magnitude vs Color Index Red Magnitude (AB) Red Magnitude (AB) V - R (a) Red AB Magnitude vs V - R color index B - R (b) Red AB Magnitude vs B - R color index. 9 Magnitude vs Color Index Visual Magnitude (AB) B - V (c) Visual AB Magnitude vs B - V color index. This is also the plot that was selected for further analysis into estimating the age of the cluster in Section 5.2. Figure 4.1: HR diagrams in observer space corresponding to specific color indexes. The vertical axes are of AB magnitudes through a given filter and the horizontal axes are of color indexes. Each of the plots shown in Figure 4.1 clearly shows the Main Sequence in the diagonal trend of stars decreasing down and to the right. An additional similarity is in the wide error bars that develop at dimmer magnitudes. This is 15

16 most likely because of the way in which SExtractor makes magnitude determinations, which relies on objects being above some threshold (which was set at 3σ for this experiment) to be detected and analyzed. Dimmer objects are closer to this threshold, and thus the software will assign larger and larger relative errors to the magnitudes. 5 DISCUSSION 5.1 PRE-LAB EXERCISES A series of calculations were conducted to better understand and gain insight into the behavior of the Main Sequence. These calculations will now be outlined and discussed: RANDOM WALK In the sun, the mean density ρ is approximately 1.4 g/cm 3 and the average temperature T is roughly K. Due to these conditions, photons in the sun will only travel l 0.5 cm before scattering off of a particle. Assuming that photons originate in the center of the Sun and scatter with random directions the following assertions and calculations can be made: (a) Assuming only 1 dimension, there is an equal probability of the photon moving one way a distance of l as compared to the opposite direction. Iterating this over many scatterings, it is obvious that these opposing directions will cancel each other, giving the expected result that the average distance of a photon from the center after N scatterings is 0. Expressing this mathematically where d is the distance from center and s i is the distance traveled in the i th step (always ±l, thus the average of s i over many iterations is always 0) yields: d = s 1 + s s N (5.1) Taking the average of this expression over many iterations yields the expected result: < d >=< s 1 + s s N >=< s 1 > + < s 2 > + + < s N >= = 0 (5.2) (b) Still assuming only 1 dimension, the average of the square of distance traveled by a photon from the center in N scatterings is expressed mathematically as follows: < d 2 >=< (s 1 + s s N ) 2 >=< s1 2 + s s2 N > +2(cross terms) (5.3) Noting that all cross terms in the above equation will become 0 when averaged over many iterations, the equation is simplified and rewritten as follows: < d 2 >=< s 2 1 > + < s2 2 > + + < s2 N > (5.4) The probability of each term in the above equation is l 2 because (±l) 2 = l 2, thus: Next, the RMS distance can be found as follows: < d 2 >= l 2 + l l 2 = Nl 2 (5.5) d RMS = < d 2 > < d > = Nl 2 0 d RMS = l N (5.6) Moving to 3 dimensions, the radial RMS distance becomes: N d RMS = l 3 (5.7) 16

17 (c) Setting the RMS distance d RMS found above for the radial distance from the center of the Sun to the solar radius R allows for the number of scatterings to be determined as follows: N R = l 3 N = 3 ( R l ) 2 = scatterings (5.8) Next, knowing that light travels at the speed of light c and that it must travel a total distance of Nl to escape the Sun, the total time t for a photon to escape from the Sun can be found as follows: c = Nl t t = Nl c = 3R2 lc = s = years (5.9) LUMINOSITY OF THE SUN Using the random walk approximation developed in Section 5.1.1, the luminosity of the Sun will be determined: (a) It is assumed that the energy density ɛ, that is the energy per unit volume is given by: Where a is the radiation density constant ( erg cm 3 K 4 ). ɛ = at 4 (5.10) The total radiative energy E of the Sun can be found by multiplying the energy density by the volume of the Sun as shown in the following equation where the average temperature T is used as an assumption from the random walk approximation: E = ɛv = 4 3 πr3 at 4 (5.11) The luminosity of the Sun L can be estimated by then dividing the energy found above by the photon escape time t found in Section as follows: L = E t = 4πR3 at 4 lc 9R 2 = 4 9 πaclr T 4 (5.12) (b) Now, the expression found for the luminosity of the Sun above will be evaluated: L = 4 9 πaclr at 4 = 4 9 πac(0.5cm)r ( K ) 4 33 erg = s (5.13) Where the known physical constants have not been explicitly written in numerical form, but were used in the calculation. The result for the solar luminosity compares very well with the accepted scientific value of erg/s, yielding a percent difference of just 14.8% PRESSURE A star of mass M and radius R in hydrostatic equilibrium is assumed for the following calculations: (a) The scaling relation for the density ρ and the pressure P in terms of the mass and radius of the star will be found. Hydrostatic equilibrium is the situation in a star when the outward pressure balances with the gravitational pull and an equilibrium is reached. This can be expressed mathematically as follows: dp dr = Gm(r )ρ(r ) r 2 (5.14) 17

18 Next, the following assumptions are made to simplify the problem Differentiating both sides of this result yields: ρ(r ) = ρ = m(r ) 4 3 πr πρr 3 = m(r ) = m (5.15) dm dr = 4πρr 2 dr dm = 1 4πρr 2 (5.16) Multiplying both sides of the hydrostatic equilibrium equation given in Equation 5.14 by the above result and simplifying gives the Lagrangian form of the equation which is presented as follows: dp dr dr dm = Gmρ(r ) dr r 2 dm dp dm = Gm 1 r 2 4πρr 2 = Gm 4πr 4 (5.17) Now separating variables and integrating yields 5 P(r =R) M Gm dp = P(r ) 0 4πr 4 dm P(r = R) P(r ) = Gm2 M 8πr 4 P(r ) = GM2 8πr 4 P(r ) = GM2 8πr 4 Applying this result to the case of the entire star yields the desired scaling relation for the pressure: 0 (5.18) As was alluded to previously the density scaling relationship is given by: P(R) = GM2 8πR 4 P M 2 R 4 (5.19) ρ M R 3 (5.20) (b) The total pressure in a star is a combination of gas pressure P g, degeneracy pressure P NR, and radiation pressure P r. For low to high mass stars, gas pressure dominates. According to the ideal gas law, pressure depends directly on the product of density and temperature. Thus: P ρt (5.21) For very-high mass stars, radiation pressure dominates. Thus by the relation for radiation pressure [3]: 5 It assumed the pressure at the surface of the Sun is 0, that is P(r = R) = 0. P r ad = 4σ 3c at 4 P T 4 (5.22) 18