The light curve of the exoplanet transit of GJ 436 b

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1 The light curve of the exoplanet transit of GJ 436 b Dana Simard Phys 315 Term Project Lab Partner: Natasha Urbancic April 5th, 2013 Abstract The transit of the exoplanet GJ 436 b was observed using a Deep Field telescope on the itelescope remote observing network. Using the Exoplanet Transit Database curve fitting software, a parametric model was fit to the observed transit, returning some parameters of the transit. These parameters, along with assumptions concerning the mass-radius relation of the host star GJ 436 and the period of orbit of GJ 436 b, physical characteristics of the stellar system were calculated, including the mass of the host star, found to be 0.42 ± 0.10 solar masses, the radius of the host star, 0.47±0.10 solar radii, and the radius of the planet, 4.8± 1.1 Earth radii. These agree well with previous work (eg. Bean et al. 2008, Gillon et al. 2007, Shporer et al. 2008), however many of the physical parameters have very large percentage uncertainties. To reduce these uncertainties and better constrain the physical parameters, observation of multiple consecutive transits and parametric fitting of the physical parameters (for example through a Monte-Carlo method) is suggested. I. Introduction Detection of extrasolar planets has become one of the fastest-growing areas of astronomy research. An understanding of the number and physical properties of extrasolar planets is important not only for understanding the formation of stellar systems, including the Solar System, but also for determining how common earth-like planets are in the Milky Way and the probability of finding other planets in our galaxy that may support life. Some planets are bright enough and far enough away from their host star that they can be imaged directly; however the low luminosity of most planets in comparison to the host star makes direct detection very difficult and the majority of extrasolar planets have been detected through indirect methods. One such method, Doppler spectroscopy, takes advantage of the fact that the observed radial velocity of the planet changes as it orbits its host star due to Doppler shifting. A second indirect method of detecting extrasolar planets, known as the pulsar timing method, searches for small anomalies in the timing of a pulsar s radio pulses. A third common method for indirect detection of extrasolar planets, a photometric method known as the transit method, was employed in this study. When a planet passes in front of its host star, it partially occludes the star, and the observed brightness of the host star decreases. If the host star is observed over a period of time during which the planet passes in front of the star, a dip in the light curve of the star is therefore ob- 1

2 served. Many physical parameters of the system can be determined from the transit light curve. For example, the depth of the transit dip depends on the relative sizes of the star and the planet (Seager & Mallén- Ornelas 2003). If consecutive transits are observed, the period of the planet s orbit can be determined. From the period, and with some assumptions of the stellar radius-mass relation, many further parameters of the system can be characterized, including the impact parameter, stellar density, stellar mass, stellar radius, system inclination, planetary radius, and the semi-major axis of the planet s orbit. For details on this type of analysis, see Section III. In this analysis, the transit of the extrasolar planet Gliese 436 b (GJ 436 b) across the host star Gliese 436 (GJ 436) was observed remotely on the night of January 31st, 2013, from the AstroCamp Observatory in Nerpio, Spain using itelescope s Deep Field Telescope 7. According to the Exoplanet Transit Database and the NASA Exoplanet Archive, GJ 436 is a red dwarf star located approximately 10.1 parsecs from the Solar System in the constellation Leo. GJ 436 is classified as an M2.5V star, placing it on the low-mass end of the main sequence (M2 main sequence stars are generally found to have masses near 0.4 M Sun ). GJ 436 b is a Neptune-sized exoplanet discovered via the radial velocity method by Butler and Marcy in 2004, and is one of the smallest exoplanets observed before the discoveries made using the Kepler Space Telescope. GJ 436 b orbits at a distance about fifteen times closer than Mercury s distance from the Sun as stated by the NASA Planetary Data System. This system was chosen mainly because the host star, with a magnitude of 10.68, is easily visible using the available 0.43 m deep field telescope and the depth of the transit, expected to be magnitudes according to data from the Exoplanet Transit Database, is large enough to be detected using this telescope. In addition, GJ 436 is a well-studied system, and therefore parameters required in the analysis are well-defined. Refer to Appendix A for tabulated information on the GJ 436 system. Further details on the observations made are presented in Section II. The data was reduced through a combination of itelescope s automatic reduction process and Exoplanet Transit Database s curve fitting routine, as explained in Section III. The results of this fit were used to determine the physical properties of the GJ 436 system, which are derived and presented in Section III. These results are discussed and compared with previous studies of GJ 436 b in Section IV. Finally, important conclusions are drawn in Section V. II. Observations The system GJ 436 was observed remotely between 22:53 January 31st, 2013, and 01:18 February 1st, 2013 (UTC) using the itelescope Telescope 7 at the AstroCamp Observatory in Nerpio, Spain. Telescope 7 is a Deep Field 0.43m telescope capable of long exposures for science over a wide variety of bands, including the photometric V band. Thus, it was ideal for imaging a bright extrasolar system, such as GJ 436. See Appendix A for more information on this telescope. Nerpio, Spain, was chosen as the observation location as it experienced very clear weather in the weeks during which data acquisition was planned and executed. Sixty-four sixty-second exposures were taken with a broadband V filter using Telescope 7 over 142 minutes which included approximately sixty minutes prior 2

3 to the transit, sixty minutes during the transit, and twenty minutes after the transit had occurred. While observing, the sky remained clear and the seeing varied between and mag/arcsec 2. The telescope was repositioned every sixty minutes and refocused every thirty minutes. While imaging, the moon was at maximum brightness and at a high altitude of degrees. The brightness of the moon may have contributed to trends observed in the data, which were corrected for during the data analysis as detailed in Section III. See Appendix A for more detailed information on the observing conditions. III. Results and Analysis The images were obtained along with their Julian dates from the itelescope network in the form of FITS files after the removal of dark, flat and bias. IRIS, software for processing astronomical images, was used to further reduce the data. The images were aligned through stellar registration of a field of approximately thirty recognized objects. The use of a large field allowed for a more accurate correction of not only translational offsets but also rotational offsets between images. After the images were aligned, three reference stars were chosen in proximity to GJ 436 through two main criteria: First, reference stars similar in brightness to GJ 436 b were chosen, as these stars were bright enough to be easily located in the images but did not saturate the CCD. These stars also required aperture photometry rings of the same radii as the target star. Secondly, stars in very close proximity to the target were chosen so that changes in the background flux of the images, especially that of the moon, had similar effect on both the reference stars and the target star. It is required that the reference stars for this type of analysis are non-variable stars. This was assumed throughout the analysis and no evidence to suggest otherwise (such as the flux of the target changing significantly relative to one reference star but not the others) was found. See Figure 1 for a starchart of the target star and the three reference stars chosen. Using IRIS, automatic aperture photometry was performed on the data. For automatic photometry, one must choose the radii of three rings. The first ring contains the area within which the counts are integrated to determine the flux of the star, while the area between the second and third rings is used to determine the background sky level. The radii of these rings were chosen to ensure that all stars were enclosed within the smallest ring and that no background stars were enclosed in the region between the second and third rings. Since the magnitudes of the reference stars were unknown, this analysis did not allow the calculation of the magnitude of the target star. However, the counts of the reference star were converted to magnitudes by assuming that the reference stars had magnitudes of 11. While the magnitudes obtained in this manner are incorrect, they allowed for the calculation of changes in magnitude of the target star. The conversion to magnitudes was performed using Equation 1: m = m re f log 10 ( C C re f ) (1) where m and m re f are the magnitudes of the target and reference star, respectively, and C and C re f are the counts of the target and reference star, respectively. This conversion was done for all three reference stars, resulting in three different magnitudes for the target star for each image. These magnitudes were then averaged to 3

4 Figure 1: A starchart showing the location of the three reference stars (Ref1, Ref2, Ref3) relative to the target star, GJ436, (V), taken from one of the images used in this analysis. find the average magnitude of the target star compared to the reference stars. All four light curves (one in reference to each reference star in addition to the average) were compared to ensure that no anomalous behaviour was evident in any of the light curves. Uncertainties were estimated by examining the average magnitude of the target star after the transit. The magnitudes of the target star tends to vary by up to approximately between images. Therefore, the uncertainty in the magnitudes was estimated as 0.003, half of this value. This takes into account random uncertainties in the data, but does not take into account systematic uncertainties, as only a small subset of the data near the end of the observation was used to determine the uncertainties. As well, it should be noted that this uncertainty applies only to the change in magnitude during the transit; the true magnitude of the target star is still unknown. Using the Exoplanet Transit Database curve-fitting software, a parametrized model was fit to the observed transit light curve. The model is based on that proposed by Pejcha (2008), reproduced here: m(t i ) = A 2.5 log 10 F(z[t i, t 0, t T, b]p, c 1 ) (2) +B(t i t mean )+C(t i t mean ) 2 where F(z, p, c 1 ) is the decrease of the flux of the target star due to the planet transit, A, B, and C describe systematic (constant, linear or quadratic, respectively) trends in the data, p is the ratio of the radii of the planet and the star, z is the projected relative separation of the planet from the star, c 1 is the coefficient of the linear law modelling limb darkening of the star, t T is the duration of the transit, t 0 is the time of closest approach, b is the impact parameter (corrected for inclination), and t mean is the mean time of observations. This model makes some approximations. First, the planet is assumed to be much smaller than the star (p < 0.2). This small-planet approximation is expected to be valid for GJ 436 b based on the previous literature, (eg. Bean et al. 2008, Gillon et al. 2007, Shporer et al. 2008, Southworth et al. 2008). Secondly, the planet s trajectory is modelled as a straight line across the stellar 4

5 Figure 2: The transit light curve of GJ 436 b (points) and the parametrized fit (line) before the removal of the linear trend and offset Magnitude JD disk, which is valid even for highly inclined systems as long as the transit duration is small in comparison to the period of the planet s orbit. This model was fit using a Levenberg-Marquardt non-linear least squares fit; for details refer to Pejcha The initial conditions for this fit were taken from previous studies of GJ 436 b in the Exoplanet Transit Database. From this curve fitting, the fit shown in Figure 2 was obtained. As can be seen from this plot, strong evidence for an exoplanet transit was observed. In addition, a strong linear trend, possibly due to the bright moon present during the observations, was fit. Since all of the reference stars were on the same side of the target star, averaging of the stellar magnitudes in reference to the reference stars would not eliminate a systematic trend due to the brightness of the moon. Fitting a quadratic trend did not significantly improve the fit (compared to fitting a linear trend); therefore the quadratic coefficient C was set to zero. Figure 3 shows the light curve and fitted model after the removal of the linear trend. As can be seen from the plot of residuals shown in Figure 4, the model agreed very well with the original data within experimental uncertainties. The fit returned some parameters of the system, including the JD and HJD time mid-transit and the duration, t T, and depth, m, of the transit, which are displayed in Table 1. In addition, t F, the duration of the flat part of the transit (in between ingress and egress) can be calculated from the fit. Since data during ingress is scarce, this was done by determining the time between the centre of the transit and the beginning of egress by eye; the result is given in Table 1. The uncertainty of the time of the beginning of egress was estimated as the time difference between the point chosen as the beginning of egress and the neighbouring data point. The uncertainty in the duration of the flat portion of the transit 5

6 Figure 3: The transit light curve (points) and the fit (solid line) after the removal of the linear trend (including offset) Magnitude JD Figure 4: Residuals from the fit of the transit light curve of GJ 436 b shown in Figure Magnitude JD

7 Table 1: The best-fit parameters for the model shown in Equation 2 with C = 0 when fit to the exoplanet transit light curve of GJ436 b. These are compared to the predicted values given by the Exoplanet Transit Database. See Equation 2 for a definition of the variables. Fit Parameter Experimental Value ETD Prediction JD mid-transit ± ± HJD mid-transit ± ± t T (min) 68.2 ± ± 1. m (mag) ± ± t F (min) 36 ± 5 Table 2: Physical parameters of the star GJ 436 and its orbiting planet GJ 436 b determined from the transit light curve. Values stated without uncertainties have uncertainties of 1 times the smallest figure in the value. This Work Alonso et Bean et Gillon et Shporer Southworth al al al et al et al ρ (g/cm 3 ) 6.±2. M (M sun ) 0.42± ±0.9 R (R sun ) 0.47± ± ±0.017 b 0.82± ±0.01 R P (R earth ) 4.8± ± ±0.16 a (AU) 0.028± i (deg) 86.3± ±0.18 was then found from the uncertainty in the mid-transit time and the beginning of egress time using standard error propagation methods (see Appendix B). From the parameters of the fit shown in Table 1, many physical characteristics of the GJ 436 system can be calculated, as described by Seager & Mallén-Ornelas (2003). For this analysis, F, the ratio of fluxes was first calculated using Equation 1 of Seager & Mallén-Ornelas (2003): F = F notransit F transit F notransit = m/5 (3) From F, t T and t F, the impact parameter b can be calculated according to Equation 6 of Seager & Mallén-Ornelas (2003), reproduced here: ( ) (1 F) ( 2 tf 2(1+ tt F) 2) 1/2 b = ) 2 1 ( tf tt (4) In addition the density of the host star, ρ, can be found from a rearrangement of Equations 1, 2, 3 and 4 of Seager & Mallén- Ornelas (2003): ρ = 24P F3/4 Gπ 2 1 (t 2 T t2 F )3/2 (5) where P is the period of the orbit of the planet about the star. As this analysis did not take images over one full period of the planet s orbit, it did not allow a direct determination of the period; the period was adopted to be ± days, as 7

8 found by Alonso et al. (2008). The mass of the host star can then be determined, if the stellar mass-radius relation is known or can be assumed. In this analysis, the empirical relation found by Demircan & Kahraman (1989) was adopted: R = km α (6) Where k = 1.054±0.001 and α = 0.935± 0.001, a relation valid for main-sequence stars with masses less than 1.66 M sun. The mass of the star, M was found in units of M sun using Equation 8 of Seager & Mallén- Ornelas (2003): ( M = k 3 ρ ) 1 1 3α (7) ρ sun Where ρ sun = 1.41 ± 0.01g/cm 3 (Irwin 2007). The radius of the star, R, in units of R sun was then found from the mass-radius relation. The semi-major axis of the orbit of GJ 436 b was found via Equation 10 of Seager & Mallén-Ornelas (2003): ( P 2 ) 1/3 GM a = 4π 2 (8) The inclination of the orbit was then found via Equation 11 of Seager & Mallén- Ornelas (2003): ( ) br i = arccos (9) a Finally, the radius of the planet GJ 436 b was found from the relation between the depth of the transit and the ratio of the radii of the planet and the star, Equation 1 of Seager & Mallén-Ornelas (2003): ( Rp ) 2 F = (10) R In all of these calculations, the uncertainties were estimated by error propagation; see Appendix B for further information. The physical parameters derived from these calculations are displayed in Table 2. IV. Discussion As Figure 3 shows, a light curve was obtained showing significant evidence of an exoplanet transit across the star GJ 436. Parameters found directly from this light curve and from further analysis are displayed in Tables 1 and 2. The mid-transit HJD of the transit observed was slightly earlier than the prediction obtained from the exoplanet transit database. However, all other parameters of the fit agreed with prediction, including the duration and the depth. The NASA Exoplanet Archive predicts a midtransit JD of ± , which corresponds to an HJD of ± This prediction does not agree with that made by the Exoplanet Transit Database, and the observed value falls in between these two predictions. This suggests that the HJD predicted by the Exoplanet Transit Database and the NASA Exoplanet Archive are less certain than stated. These predictions are obtained from a selection of previous light curves of the GJ 436 b transit, which have a wide variety of quality and depth; therefore it is possible that the method of extracting predictions from these light curves yielded underestimated errors. However, without knowledge on how these predictions were determined (which is not readily available), little more can be said on the accuracy and precision of the predictions. The host star was found to have a density of 6. ± 2. g/cm 3, a mass of 0.42±0.10 solar masses and a radius of 0.47 ± 0.10 solar radii, in agreement with many previous works, including Alonso et al. 2008, Gillon et al. 2007, Shporer et al. 2008, and Southworth The determined mass of the star agrees with the expectation that 8

9 GJ 436 is a low-mass star. Physical parameters of the planet GJ 436 b also agreed well with previous literature. The planet was found to have a radius of 4.8 ± 1.1 Earth radii at a distance of ± AU from its host star. Therefore, the planet GJ 436 b is a "hot Neptune". The system is highly inclined, as expected for a system detectable via the transit method. While some parameters, including the impact parameter, semi-major axis and inclination, were well constrained, the stellar and planetary radii and the stellar mass found through this method had large associated uncertainties. These uncertainties may be overestimated, as the errors in the individual data points of the transit light curve were estimated prior to removing the trend from the data. Therefore, the trend itself may be included in this estimation of random uncertainties. A better estimation of uncertainties could be determined by taking multiple image sequences of consecutive transits. After removal of any systematic trends in each dataset, the data from all transits could be combined. With more data points, the effects of random error on the data could be more effectively determined through statistical analysis. As well, as can be seen from Figure 3, recalibration of the telescope caused some sections of the transit to be missed, while the time required for the telescope to prepare to take each image resulted in data points that were often spaced apart by 1.5 to 2.0 minutes. Taking data from multiple transits would also eliminate this problem. Analyzing multiple transits would allow for an independent calculation of the period of the planetary orbit, decreasing the reliance on previous work in determining the parameters. It is expected that all of these benefits of imaging multiple transits would yield better constraints on the physical parameters of the GJ 436 system. It should also be noted that the physical parameters were found using a series of equations after the initial fit to the transit light curve was performed. Using numerical methods, such as Monte-Carlo methods, to determine these parameters while performing the fit may also have resulted in betterconstrained parameters. V. Conclusions In this analysis, the exoplanet transit of GJ 436 b, a member of the GJ 436 system in the constellation Leo, was observed from the remote itelescope network. Using software provided by the Exoplanet Transit Database, a parametric model was fit to the observed light curve, yielding some parameters of the transit. From these parameters, as well as some parameters from previous studies, the physical characteristics of the GJ 436 stellar system were calculated. The calculated parameters agreed well with previous studies. The host star was found to have a mass of 0.42 ± 0.10 solar masses and a radius of 0.47±0.10 solar radii, which agrees well with the classification of GJ 436 as a low-mass star. The exoplanet GJ 436 b was found to have a radius of 4.8 ± 1.1 Earth radii and a semi-major axis of ± AU, making it a "hot Neptune". This work allowed the characterisation of the exoplanetary system GJ 436. However, the physical parameters found through this analysis were not well constrained. To build upon this work, imaging multiple consecutive transits of GJ 436 b is recommended. Compiling multiple light curves would allow the reduction of random noise in the obtained light curve and would ensure that all portions of the light curve were imaged. In addition, using numerical (eg. Monte-Carlo) methods 9

10 as opposed to calculations to determine the physical parameters may also have resulted in better and more constrained parameters. Therefore, in further work, it may be beneficial to examine this method of characterizing the physical parameters of GJ 436 b. References [1] Alonso R., Barbieri M., Rabus M., Deeg H. J., Belmont J. A., & Almenara J. M., 2008, A&A, 487, L5 [2] Bean J. L., Benedict G. F., Charbonneau D., Homeier D., Taylor D. C., MacArthur B., Seifahrt A., Dreizler S., & Reinsers A., 2008, A&A, 486, 1039 [3] Coughlin J. L., Stringfellow G. S., Becker A. C., López-Morales M., Mezzalira F., & Krajci, T., 2008, ApJ, 689, L149 [4] Demircan O., & Kahraman G., 1989, Commun. Fac. Sci. Univ. Ank. Serie B., 38, 9 [5] Exoplanet Transit Database, predictions.php, Accessed January 30, 2013 [6] Gillon M., Demory B.-O., Barman T., Bonfils X., Mazeh T., Pont F., Udry S., Mayor M., & Queloz D., 2007, A&A, 471, L51 [7] Irwin J., 2007, Astrophysics: Decoding the Cosmos, West Sussex, England: John Wiley & Sons [8] NASA Exoplanet Archive, 2012, GJ 436 b, cgi-bin/exotables/nph-exotbls, Accessed February 5, 2013 [9] NASA Planetary Data System, 2013, GJ 436, viewtargetprofiles.jsp? TARGET_NAME=GJ+436, Accessed February 5, 2013 [10] Pejcha O., 2008, FitProcedureDescription- Pejcha2008.pdf, Accessed April 1, 2013 [11] Poddany S., Brat L., & Pejcha O., 2010, New Astronomy, 15, 297 [12] Seager S., & Mallén-Ornelas G., 2003, ApJ, 585, 1038 [13] Shporer A., Mazeh T., Pont F., Winn J. N., Holman M. J., Latham D. W., Esquerdo G. A., 2008, ApJ, 694, 1559 [14] Southworth J., 2008, MNRAS, 386, 1644 [15] Southworth J., 2009, MNRAS, 394, 272 [16] Southworth J., 2010, MNRAS, 408, 1689 [17] Southworth J., 2011, MNRAS, 417, 2166 VI. Appendix A In this appendix, tables of parameters of the observed system, the telescope used for observation and the observing conditions are presented for reference. Previously tabulated data on the extrasolar system GJ 436 is presented in Table 3. Table 4 presents the parameters of the location, telescope and CCD used in the observations of GJ 436. Table 10

11 5 presents information on the conditions during observation. Table 3: Tabulated data on the extrasolar planet GJ 436 b and its host star GJ 436 from the NASA Exoplanet Archive and the NASA Planetary Data System. Planet ID GJ 436 b Host name GJ 436 Discovery method Radial velocity Orbital period (days) ± Semi-major axis of orbit (AU) ± Eccentricity of orbit ± Planet mass (Jupiter masses) ± Planet radius (Jupiter radii) ± Planet density (g/cm 3 ) ± Inclination (deg) ± 0.17 Stellar V magnitude Transit depth Distance to system (pc) 10.2 ± 0.2 T e f f (K) 3350 ± 300 Stellar mass (solar masses) ± Stellar radius (solar radii) ± VII. Appendix B For the reference of the marker, the Mathematica code used in this analysis is attached as the last pages of this report. 11

12 Table 4: Parameters of the location, telescope and CCD used in the observation of the exoplanet transit of GJ 436 b. Location Parameters Location Nerpio, Spain Latitude North 38 deg 09 arcmin Longitude West 002 deg 19 arcmin Elevation (m) 1650 Minimum target elevation (deg) 35 Telescope Parameters Telescope itelescope Telescope 7 (Deep Field) Focal length (nm) 2939 Aperture (mm) 431 F/Ratio f/6.8 Guiding External Mount Paramount PME CCD Parameters CCD SBIG STL-11000M QE 50% Peak Full well electrons, Anti Blooming Gate Dark current 0 deg C 0.5 Pixel size (µm) 9 Resolution (arcsecs/pixel) 0.63 Sensor Frontlit Cooling (deg C) -20. Array 4008 by 2672 Field of view (arcmin 2 ) 28.2 x

13 Table 5: Observing conditions and parameters. Expected transit start time and date (UTC) 23:53 31/01/13 Expected transit end time and date (UTC) 0:55 01/02/13 Expected transit duration (min) 68 Observation start time and date (UTC) 22:55:53 31/01/13 Observation end time and date (UTC) 01:17:36 01/02/13 Imaging start time and date (UTC) 23:01:58 31/01/13 Imaging end time and date (UTC) 01:15:30 01/02/13 Number of exposures 64 Filter V Average imaging rate (hour 1 ) 28 Exposure time (s per image) 60 CCD binning 1 Seeing (mag/arcsec 2 ) to Cloud cover Clear Moon altitude (deg) Moon brightness (%) Azimuthal coordinates of target 61 deg North RA of target 11h42m10.01s Dec of target +26d42 37 RA of centre of field 11h42m10s Dec of centre of field +26d42 37 Telescope repositioning Every 60 min Telescope refocusing Every 30 min 13

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