Deriving new magnitude system

This web page describes some basic concepts required for the creation of a new magnitude system, specifically intended as support for the Dawn Framing Camera 7-band magnitude system that I created for calibrating Vesta and Ceres geometric albedo in 2014 (link).

When using a filter that does not belong to one of the standard filter bands it is important to characterize measurements made with it in terms that relate to physical phenomena, such as energy flux [watts per m² per micron]. If this can be accomplished then measurements with the new filter can be used to fill gaps in the “spectral energy distribution” (SED) spectrum of stars. In addition, such measurements can be used to add detail to an asteroid’s albedo. This write-up describes an invention of a new magnitude system using a standard filter as a test case, which can serve as a model for doing the same for other filters (such as the Dawn FC filters). The underlying philosophy for a magnitude system is that any observer who processes his observations properly can state that his magnitude is what would have been measured by the telescope system used to create the magnitude system, and that such magnitude can be converted to a physically useful property such as flux at an effective wavelength.

Figure 2. Solar spectrum for the wavelength region where CCD cameras respond, showing spectral location of a standard filter (V).

The solar energy spectrum is approximately constant across V-band. The effective width of the V-band shape is ~ 0.10 micron, and the average solar flux within this region is ~ 1850 [watts/m²/micron]. (Note my use of “watts/m²/micron” to be the same as “watts per m² per micron.”) Therefore, a V-band filter above the atmosphere would transmit ~ 185 [watts/m²].

How many photons is that? Well, the energy of a 0.55 micron photon is hν, where h is Planck’s constant and ν (frequency) is c/λ (c is speed of light). A 0.55 micron photon has an energy of 3.61e-19 [watts]. Therefore, 185 [watts/m²] within the V-band corresponds to 5.12e+20 photons per second per m². At other wavelengths photon energy will differ: energy/photon = 5.12e+20 × 0.55 / λ[micron]. This information allows us to convert Fig. 2 to a plot of photon flux, shown as Fig. 3.

Using Fig. 3, and noting that V-band is ~0.1 micron wide, we can estimate that a V-band filter above the atmosphere would transmit ~5e+20 [photons per second per m²]. This agrees with the previous estimate.

For wideband filters, such as a clear filter, it is appropriate to work with photon flux spectra since a CCD camera produces data numbers (DN) proportional to the number of photons intercepted by pixels and available for conversion to photo-electrons; in other words, DN is not proportional to “energy per unit are per unit wavelength” and using an energy spectrum would be misleading. It is common practice to convert an “energy per unit area per unit wavelength” spectrum, referred to as F_λ, to something proportional to “photons per unit area per unit wavelength” by simply multiplying F_λ by λ, yielding λF_λ [watts/m²]. Such a spectrum will have the same shape as Fig. 3, and will be proportional to “photons per unit area per unit wavelength” but the invented parameter has a difficult to understand meaning and the units are meaningless to me (since the “per micron” unit is missing, yet is implicit in λF_λ). For filters with narrow pass-bands it will be acceptable to use the “energy per unit area per unit wavelength” spectrum (e.g., Fig. 2), so that’s what I will do in the remainder of this document.

Fig. 4 shows the effect of hardware on filter pass-bands; the response loss is due to telescope optical transmission and CCD QE response.

In creating a new magnitude system it is necessary to use the pass-band response functions of the entire telescope hardware system: telescope optical transmission, filter response and CCD QE. After all, if the system that uses a filter doesn’t respond to a wavelength region then nothing related to that region should enter into consideration of a magnitude system for that filter band. This has been true for the calibrations of the BVRcIc and u’g’r’i’z’ standards.

Atmospheric extinction requires a different treatment from the hardware losses and responses. Extinction effects must be removed before any magnitude determinations can be made. In effect, we are asking what the standard telescope system would observe if it were above the atmosphere, and anyone else who wants to use the magnitude system must do the same. Since each observer will have a different telescope system spectral response a method has to be devised for converting an observer’s measurement to an equivalent of what the standard telescope system would have observed (above the atmosphere). Atmospheric extinction consists of four principle components: Rayleigh scattering, aerosol scattering and absorption, ozone absorption and water vapor absorption. This is shown in the next figure.

Figure 5. Atmospheric extinction at zenith [magnitude] at my observing site on a typical winter night.

Converting measured magnitude to “above the atmosphere magnitude” is straightforward, and will not be described here. Instead, I will assume that all measurements will be converted to above the atmosphere magnitudes before any attempt is made to convert an “instrumental magnitude” to a standard magnitude. (Details may be given later.)

Referring back to Fig. 4, notice that a V-band filter for my telescope system responds to photons between 480 and 640 nm, and the effective wavelength for this filter response function is 546 nm (detailed calculation takes into account that CCD responds to photons). The “filter response weighted solar flux” is 1853 [watts/m²/micron]. If we want to devise a magnitude system for this filter response function then 1853 [watts/m²/micron] can play an important part in such a creation. For example, if an object was measured to be 1% as bright as the sun, 5 magnitudes (by definition), we could state that it’s “filter response weighted flux” was 18.53 [watts/m²/micron]. Another way of expressing this is to state that:

where “delta-magnitude = magnitude of unknown target – magnitude of sun.” If we arbitrarily adopt a magnitude for the sun then we will have established a new magnitude scale. However, it is impractical for any observer to use the sun as a reference standard, so another choice should be adopted for a non-variable star to be the reference.

Vega has traditionally been used to establish magnitude scales, and I will do the same. I will define magnitudes such that Vega has a zero magnitude for every filter (referenced to above the atmosphere).

This energy spectrum for Vega is well-established. Convolving my V-band response function with the Vega spectrum shows that if my telescope were above the atmosphere it would measure a flux of 3.5e-8 [watts/m²/micron] at an effective wavelength of 0.541 micron. (This effective wavelength is different from the one given above due to Vega being bluer than the sun.) Measurements of any other star (having the same blueness as Vega) could then be converted to flux at the same effective wavelength using a standard magnitude equation.

For this magnitude system to be useful a network of stars should be calibrated using Vega as a primary standard. Whenever filters are used that have response widths as large as V-band, for example, it is necessary to correct for star color effects. This means that it is important to establish magnitude scales at other effective wavelengths (preferably surrounding the wavelength of greatest interest), and to establish calibrated magnitudes for these other bands for stars having colors different from Vega.

As a check on the above, let’s calculate the sun’s flux at V-band. Since the sun’s V-mag = -26.75 we have “sun’s flux at 541 nm” = 3.66e-8 / 2.5119^-26.75 = 1837 [watts/m²/micron]. This differs by < 1.0% from the 1853 [watts/m²/micron] value determined above.

The above procedure can be applied to any non-standard filter for the purpose of determining an object’s flux in useful physical units, e.g., [watts/m²/micron]. For the task of measuring asteroid albedo this capability is useful; since we know the flux incident upon an asteroid (since we know the sun’s spectrum), the reflected flux can be used to determine albedo.

Suppose that on 2014.01.28 I observed Vesta and found V = 7.28 (the ephemeris value). On this date the sun-Vesta distance was 2.288 a.u., the Earth-Vesta distance was 1.869 a.u. and the “phase angle” α (same as STO, sun-target-observer) was 25 degrees. I’ll assume Vesta has a radius of 260 ± 5 km. Suppose surface roughness is sufficient to reflect light in a manner that is 90% of the way between uniform and Lambertian (for which disk average = 2/3 of flat disk reflection, for α = 0). I’ll define a parameter for this, and give it the value 0.90 ± 0.10. Let’s assume a phase function slope (beyond 7 degrees α) = 0.035 ± 0.010 [magnitude/degree]. Combining these assumptions yields an albedo of 15 ± 4 %. I don’t know enough about albedos to know if this is reasonable (if it’s a Bond albedo, or geometric albedo, etc.) so I’ll simply take the position for now that the magnitudes and fluxes derived from my invented magnitude scales will be useable by someone knowledgeable in how to use them.

The previous material demonstrates a method for creating a magnitude system that has physical significance, which for example can be used to assess an asteroid’s albedo from ground-based measurements (if its size is known, and other assumptions are made). Implementing such a procedure using specific hardware is definitely feasible, but it not trivial. It will require some skills that few professionals are practiced in; namely, all-sky photometry calibration. Fortunately I’ve been doing all-sky photometry for ~ 8 years.

The first task will be to transfer the magnitude of Vega to a fainter star, such as one at 6^th magnitude. This will require using an aperture mask because Vega would saturate my CCD at all bands using a full aperture. Any fainter stars with transferred magnitudes, using the new magnitude systems, will be secondary standards, while Vega will be my primary standard. The secondary standard stars will be located in the vicinity of Ceres during its 2014 opposition, RA/DE = 13:52+03. The star 83-Tau Virginis is a mere 3 degrees away from Ceres at opposition (Mar 20), and it’s a main sequence star with a similar spectral type to Vega (A3 vs A0). Sun-like stars will also be important to include in the secondary standard list because I’ll need to verify the accuracy of star color corrections. The shape of flux vs wavelength for the sun-like secondary standards should resemble the solar flux spectrum, so this can serve as an internal consistency check. I’ve already identified a half dozen sunlike stars near the Ceres opposition location, using the criteria B-V and g’-i’ colors must both be within 0.05 mag of the sun’s colors. Fig. 7 is a color/color scatter plot showing that ~ 13% of stars meet this criterion.

A second task, which can be done in parallel with the first one, is to calculate pass-band shapes for my hardware and then calculate the fluxes that each filter would intercept from Vega (above the atmosphere) for conversion to photo-electrons. This will permit the creation of a set of equations, one for each filter i (analogous to eqn. (1), above):

Another check for the C_i coefficients will be to compare SEDs (spectral energy distributions) for the secondary sun-like stars with their known BVRcIc and u’g’r’i’z’ SEDs. The SED spectrum derived from eqn (2) should overlap with the SED from known magnitudes (APASS, etc). When this can be demonstrated then it is likely that any measurements of Ceres, or any asteroid, using the eqn (2) C_i set and the FC filters should be correct.

This site opened: 2015.01.14 by Bruce L. Gary (B L G A R Y at u m i c h dot e d u).