Determining an Asteroid's Albedo
Using the SA-100 Transmission Grating
Bruce L. Gary, Last Updated 2014.09.08

This web page demonstrates that using a SA-100 transmission grating with a 12.5-inch Schmidt-Cassegrain telescope is adequate for deriving the flux spectrum of stars (and asteroids) as faint V-mag = 14.0 (and probably fainter) for the wavelength interval 470 to 800 nm. For V-mag = 12 the longest useable wavelength is 900 nm, while brighter objects can provide useful flux spectra to ~ 1000 nm. Flux spectra can be processed to produce standard magnitudes BVg'r'i'z', and the necessary procedure for doing this is given. Two SA-100 calibration methods are compared: all-sky photometry using a well-calibrated standard star and differential photometry using another star in the same FOV. Both methods are feasible, but the best results are obtained using all-sky photometry. The only exception is when the atmosphere is very humid and spatial structure of line-of-sight water vapor is likely; for that situation calibration using a background star in the same FOV is preferred. This method requires the conversion of a star's magnitudes (BVg'r'i'JHK) to a Spectral Energy Distribution (SED), with a blackbody fit for use in the SA-100 calibration. Converting SA-100 flux spectra to geometric albedo is demonstrated for Vesta, 23 Thalia and Scythia.

The SA-100 transmission grating is designed for use by amateurs for obtaining low resolution (1%) spectra of stars, asteroids, planetary nebulae, etc. It costs $200 and screws into a typical 1 1/4-inch filter wheel. This web page demonstrates how to use the SA-100 for determining as asteroid's albedo. Geometric albedo can be determined if sufficient brightness versus phase angle information is available. The procedure to be described is novel, as far as I know, for it employs a "differential photometry technique" (instead of the more difficult all-sky photometry technique) for calibration, which relies upon a SED (Spectral Energy Distribution) for a nearby star in the CCD's FOV instead of an already calibrated secondary standard star.  I will illustrate use of both calibration techniques on this web page.

Links on This Web Page
    Example 1: Geometric albedo spectrum using background star for SA-100 calibration
    Example 2: All-sky calibration alternative, Vesta geometric albedo spectrum using 59 Vir
    Dangers of all-sky calibration (due to water vapor inhomogenities)
    Equation for calculating geometric albedo
    Example 3: Faint asteroid (V=14.1) using all-sky calibration (51Peg); all-sky and background star calibration methods compared
    Converting flux spectrum to standard magnitudes
    Standard filter band measurements; comparing magnitudes from SA-100 with those using standard filters

   

Example #1: Geometric Albedo Spectrum Using Background Star for SA-100 Calibration

The following image was made with a Celestron CPC-100 (11-inch) telescope and a SBIG ST-10XME CCD. A "clear with blue blocking" was included in the optical path to reduce the 2nd order spectrum from interfering with the long wavelength portion of the 1st order spectrum (described later). The image is a median combine of 19 10-second exposures.


Figure 1. Image of asteroid 23 Thalia (indicated by target lines) and its 1st-order spectrum to the right. On this date (2014.09.01) the asteroid's V-mag ~ 12.0. Brighter stars to the east and southeast are present for possible use as calibrators. FOV = 18 x 12 'arc, north up, east left. Median combine of 19 10-second images. The bright star near the lower-left corner is a 10th magnitude star that will be used for calibrating the spectrum of the asteroid.

Below is a "profile" of the asteroid's spectrum.


Figure 2. MaxIm DL's "Profile" tool captures the sum of counts within each column of a horizontal box for a long stretch of rows, that contain spectrum information. The horizontal box has a "vertical width" of 12 pixels.

The profile's spike is the zero-order (straight-through) image of the star (~28% of total incident light). Plenty of baseline level information is retained in this long profile, as well as some 2nd-order information. The exported CSV-file is loaded into a spreadsheet that I've prepared for such data.


Figure 3. Plot of the "profile" recorded by MaxIm DL's Profile tool, showing a baseline fit. The x-axis is pixel distance from the zeroth-order location.

The above plot is total counts for each 12 pixel wide column versus pixel distance from the 0th-order star-like image. From an analysis of other stars I know the pixel-to-wavelength conversion:  λ [nm] = 12 + 2.140 * pixel location. The two vertical dashed lines represent 1000 and 1100 nm. For bright stars (or asteroids) there is measurable flux out to ~ 1100 nm.

This profile by itself is essentially useless. We need to calibrate it using a star with a known spectrum. That's what the bright star in the lower-left corner (c.f., Fig. 1) will be used for. This star has coordinates 04:30:49.0 +17:21:16, so I'll refer to it as RefStar043049. It is ~2.02 magnitudes brighter than 23 Thalia (at the time of this image). Below is a profile of this star, made in the same way as for the asteroid.


Figure 4. Profile of RefStar043049 for use in calibrating the 23 Thalia spectrum.

Now we need to know the flux spectrum of the reference star. A detailed procedure for doing this is given at this web page:  http://brucegary.net/SED/.  Briefly, we enter magnitude values for the bands BVg'r'i'JHK in a spreadsheet that converts each magnitude to a flux value. Here's the reference star's flux spectrum with a blackbody fit: 


Figure 5. Flux spectrum of RefStar043049, based on BVg'r'i'JHK magnitudes (APASS and 2MASS) fitted to a blackbody model.

The plotted flux is a special version that is proportional to photon flux, which is what CCD counts is proportional to.

In the spreadsheet a ratio is calculated for "23 Thalia Counts / RefStar043049 Counts" vs. wavelength. This is multiplied by the blackbody function that fits the flux spectrum of the reference star. Notice that by representing the reference star's flux spectrum by the blackbody fit it is possible to calculate flux values at each of the wavelengths sampled by the SA-100. We arrive at a flux spectrum for 23 Thalia, shown in the next figure.


Figure 6. Flux of RefStar043049 (top) and 23 Thalia (below) vs. wavelength. The reference star spectrum is based on a blackbody fit to the BVg'r'i'JHK magnitudes. The 23 Thalia flux spectrum is simply the ratio of the CCD counts profile for 23 Thalia to the counts profile of the reference star, multiplied by the reference star's flux spectrum.

Notice that the 23 Thalia data is noisy at short and long wavelengths, where SNR is low. The short wavelength region has low SNR due to the use of a "clear with blue blocking filter," which turns on at ~ 480 nm. At SNR at longer wavelengths is low for several reasons: 1) the CCD's QE is low and decreasing steeply with wavelength, 2) the telescope's corrector plate has a transmission function that decreases steeply with wavelength, and 3) the asteroid's reflected sunlight is is decreasing with wavelength.

We now know the asteroid's flux spectrum, and since the sun's flux spectrum is well known we can simply take their ratio to obtain a shape for the asteroid's albedo. Here's the solar flux spectrum.


Figure 7. Solar flux, above the atmosphere, smoothed.

After dividing the asteroid's flux spectrum by the solar flux spectrum we arrive at a shape for the asteroid's albedo, shown below.


Figure 8. Albedo of 23 Thalia based on 19 10-second images with a Celestron 11-inch telescope and SA100 transmission grating. Converting albedo to geometric albedo assumed a phase function, asteroid diameter, asteroid sun distance and asteroid Earth distance.

Actually, the above plot is "geometric" albedo, which required an assumption about the asteroids albedo dependence upon phase angle (sun-target-observer angle, 25 degrees at the time of these observations). This asteroid's radius is known (107 km), as is its sun distance and Earth distance at the time of these observations (2.42 and 2.20 a.u., respectively). If the phase function wasn't known we could have at least obtained an albedo shape spectrum (reflectivity spectrum) by normalizing to unity at the highest value.

Example #2: All-Sky Alternative Procedure for Determining Asteroid Flux Spectrum

As an alternative to the "differential photometry" method described above, it is also possible to determine the asteroid's flux spectrum by employing all-sky observing techniques. Vega has a well-established flux spectrum, but it is unmanageably bright for comparing with an asteroid 12 magnitudes fainter, for example. I've used an aperture mask with a throughput of ~5% to transfer Vega's spectrum to secondary standard stars (e.g., 59 Vir and 51 Peg). These secondary standards are sun-like, which has advantages, and they are ~ 5.3 and 5.5 magnitudes fainter than Vega. The all-sky observing requirement is to observe a secondary standard star at the same air mass and approximately same time as the asteroid.

The all-sky calibration method will be illustrated for obtaining the albedo spectrum of Vesta using for reference the sun-like star 59 Vir, with V-mag = 5.26. The Vesta observations were made on the same date as the 23 Thalia observations, above. In order to use 59 Vir for reference it was necessary to determine its flux spectrum. There are two ways to do this: 1) convert the star's magnitudes at the standard bands BVRcIc etc to flux, and 2) measure the star's spectrum with the SA-100 and compare it with Vega's, which is well-known. The following graph shows both results for 59 Vir:


Figure 9. Comparison of obtaining fluxes two ways: using Vega as a standard (square symbols and red trace fitted) and using catalog magnitudes BV & JHK (green symbols and dotted blackbody fitted trace). The two methods differ by ~8% (I favor the Vega-based spectrum). (In the figure title the term "S2" refers to 59 Vir.)

The 59 Vir flux spectrum based on magnitudes relies upon only BV and JHK bands since 59 Vir is too bright for APASS to have produced g'r'i' mag's. The 8% difference might be due to the fact that 59 Vir is so bright that most surveys don't measure it. The two spectra could be reconciled by faulting only the B and V magnitudes (B and V couldn't be measured by APASS because the star is too bright).

I adopted my Vega-based SED for 59 Vir and observed it on the same date that I observed asteroid Vesta (which was nearby). The Vesta and 59 Vir observations were made with the SA-100 close in time, and at nearly the same air mass. The Vesta SA-100 spectrum was divided by the 59 Vir spectrum, and this ratio was multiplied by the adopted 59 Vir flux spectrum, to arrive at Vesta's flux spectrum. This was then compared with the sun's spectrum to derive an albedo for Vesta (see section below for doing this).  Here's the resulting geometric albedo spectrum for Vesta.


Figure 10. Geometric albedo of Vesta based on 35 10-second images with a Celestron 11-inch telescope and SA100 transmission grating. Converting albedo to geometric albedo assumed a phase function, asteroid diameter, asteroid sun distance and asteroid Earth distance. The reference star was not in the FOV of the Vesta images, but images of 59 VIr were made at a similar air mass and close in time to the Vesta observations (i.e., all-sky photometry calibration). This introduces the possibility of an additional error possibility, but the error is likely to be of the offset type, such as when atmospheric extinction is not homogeneous across the sky.

The observed structure and overall value for the Vesta geometric albedo agrees with measurements made a few months earlier using a set of 7 filters (a spare set of the Dawn FC color filters). The sharp minimum at ~ 915 nm is due to mineralogy (Band I). This result supports the case I'm making for using blackbody fits to a nearby calibration star's BVg'r'i'JHK magnitudes.

Dangers of All-Sky Calibrations
  
In this section I illustrate how an all-sky flux spectrum transfer can go wrong!

If there are no strong temporal trends, or spatial gradients, then the ratio of measured spectra should be useable for determining the asteroid's flux spectrum. In my experience, however, atmospheric extinction is very difficult to deal with in the region of the broad water vapor absorption region, 920 nm to 960 nm, when the atmosphere is humid. This was not a problem during the dry season at my observatory (October through June), but during the monsoon (July to mid-September) it is a big problem. During the year at my site the water vapor burden (also called "precipitable water vapor") can vary from ~ 0.3 cm to ~6 cm. During the 2.5 month monsoon it averages 4 or 5 cm. For most of the rest of the year the water vapor burden is ~ 1 to 2 cm, and during the winter it is probably ~ 0.5 cm (my lowest measurement is 0.3 cm). During March through June I was able to measure geometric albedo of Vesta and Ceres without a problem from water vapor. But during the few clear nights of 2014 August and September the water vapor was so high that the all-sky technique for determining asteroid albedo was unfeasible (described below), which meant that the differential photometry technique in the previous section was required.

Here's an example that illustrates a shortcoming of using the all-sky technique for establishing the spectrum of another star during high water vapor burden conditions (varying between 3.7 cm and 6.5 cm during 30 hours during previous & following daytimes).


Figure 11. SED for 51 Peg, based on observations of Vega taken 30 minutes before 51 Peg using an aperture mask passing 5.613(26) %. The square symbols are from a published article by Schroder et al (2013) about the Dawn spacecraft Framing Camera post-launch calibration.

This comparison shows good agreement from ~ 520 nm to 925 nm; the "artifact" at ~ 960 nm has the shape of the water vapor absorption complex of lines, so it must be produced by a difference in the water vapor for the two line-of-sights (Vega and 51 Peg, at same air mass, but different parts of the sky). The loss of data quality longward of ~ 1020 nm is due to small flux levels for the Vega observations (1-second exposures, to avoid saturation at short wavelengths, with a 5% mask).

This is why I prefer to use the "differential photometry" technique for calibrating SA100 spectra, at least when water vapor burden is high and potentially variable.

Calculating Geometric Albedo

In case you're interested in calculating geometric albedo, here's the equation.

    Ag[%] = 100 × GeometryRatio × FluxRatio,

    where GeometryRatio = ( r[a.u.] × d [km] / Ra [km] )2 
    FluxRatio = FluxAsteroid [W/m2] / FluxSun [W/m2]
    and r [a.u.] is sun/asteroid distance, and d [km] is asteroid/Earth distance = d [a.u.] × 1.496e8 [km]
    and Ra [km] is the asteroids radius

Example #3: Faint Asteroid Geometric Albedo Determination

When a Near Earth Asteroid passes close to Earth it may not be possible to obtain the necessary SA-100 observations when the asteroid is bright due to cloudiness or the asteroid's fast motion. Days after closest approach observing opportunities might be more favorable, but by then the asteroid will be fainter. In this section I show a successful measurement of a 14th magnitude asteroid's geometric albedo using the SA-100 mounted to the prime focus of a Meade 14-inch telescope.

On 2014 Sep 20 I observed the main belt asteroid 1306 Scythia (1930 OB) using my Meade 14-inch with a prime focus configuration that employed a HyperStar lens, a SBIG ST-10XME CCD and a SBIG CFW10. The 10-slot CFW obscured 31% of the collecting area, rendering the photons intercepted at the CCD equivalent to a 11.3-inch refractor or a 12.5-inch typical Schmidt-Cassegrain reflector (with 13% blockage).  A few days earlier I had calibrated the sun-like star 51 Peg using a Celestron 11-inch telescope to compare its SA-100 flux spectrum to Vega's SA-100 spectrum using a 5% aperture mask. Here's the SED I derived for 51 Peg:


Figure 12. Flux spectrum of 51 Peg derived from SA-100 spectra of Vega with a 5% aperture mask (red symbols), compared with readings of the 51 Peg spectrum given in a publication by Schroder et al, 2013 (blue symbols).

There is excellent agreement between the 51 Peg SED adopted by Schroder et al (2013) and the one I measured on Aug 31 using the SA-100. 

My observations of Scythia on 2014 Sep 20 with a Meade 14-inch were sub-optimal because the Meade's prime focus FOV was 71 x 48 'arc, with an image scale of 1.95 "arc/pixel, which produced a crowing of stars and their SA-100 spectra, as shown by the next sample image:


Figure 13. Median combine of 9 30-second exposures of the asteroid Scythia (target cross hairs) using a Meade 14-inch telescope with a HyperStar prime focus lens and SBIG ST-10XME CCD, with a FOV = 71 x 48 'arc. First-order spectra are to the left of the zero-order star images (28% throughput). The asteroid's first-order spectrum is indicated by a rectangle.

Several 9-image median combine images were measure using the MaxIm DL profile tool, and each was processed in a spreadsheet to produce "raw spectra" shown here:


Figure 14. "Raw spectra" of asteroid Scythia from 4 median combined images (the red trace is from a median combine of 5 images instead of 9).

Cirrus clouds were present so one of the raw spectra (red trace) may be low due to transient cloud losses.

Path 1: At this point there's a divergence of analyses of the Scythia SA-100 data. The first path will make use of the traditional "all-sky calibration method, and later I'll resume from this stage of analysis using the differential calibration method.

Scythia was only 3.3 degrees from 51 Peg, so my observations of both fields were automatically at about the same air mass (all at an elevation of ~ 71 degrees). 

The ratio of Scythia to 51 Peg was multiplied by my adopted SED for 51 Peg to produce the Scythia SED in the following:


Figure 14. SEDs for 51 Peg and asteroid Scythia, taken at the same air mass and approximate same time with the Meade 14-inch telescope. 

Converting an asteroid's SED to geometric albedo involves dividing asteroid SED by the sun's SED, and entering the result in an equation described in the previous section. For Scythia, using 51 Peg for an all-sky calibration just described, the geometric albedo result is shown in the next figure.


Figure 14. Geometric albedo of Scythia, using a radius of 34 km and G = 0.15.

The derived geometric albedo of ~ 5.7% at 550 nm (V-band) is in pretty good agreement with the oft sited 5.1%. Further, there is no evidence for a Band I mineralogy absorption feature.

Good! QED! I've just shown that with a SCT having an equivalent aperture of a 12.5-inch telescope, using a SA-100, it is possible to measure the albedo shape (and geometric albedo if asteroid size is known) when the asteroid is as faint as V-mag 14.1. With larger apertures fainter asteroids will be measurable. For example, with a 32-inch telescope it should be possible to measure albedos for asteroids as faint as 16.3, using a SA-100, provided the background of stars permits it.

Path 2: Now, let's return to the place above, Path 1, and resume analysis without use of 51 Peg for calibration. Instead, we'll use one of the stars in the asteroid image (Fig. 13) to serve in the calibration role.

Consider star "10012" in the following figure:


Figure 15. Star 10441 (i.e., V-mag = 10.441) will be used to calibrate the Scythia spectrum.

This star has APASS magnitudes that can be used to produce a SED, shown in the next figure:


Figure 16. SED for "Star 10441" using APASS BVg'r'i' and 2MASS JHK magnitudes, fitted by a 4200 K blackbody curve.

The "straight line segment" representation was adopted for use for calibrating the star's SA-100 spectrum, which in turn was used for calibrating the asteroid's SA-100 spectrum. Both are shown in the following log-linear SA-100 spectra:


Figure 17. SEDs for Scythia and Reference Star 10441 (based on straight-line-segment fit to BVg'r'i' and JHK magnitudes).

With this SED for Scythia the following geometric albedo was determined:


Figure 18. Scythia geometric albedo based on use of Reference Star 10441.

This geometric albedo for Scythia is almost the same as the one derived from the use of 51 Peg for reference (c.f., Fig. 14). The quality is similar, and overall shape is similar in that there is no evidence for a Band I feature and geometric albedo s very low (~ 5.5%). The advantage of this albedo spectrum is that it did not require observations of another part of the sky including a secondary calibration star (e.g., 51 Peg). The point of this exercise is to show that a star within the FOV of the asteroid can be used to produce a calibration that gives approximately the same albedo as that from using a secondary calibration star that is based on Vega for its primary source.

Converting SA-100 Flux Spectrum to Magnitudes

There's a relationship between flux and magnitudes, given by the following equation and table (described in detail at link):

    λFλ [W/m2] = SED Constant × 10-0.4 Mag

Band
 Effective
Wavelength [micron]
 SED Constant
[W/m2]
B
 0.433
 2.954E-8
V
 0.550
 2.091E-8
Rc
 0.640
 1.443E-8
Ic
 0.790
 0.968E-8
g'
 0.481
 2.48E-8
r'
 0.670
 1.75E-8
i'
 0.790
 1.40E-8
z'
 0.910
 1.22E-8
J
 1.235
 3.869E-9
H
 1.662
 1.847E-9
Ks
 2.159
 0.926E-9

Since we know the flux spectrum for asteroid Scythia (Fig. 14) we can determine magnitudes at these standard bands (by solving the λFλ equation for "Mag"). Below is a plot of filter response functions for these bands, which we must convolve with the asteroid spectrum in order to get a response-weighted flux.


Figure 19. Filter response functions and Scythia's flux spectrum (running average). Trends are adopted shortward of 400 nm and longward of 936 nm because the measured values are very noisy in these regions.

Notice that the filter response functions take into account CCD QE function and telescope corrector plate transmission function.

The filter response weighted asteroid fluxes yield the following magnitudes:

    B = 14.52
    V = 14.09
    g' = 14.31
    r' = 13.93
    i' = 13.67
    z' = 13.52

Here's another way to present these mag's:


Figure 20. Scythia SED compared with a blackbody SED for the sun's temperature.

This plot is another way of showing that Scythia has increased albedo at longer wavelengths 'i & z').

Standard Filter Measurements: Evaluating SA-100 Magnitudes with Conventional Measurement Method

On the same night that the SA-100 was used to observe Scythia a set of measurements was also made with filters g'r'i'z'. The magnitudes determined using these filters can be compared with the SA-100 magnitudes determined from the flux spectrum in order to evaluate the quality of magnitudes inferred from SA-100 observations.

    g' = 14.28 ± 0.01 vs 14.31 SA-100 (dif = +0.03)
     r' = 13.75 ± 0.01 vs 13.96 SA-100 (dif = +0.21)
     i' = 13.48 ± 0.01 vs 13.87 SA-100 (dif = +0.39)
     z' = 13.42 ± 0.02 vs 13.77 SA-100 (dif = +0.35)

The g' magnitudes agree well, but the others differ by ~ 1/3 magnitude. I think this is probably going to be a realistic assessment of the much lower quality of SA-100 based magnitudes compared with the traditional ones using filters.

Here's a SED for Scythia based on the high quality g'r'i'z' magnitudes:


Figure 21. SED for Scythia based on my differential photometry observations calibrated using APASS magnitudes.

Comparing Scythia's g'r'i'z' magnitudes with the solar spectrum shows that Scythia has increased albedo at i' and z' bands, which confirms the SA-100 result.

To satisfy my curiosity about the suspicious-looking SED for the Reference Star 10441 (c.f., Fig. 16), I derived good quality measurements of it using the g'r'i'z' filters. Here's how my high quality magnitudes compare with the APASS ones in a SED plot:


Figure 21. Revised SED for Reference Star 10441, using my high quality g'r'i'z' magnitudes plus the APASS BV and 2MASS JHK magnitudes.

I conclude that this star simply departs from a blackbody spectrum in a way that renders it problematical for use as a reference star for calibrating SA-100 spectra. This underlines the misgivings I have for using background stars for differential photometry calibration. The safest procedure for calibrating SA-100 observations is to use a well-calibrated secondary standard stars using the all-sky photometry method. So far I've used Vega as a primary standard to calibrate 59 Vir, beta CB, tau Vir, 109 Vir and 51 Peg. I need to emphasie stars near the ecliptic for the winter season. When this project has progressed I'll present their spectra at a web page devoted to reference stars for SA-100 calibration, and link to it at this web site.


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