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.
Here are some fun factoids about sunspots:
Starspots, like sunspots, are regions of intense magnetic
activity (2,000-3,000G) as evidenced on the sun
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
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
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.
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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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.
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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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.
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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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.
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.
AP063 through the B & V
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.
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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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
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
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.
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: