Asteroid "46053 Davidpatterson" Light Curve
Bruce L. Gary & Dave Healy
Links Internal to this Web Page
This web page describes the
determination of a light curve for a faint asteroid. It is not unusual
for professionals to establish light curves for main belt asteroids
having opposition magnitudes
of ~20 to 21; however, the observations and analysis described here were
performed by amateurs. Special data analysis procedures were devised to
reduce the effects of background field stars. The principal value of
this web page is a description of these data analysis techniques.
Observations on 2005.09.01
Results for 2005.09.01
Observations on 2005.09.26
Combining Data for Both Dates
This observing project is meant to show that amateurs are
capable of measuring the light curve of asteroids as faint as magnitude
Of course, not every amateur has a high-quality 32-inch Ritchey
Chretien telescope system like the one used for these observations. We
use the Junk Bond Observatory's Optical Guidance System 32-inch
telescope and fork-mount (owned and operated by author DH). It is
housed in a computer-controlled 16.5-foot dome. The CCD camera is a
SBIG model STL-6303E. It is a large format CCD (27.7 x 18.5 mm main
chip) that employs a KAF-6303E main chip having a 3072 x 2048 pixel
layout (9 micron square pixels size) and a TC-237 autoguider chip (not
used for these observations). All components are computer-controlled
except the focusing, which is reliably constant due to the telescope's
low-to-zero expansion invar cage.
Observations on 2005.09.01
Observations were made with 2x focal reducer lens to
increase the field of view (FOV), which was ~27 x 18 'arc. Unfiltered
exposures of 120-second duration were made during a 2.7-hour observing
period on the evening of 2005 August 31 (2005.09.01 UT). Autoguiding
was not used since the OGS mount has excellent tracking. All CCD camera
control and image analysis was performed using MaxIm DL (ver. 4.0).
Binning was set to 2x2, which provided 3.5 pixels per FWHM during the
best seeing for the observing session. Eleven flat field images were
made near sunset and were median combined using automatic level
normalization. The CCD temperature of -8 C was maintained during most
of the observations. A set of 10 1-minute dark frames were median
combined to produce a master dark frame. Dark frame calibration used
the "auto-optimize" feature to compensate for the difference in
exposure time and small differences in CCD temperature. An observing
sequence was used to automatically expose and record images. The
limiting magnitude for a 2-minute, unfiltered image is Cv = 21.3 when
the seeing is FWHM = 4.0 "arc (where Cv represents the V-magnitude
equivalent using a "clear" filter).
A total of 66 exposures were made of the asteroid field during a
2.7-hour observing period. Following this about 35 minutes was spent
performing photometry calibration observations of a star field near M27
for which Arne Henden had recently produced a BVRI photometric sequence
for 20 stars.
The following is a typical example of a 2-minute exposure that has been flat field and dark frame calibrated.
Figure 1. Example of a single 2-minute exposure. FOV = 26.6 x
17.7 'arc, northeast at upper-left, reduced in apparent size to fit
this web page. The asteroid location is at the intersection of the two
line segments (not visible here).
Figure 2. Two-times zoom of the center of the above image. The asteroid is barely visible, and has SNR ~7. FOV = 13.3 x 8.9 'arc.
The asteroid's SNR is insufficient for use in aligning since low SNR
objects have centroid locations that are influenced by random noise for
each pixel. The following section describes how the low SNR situation
was overcome for the purpose of creating alight curve.
The asteroid's motion was 0.12 "arc/minute (northward). The image scale
was 1.04 "arc/pixel (where a pixel is a 2x2 binned pixel). During the
course of exposing and downloading 4 images the asteroid moved 1.1
"arc, or 1.0 pixel. The FWHM for stars was typically 4.4 "arc after
median combining 4 images, so the enlargement of the asteroid's FWHM in
a 4-image median combine image was very small (4.5 versus 4.4 "arc).
The following image is a median combine, using stars for alignment, of
4 calibrated 2-minute images.
Figure 3. Crop and zoom of a 4-image median combine, showing the asteroid having SNR ~14.
Figure 4. Median combine of 4 images each for the
beginning (left) and end (right) of the 2.7-hour observing period. The
asteroid is located within the yellow circles. The blue cirlces show
where the assteroid was located at the other end of the observing
period. The asteroid's motion is mostly northward (upward) along a path
34 "arc long.
Notice the presence of faint background stars near the asteroid's
track. The reason the asteroid appears especially bright at the end of
the observing session (upper position) is because the asteroid is very
close to background stars (with Cv ~ 21). The problem of background
stars is worst near the middle and end of the observing period.
Background stars is often the most important problem to solve when
attempting light curves for faint asteroids.
One method used to correct for the influence of background stars on an
asteroid's apparent brightness in an image involves the use of an image
taken when the asteroid was at a different location. For example,
consider images taken at the beginning and end of the observing
session. Start by assuming that the first image can be used to locate
the asteroid's RA/Dec location. The aperture circles can be placed at
the sameRA/Dec location in the last image to obtain a flux that is
produced by the background stars. If there are no background stars then
the flux reading will be close to zero (and dominated by CCD thermal
pixel noise and sky background noise). If there's a background star
within the signal aperture it will produce a positive signal, and this
will have to be subtracted from the flux reading of the asteroid when
it was at the same loaction. If there's a background star within the
sky reference annulus it will produce a negative flux, and that will
have to be used to add to the asteroid flux reading. One limitation of
this technique is that it is difficult to accurately establish the
asteroid's track. This is due to the influence that background stars
have upon the centroid pixel location of the asteroid. The fainter the
asteroid, the worse is this problem.
An alternative method for removing the effect of background stars is to
subtract one image from another, after careful alignment using the
stars, and proceed to measure the asteroid flux. This method was
adopted for the present analysis. When an image is subtracted from
another the resultant image will in theory register nothing where stars
are present but register the complete brightness distribution of the
moving asteroid. In a real-world situation the stars will not exactly
cancel since they will have a slightly different point spread function
(due to seeing changes) in different images and they will have a
slightly different intensity due to atmospheric extinction changes
(caused by air mass differences). Nevertheless, the intensity of the
star field can be reduced by a coule orders of magnitude. The
brightness of faint stars in the subtracted image along the asteroid's
track will also be reduced in amplitude by a couple orders of
magnitude. This, in effect, should remove the effect of a field of
faint background stars in an objective manner and allow for an accurate
measurement of the asteroid's flux.
Subtracting images increases the noise level at each pixel location. It
is therefore important to use a "reference" image, defined as an image
where the asteroid is "out of the way," that has a longer total
exposure than the signal image. By averaging several images when the
asteroid is "out of the way" to produce a reference image the
subtracted image will have only a small amount of additional noise. In
the case of this 2.7-hour observing session the asteroid didn't move
much, just 34 "arc. Care was taken in producing reference images for
each of several time intervals of the asteroid's motion. The following
figure shows the before and after subtraction for an image near the end
of the observing session, when the asteroid was located among several
Figure 5. Demonstration of "removal" of background stars
without affecting the asteroid's presence in the subtracted image
In this image the bright stars were not completely subtracted because
of an imperfect pixel alignment (~1/2 pixel error, which is sometimes
unavoidable) and different star intensities and PSFs (point-spread
functions). Nevertheless, the asteroid is clearly evident (using an
objective image analysis procedure) and it is clear of nearby
background star artifacts - which permits the aperture signal circle
and sky reference annulus to be used in a straightforward (free of
subjective user decisions) manner, as the next figure illustrates.
Figure 6. Same as right panel of previous image, with
aperture circle pattern centered on the asteroid for making a flux
"reading." The asteroid flux has SNR = 22 for this set of eight
2-minute images (and corresponds to Cv = 19.71).
Using the the above image as a guide (asteroid's SNR = 22, Cv = 19.71)
it should be possible to detect asteroids with Cv in the range of 21.9
(SNR = 3) to 23.1 (SNR = 1), even in crowded star fields.
A set of these images were used to measure the flux of the asteroid for
sets of eight 2-minute images. These asteroid fluxes were converted to
Cv magnitudes using telescope photometry coefficients derived on the
same night and described in a later section. The plot of Cv versus time
is shown in the next section.
Results for 2005.09.01 Observations
The following graph shows the light curve for the asteroid during the 2.7-hour observing period of 2005.09.01 UT.
Figure 7. Light curve for the first night of
observations. (The faint gray symbols are from an alternative
analysis method.) The error bars are stochastic SE and do not include
This Minor Planet Center lists this asteroid's H value as 16.2.
According to the ephemeris it should have had a V-magnitude of 19.65 at
the time of these observations. Instead, it is 0.40 magnitude fainter,
implying that H = 16.60.
The asteroid's average diameter is estimated to be 1740 meters (range
1230 to 3900 meters). These diameters are based on it brightness and an
albedo assumption of 15 % (range 3 to 30%). For an asteroid this size
the most likely rotation period is 5.3 hours (range 4.8 to 6.0 hours,
SE). The measured value of 4.6 hours is near the short end of the expected
The brightness peak-to-peak variation of 0.90 magnitude (twice the
"amplitude") is slightly larger than for asteroids of this size.
Asteroid light curves are often non-sinusoidal. It is possible that the
extra brightness of the first point is real. After all, at ~3 UT and
5.3 UT we are viewing opposite sides of the asteroid (the largest solid
angle perspective of the "potato" shaped obsject). The end-on view at
~4.0 UT can have a different brightness from the other end-on view that
we would have had at ~6.5 UT. More observations are needed to clarify
the shape of this asteroid's light curve.
Here's an animation of the asteroid's motion.
Figure 8. Animation of asteroid motion during the 2.7-hour observing period.
Observations of 2005.09.26
The second observing session was 25 days
later when the asteroid was predicted to be 0.47 magnitude fainter than
during the first observing session (20.12 versus 19.65). Given
that the previous observations called for an H-value fainter
by 0.4 magnitude the ephemeris value of 20.12 meant that we should
expect that the asteroid would actually average ~20.5.
Here's an example of image subtraction for a set of 4 images.
Figure 9. Before image subtraction. Note the barely visible
asteroid at the center of the photometry aperture circles. The stars to
the upper-right of the asteroid have V-magnitudes of 16.2.
Figure 10. After image subtraction. In this image the asteroid is easily visible.
Figure 11. Light curve for 2005.09.26. Each green diamond is
based on 4 images, each having an exposure of 2-minutes. The red
diamonds are based on 8 images. The error bars are stochastic SE and do not include estimated uncertainties.
brightness of 20.4 is close to what was expected based on the 2005.09.01
observations. The amplitude is also the same as 25 days earlier,
corresponding to a 0.90 magnitude peak-to-peak variation. The period is
also approximately the same.
It is noteworthy that the photometric calibration for the second
observing session was quite different from the first one. The next
section describes how a set of stars near M27 were used to establish
the JBO telescope photometry constants. For the second session another
telescope was used to establish magnitudes for stars near the
asteroid's location on 2005.09.26. A 14-inch Celestron was used on
2005.09.29 to observe a set of 14 stars in the Skiff catalog with B and
V magnitudes. The star region was 1/4 degree to the east of the
2005.09.26 asteroid location, which made it unecessary to refine the
zenith extinction value for transferring Skiff brightnesses to the
stars near the asteroid field. The 14-inch Celestron's telescope system
photometry constants had values similar to those on previous occasions
when Landolt regions were observed, and the 14 Skiff stars exhibited a
0.018 magnitude RMS scatter about the color-correcting solution.
Combining Data From Both Dates
A rotational light curve solution must satisfy the night's light curve
for both observing dates, as illustrated in the following figure.
Figure 12. Light curve for 2005.09.01 (red) and 2005.09.26
(blue), with a 236 period offset for the latter. The model trace is a
sinusoid with parameter values given in the figure. The model also
includes ephemeris values for average magnitude for the two dates but
including an offset of +0.4 for 2005.09.01 and +0.2 for 2005.09.26. The error bars are stochastic SE and do not include systemtic uncertainties.
Since the two observing dates were 25 days apart, during which the
asteroid rotated ~236 times, it is not possible to unambiguously solve
for a period. Periods of 2.5261 or 2.5477 are equally acceptable.
Indeed, the period solution should be stated as P = 5.05232 + N *
0.01075 where N = any integer between -5 and +5. To resolve the period
it would be necessary to observe on two closely separated nights.
The standard brightness in the ephemeris is off by about 0.3 magnitude,
should be 16.5 (instead of 16.2). There is a slight difference between
the H values required for 2005.09.01 (H = 16.6) and 2005.09.26 (H =
16.4). This could be due to errors in either of the observing session's
all-sky photometric calibration (completley different photometric
calibration procedures were used for the two observing sessions), or
maybe it is due to the changed observing geometry (i.e., G differs from
the assumed value 0.15).
I should comment about the SE error bars and their apparent Bayesian
incompatibility with the model fit. It is extremely unlikely that the
error bars are correct since there are many data points that depart
from the model by several SEs. The plotted SE bars are stochastic,
based solely upon SNR - specifically, SE = 1/SNR. Systematic
uncertainties are undoubtedly present but they are very difficult to
impossible to evaluate prior to comparing the data to a reasonable
model. There are two cattegories of systematic SE: those that are
shared by all data, such as a photometric calibration error, and those
that casn change during the observing session. This latter category is
obviously present as evidenced by the large differences between points
that are spaced closer in time than any light curve variation could
accomodate; the first category is probably present but it can't be
proven from anything evident in this data set. The "varying systematic
error" must be approximately 0.15 magnitude since this amount of error
would have to be orthogonally added to the stochastic SE in order for
the total SE to be compatib le with the above model. This 0.15 varying
systmatic error, or ~15%, could easily be produced by an incomplete
star field subtraction. Typically, the star field subtraction ios only
99% effective, leaving a residual ~1% feature in the subtracted image.
If a nearby star is 100 times brighter than the asteroid then the
residual feature would have the same amplitude as the asteroid. This
residual star feature could be within the signal aperture or in the sky
background annulus. In one case the apparent asteroid brightness would
be increased, while in the other case it would be decreased. The
amplitude of the increases would be larger than the amplitude of the
decreases, but there should be fewer increases than decreases since the
pixel area for the signal aperture is smaller than that for the sky
background annulus. Another predicted behavior for these residual
background features is that they should become important as the
asteroid becomes less bright. At some faint level the signature of
residual background stars should become the dominant component for all
measurements. The level where this happens will depend upon how many
background stars there are. These observations were made at a galactic
latitude of -18 degrees and a longitude close to galactic center.
Therefore the star density was greater than for a typical region of the
sky. Presumably, it should be possible to observe asteroids fainter
than this one (CV= 20.4) for determining light curves provided they are
at greater galactic latitudes.
Photometric Calibration of JBO 32-inch Telescope System
Figure 13. M27 with a square showing where Arne
Henden's photometric sequence stars are located. FOV = 26 x 17 'arc.
RGBC exposure times are 6, 6, 6 and 10 minutes.
Figure 14. Crop and zoom of above image 27 showing a blue
nova (cross hairs) and 8 bright reference stars used to calibrate the
JBO 32-inch telescope system.
Eight of the calibrated stars in the above image were used to
establish the following JBO 32-inch telescope system photometric
Cv = 23.30 -2.5 * LOG (S / g ) - 0.15 m + 0.86 * C
where Cv = V-magnitude equivalent using a "clear" filter,
S = star flux using a large aperture [data number counts],
g = exposure time [seconds],
m = air mass,
C = star color,
defined as either 0.57 * (B - V) - 0.30 or (V-R -0.31), whichever is
Note that star flux S can be calculated using a small aperture
provided it is adjusted for the flux ratio corresponding to small and
large signal apertures (using a bright star with no nearby background
stars). A small signal aperture is always better to use with faint
objects in order to maximize SNR. For the asteroid observations I used
two small signal aperture radii, 5 and 6 pixels, and a large aperture
radius of ~12 pixels, which led to a flux ratio correction f5 =0.85 and
f6 = 0.91.
First created: 2005.09.03 Last updated: 2005.10.04