The Study of Starspots

Stellar Surface Imaging Via Matrix Light Curve Inversion

Elizabeth A. Cademartori, Ohio Wesleyan University
Advisor:  Dr. Robert O. Harmon, Ohio Wesleyan University
Collaborator:  Dr. Donald M. Terndrup, The Ohio State University

Sunspots are familiar solar phenomena, and starspots are an analogous occurrence seen on certain types of stars. Starspots themselves cannot be directly imaged due to the great distances involved; stars (except Betelgeuse and our own) are always seen as mere pinpoints, even by the Hubble Space Telescope. As a result, different imaging methods must be used. Matrix light curve inversion (MLI) is a technique that examines variations in the light output of stars. These variations are due to the rotation of dark spots into and out of view.  MLI allows us to determine the number, size, shape, and distribution of the spots on any given stellar surface. The aim of this project was to reconstruct the surface features of eight stars from two adjacent regions in the alpha Persei star cluster by using MLI.  The IRAF (Image Reduction and Analysis Facility) software suite was used to analyze several hundred CCD frames which were taken over eight nights in a nine day period at the Michigan-Dartmouth-MIT Observatory in September 2001. The target stars were chosen because they are young, fast rotators making them likely candidates for having imageable spots. Surface images have been successfully constructed for AP063 and AP117, two of the eight original stars, by inverting the B and V photometric light curves simultaneously.  Inverted singly, the spot on each of the stars is significantly different in the B, V, and R photometric filters.  This may be due to the fact that the parameters (angle of inclination, spot temperature, etc.) needed for MLI are slightly off, or it may be suggesting that the underlying structure of starspots are different than sunspots.  This project started as a summer research opportunity at Ohio Wesleyan University, and was expanded into a senior capstone project.

Road Map for This Site:

This web page is part of my senior research project in astrophysics at
Ohio Wesleyan University in Delaware, OH.  The information presented
here is the culmination of a year and a half's worth of research.  I hope
you enjoy this site, and find it informative and easy to navigate.  
Thanks for taking the time to check it out!

So What is a Starspot Anyway?
What is Matrix Light Curve Inversions (MLI)?
The Procedure:  Star Fields to Stellar Surfaces
Results:  Surface Images and Light Curves
Conclusions:  Wrapping Things Up
Looking Ahead:  Ideas for the Future
Acknowledgments


So What is a Starspot Anyway?

Sunspots

    The easiest way to approach spots on stars is to look at the analogous case on our own local star: sunspots.  The movie shown here is of a sunspot.  Spots are located in the photosphere, the atmospheric layer which is the star's visible “surface.”  The umbra is the most noticeable feature on a sunspot, and is the dark central region.  The lighter, filamentary region surrounding the umbra is called the penumbra.  The movement seen in the movie is actual movement of plasma on the surface of the sun.

Sunspot movie     Here are some fun factoids about sunspots:

Astronomy Picture of the Day
(Credit:  Peter Sütterlin, DOT Team, SIU)
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The Sun's Magnetic Personality

    Starspots, like sunspots, are regions of intense magnetic activity (2,000-3,000G) as evidenced on the sun Zeeman splitting across a sunspot by the significant Zeeman splitting of spectral lines.   Spectral lines in the presence of a magnetic field are split, because the energies at which atoms transition are affected.  How much the wavelengths differ is indicative of the strength of the magnetic field.  The picture to the left is an example of this effect.  The left hand frame is of a sunspot.  The thin black line running through the center of the frame is the slit of a spectrometer.  The right hand frame shows two prominent spectral lines, one of which has been split into three parts.
National Optical Astronomy Observatory
(Credit:  NOAO/AURA/NSF)

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So Why are Sunspots Dark?

      The Sun uses two different methods of transferring energy from its core to the surface.  Picture showing plasma temperature under a starspot One mode of heat transfer is via huge convective rolls.  Plasma--hot, ionized gas--is heated in deeper layers, rises to the surface, cools, becomes denser, sinks to lower levels, and so on.  The granular pattern outside the sunspot in the movie (Sunspot in motion) are convective rolls in motion.  In strongly magnetic regions this movement of material is suppressed by magnetic field lines, thus trapping cool plasma on the surface.  The figure to the right is an artist's rendering of this.  The arrows show the direction of plasma movement, and the colors indicated temperature with red equated to hot regions and blue to cooler ones.
    Cooler means darker!  Brightness, or flux, is given by F=sigma*T4.  This is the Boltzman equation, and F is the flux, sigma is the Stephan-Boltzman constant, and T is temperature.  The strong dependence on temperature is why the spot looks dim in comparison to the surrounding surface; even a small change in temperature can have a significant affect on the brightness.  Don't be fooled, though:  if isolated, a large spot as seen from Earth would be a brilliant orange pinpoint whose total brightness would be comparable to that of the full
moon!  Imagine reading by the light of a sunspot!
Astronomy Picture of the Day
(Credit:  MIDI, SOHO Consortium, ESA, NASA)
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Spots On Other Stars

     The next great question, is how do scientists know that spots occur on other stars?  First of all, other stars are magnetically active, just like the sun.  Because of this, it is logical to assume that they will also display magnetic phenomena like spotting.  However, because of the great distances involved, the surfaces of stars can not be visually resolved.  Unlike spots on the Sun, we can not attach a camera to a telescope and expect to see a starspot.  Matrix Light Curve Inversion (MLI) solves the problem through an indirect imaging process.  MLI uses light curves which are a plots of an object's brightness in a given wavelength over a set period of time.  MLI uses variations in these curves to reconstruct the surface of the star in question.  A star's brightness can vary for many reasons, e.g. eclipses in a binary star system, but there are stars for which spots on the surface provide the best explanation.  Because starspots are cooler, and therefore darker than the rest of the surface, they decrease the overall flux of the star as they rotate into and out of view causing dips in the light curves.  This periodic variation in the star's brightness would not occur if it was an eclipsing binary.  Also, a sophisticated imaging technique known as Doppler imaging, which uses variations in the spectral lines caused by variations in the surface brightness, has convincingly demonstrated that spots do exist on other stars.
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What is Matrix Light Curve Inversion?

    Now that a little is known about the nature of starspots, it is time to tackle MLI itself.  The first step in this process is to imagine that the surface of the star has been partitioned so that way it is easier to keep track of what is happening over the Example of a partitioned star entire surface of the star.  If the surface of a star is known, i.e. if the brightness of each patch is known, then we can do the forward case.  Summing over all the visible patches gives the total brightness of the star at a particular moment in time along the viewer's line of sight.  By, allowing the star to rotate and then repeating the process, it is possible to construct the star's light curve.

    This is not what we want to do though.  In fact, we want to do exactly the opposite case:  light curves can be easily obtained through observation, and we would like to know the spot configuration.


            (Cite:  Harmon & Crews, AJ, 2000)

    The simplest case would be to find the set of patch brightnesses that creates a curve that exactly fits the measured light curve.  Problem!  If you pepper the surface of a star with small spots, approximately the same number would rise as set in a given time period, so their net effect on the light output of the star would be almost negligible.  The light curve of such a star would display a high frequency ripple.  A similar effect is produced by the inherent noise in the data.  Thus, this inversion approach yields a star surface peppered with small, fictitious spots which are produced in an effort to fit the noise in the measurements.  Obviously this technique is not going to work.

    Solution:  constrained minimization.  Want to pick out the smoothest solution possible that still differs from the real light curve by an amount equal to the noise in the data.  Choosing the smoothest result within the margins of error, avoids noise artifacts in the final surface.  A computer program is used to do this.  Imagine an n-dimensional space in which each axis is the intensity of a patch.  Every point in the positive region of this space corresponds to a possible surface appearance, a possible solution.  The computer's job is to crawl around in this space and find the point which best matches the true light curve given the error restraint.
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Star Fields to Stellar Surfaces:  The Start to Finish Story of My Project Step by Step

Gathering the Stars

    Eight stars in the alpha Persei star cluster were chosen as targets for this study.  The stars are young, spectral class G and K (similar to the Sun), fast rotators, and therefore magnetically active and good potential starspot sites.  Images were taken using a CCD camera attached to the 1.3-meter McGraw-Hill telescope at the Michigan-Dartmouth-MIT Observatory in Arizona.  Observations were made through the B, V, R, and I photometric filters over eight nights in a nine day period.  Two fields within the alpha Persei cluster were focused on, AP014 and AP072.  The star field centered on AP014 (Approximate Coordinates: 3:24:19.8, +48:47:20, 2000) is given below on the left.

Starfield centered on AP014 CCD frame of AP014 through the B filter

    The next step was to clean the images.  IRAF (Image Reduction and Analysis Facility, http://iraf.noao.edu) was used to clean the original CCD frames, and then perform measurements.  Some of the types of cleaning performed are overscan correction, zero-correction exposure, and flat fielding.  A flat is an image that has been uniformly exposed, i.e. a picture of the inside of the dome.  The purpose of flat fielding is to correct for differences in sensitivity from one part of the detector to another. This correction was unable to be applied since the research assistant who took these images, inadvertently left the shutter open while the CCD was reading out. This allowed extra photons to fall on the detector, and is why each star has a streak of light emanating from behind it (see image on right, above).  The flats appear to be the only step significantly damaged by this problem.  

    Next, a series of measurements was made of the stars in the CCD frames.  In order to tie down the magnitude of the variable, target stars, it is necessary to have a bank of comparison stars.  A set of comparison stars was selected based upon their falling within a desirable range of magnitudes.  The green circles on the figure above indicate selected stars.  Coordinate lists of these stars were created for each frame.  More accurate measurements were then made of these stars, and the associated errors were found at the same time.
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Light Curves and Inversion

    Dr. Donald Terndrup of The Ohio State University then took this information, reduced it, and returned a list of Julian dates, magnitudes, and associated errors for each star through each light filter.  The next step is to create light curves.  Before that can happen, some minor adjustments in the way the data was organized had to occur.  The Interactive Data Language (IDL, http://www.rsinc.com/idl/index.asp) is a useful application and language that enables analysis and visualization.  Using IDL, I created a program to convert the Julian dates into phases, convert magnitudes into intensities and normalize these values, find the root mean square error in the magnitudes, and lastly, create plots of the intensity versus the phase.

    Now that we have obtained light curves, we can proceed with inversion using MLI.  MLI is a Fortran program created by Dr. Harmon, and inspired by the work of Walter Wild.  Despite its name Matrix Light Curve Inversion, the program actually needs certain pieces of information, besides the light curves themselves, in order to work: estimated root mean square error for each filter, angle of inclination of the star's axis, number of patches into which the star is to be partitioned, type of limb darkening and the associated coefficients for each photometric filter, and the ratio of spot to photosphere intensities.  All of these parameters, except the error estimate, can be determined from prior observational results.  To find the error in the magnitude, we start with an initial estimate based on the rms error found from the associated errors.  The next task is to write a program which does a series of inversions stepping the value for the magnitude noise from one to three times this initial estimate in increments of 0.1.  The object is to get the magnitude error as low as possible without having the spots fall apart.  It is fairly easy to tell when this happens, as can be seen below, since the spots spread significantly as the error estimate drops.  This is because we are starting to fit the noise, and are consequently seeing the artifacts of this in the images.

AP063_B_rms.0180132 AP063_B_rms_0154399 AP063_B_rms_0141532

    These images illustrate the breakdown of a spot as the magnitude error is systematically decreased.  The star is AP063 through the B filter using magnitude error values of, beginning on the left, 0.0180132, 0.0154399, and 0.0141532.

    Once good single inversions have been made in each of the photometric filters for all of the stars, then it is beneficial to attempt simultaneous inversion of multiple light curves.  This is helpful in tying down the exact latitude of the spot(s).  Finally, I wrote and ran another program which varies the other parameters (angle of inclination, spot temperature, etc.) and carries out inversions is valuable in determining the accuracy of the spot placement.
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Results:  Reaping the Reward

    Due to time constraints, magnitude information on four of the eight original target stars was never returned to me.  Of the four remaining stars, two of them were unfit for inversion.

Light curve of AP072 through the B photometric filter
Both AP072 and AP075 display clustering of the data points, leaving large gaps in the light curves.  This clustering is a selection effect.  The periods of these stars are on the order of a couple days, where as the other stars are on the hour of a few hours.  Thus, our sampling only caught the stars at certain points in its phase.  There is simply too much information missing, and inversions are not possible for these two stars.  The light curve pictured above is of AP072 through the B photometric filter and is typical of both stars through all of the filters.
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Surface Images for AP063

B filter                                          V filter                                       R filter
rms = 0.0180132                          rms = 0.0179424                        rms = 0.0159989
Surface image of AP063 through the B filter Surface image of AP063 through the V filter Surface image of AP063 through the R filter

    Notice that the spot is significantly different in appearance based on the filter it is viewed through.  There are several reasons why the surface appearances are so different based on the photometric filter used.  The first option, is that certain parameters (ratio of spot to photosphere temp., angle of inclination, etc.) are not close enough to their actual values.  This is possible since these values are not well known and are determined by spectroscopy, and are fairly recent discoveries.  Another possibility is that according to the blackbody spectrum, one should expect the contrast between the spot and photosphere to decrease at longer wavelengths.  However, this cannot account for the radical decrease in contrast, nor the difference in spot size.  Finally, the underlying physics of spots in stars may very well be different than for those on the Sun.  These stars are rotating much more rapidly than the Sun does, so it is reasonable to assume that there will be differences between the two.

BV Surface of AP063     Simultaneous inversions of the B & V light curves worked well together.  Simultaneous inversions of multiple light curves, results in a more accurate spot placement.  Three curve inversions never converged.  This is probably due to the large difference in appearance between the various filters making it difficult to combine them.  Notice that there is only one spot on this star (the other side shows not surface features).  This is significant, as stars of this type often display two spots approximately 180° apart.  The spot is elongated slightly in longitude (spans roughly 30° in latitude And 70° in longitude).  Slight variations in inclination, spot temperature and rms yield little change in appearance.  This increases our confidence that the actual spot is reasonably close to this location on the star, and that my parameter choices are most likely close to their actual values.
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                                                                                                                Simultaneous Inversion
AP063 through the B & V

Surface Images for AP117

B filter                                          V filter                                       R filter
rms = 0.0198054                          rms = 0.0191629                        rms = 0.0175478

Surface image of AP117 through the B filter Surface image of AP117 through the V filter Surface image of AP117 through the R filter

    Similarly to AP063 there is again a large difference in surface appearance based on what filter is used.
Single inversions for the B, V, and R filters were successful.  The inversions through the I photometric filter never converged, so no surface images are available.  This is true for both stars.  The spot through the R filter is significantly lighter.  I assume that the spot through the I filter would have been even lighter, so that the contrast was too small to be able to differentiate a spot in the inversion process.  The spot through the B filter has a noticeable spread longitudinally.  There could actually be an elongated spot located at this position, or this longitudinal spread could be indicative of a more circular spot located at an higher latitude.  Inversions do not work as well at higher spot latitudes, so spots located in these regions tend to drift lower and assume a kidney-bean-like shape.  While this is not quite what we see, it is a possibility that ought to be kept in mind.  Also, there are some lighter patches creeping toward the center of the spot in the B passband.  Normally, this would indicate a magnitude error a tad too low.  However, the spot never tightens up, even at higher values of the rms.  Because of this I think some of the other parameters may be a little off.
BV Surface of AP117
    Like AP063, simultaneous inversions through the B & V filters worked well for AP117.  Similarly, three curve inversions never converged.  There is only one spot on this star.  The spot is elongated slightly in longitude (spans roughly 24° in lat. And 70° in long.).  The spot is centered at about +42° lat.  This is higher than spots are ever found on the Sun, and is consistent with the results of Doppler imaging.  It is also significant that the starspots imaged in this project are significantly larger than those found on the Sun.
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Simultaneous Inversion
AP063 through the B & V

Light Curves

    The plots in this section compare the light curves of the collected data, with the light curves created from the reconstructed surfaces.  It is important to look at how well these match.  A good agreement does not automatically mean that the reconstructed surface is correct (since you could be fitting noise), but it is a good indication that you are close.  In AP117 and AP063, alike, there is a systematic tendency for the inversion to underestimate the amplitude of the variations in light output.  This hints that some of my parameters may be a little off.

Light curve of AP063 through the B filter
Light curve of AP063 through the V filter
Light curve of AP063 through the R filter
Light curve AP117 through the B filter
Light curve of AP117 through the V filter
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Conclusion

    Using the varying light output of a star over time, it is possible to image surface features (specifically spots) by inverting the light curves.  Inversions of the B, V, and R curves were successful individually.
Two curve inversions using the B and V light curves were also successful in rendering surface images.
Three curve inversions were not successful, and none of the inversions in the I photometric band were able to converge.  Based on the existence of large, lone spots; the higher latitude placement of these spots; and the differences in appearance through the different photometric filters, it appears that the underlying physics involved in spots is different for stars vs. our Sun.

    Imaging stellar surfaces with the purpose of finding starspots is important for many reasons.  First of all, imaging stellar surfaces with MLI is a relatively new field, and not many stars have been done.  Scientists know relatively little about spots on stars, and every surface image adds to the knowledge base about them.  Imaging these surfaces is very important, for through comparison we can learn more about magnetic phenomena on our own star, the Sun.  The examination of spots on other stars helps scientists in the quest to understand the nature of the magnetic dynamo in stars, and how that arises and affects surface activity.  Matrix light curve inversion is a valuable tool in this pursuit, for it does not make any assumptions about surface conditions in order to create the image.  Unlike other techniques, MLI does not have to determine ahead of time how many spots are on the star, where they are located, or how large or dark they are.  This
makes it a powerful method of indirect imaging.  Additionally, MLI uses light curves to render surface images.  Light curves can be easily obtained through observational methods, and a great many have been published over the years.  This gives researchers interested in surface imaging the ability to compare spot activity over time by inverting light curves which were created in the past.  This is valuable in determining how starspots evolve over time:  whether individual spots have distinct life times, and whether spot locations
change with the magnetic cycle.  Matrix light curve inversion has many benefits over other indirect imaging methods, and is a practical and powerful technique for obtaining surface images of stars in order to study magnetic phenomena.

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Looking Ahead

    As with all research projects, there is always more that could be done.  Listed here are some ideas for future improvement on and expansion of what has already been done here:

  • Manipulating the values used for the magnitude error in order to get a more accurate inversion.
  • Vary parameters for all inversions instead of just the B-V cases.  Expand ranges of variations as well as the number of iterations used.
  • Inversion of the four remaining target stars.  Interesting to see if they too have only one spot, and whether spot appearance is filter dependent.
  • Examine stars outside of alpha Persei but of similar type to compare their spot behavior.
  • Compare spot behavior on other stars to that of the Sun.
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Acknowledgments

    I would like to thank Dr. Harmon for being my advisor the past year or so on this project.  He has been an invaluable source of information and has been extremely patient in teaching me about starspots, MLI, programing, and much more.  I would also like to thank Dr. Terndrup for providing the CCD frames, his work in tying together all of those magnitudes, and for taking the time to write clear and detailed instructions on how to use IRAF.  Thanks to everyone else who has been involved directly and indirectly, and especially to my friends who have had to put up with me babbling about starspots for months now.
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