A photometric system for the Foveon camera

To make comparisons easier with the UBVRI standard photometric system, we have decided to set the zero point of the red, green and blue magnitude scales of the Foveon camera such that the synthetic magnitudes of Vega (alpha Lyr.) are also 0.03 in these scales (0.03 is the magnitude of Vega in the V passband as reported by the Hipparcos catalog), thus making the non-standard photometric system we are creating for the Foveon camera also a VEGAMAG-type system (BESSELL; MURPHY, 2012, Section 7). To compute synthetic magnitudes of a star in this photometric system from its spectrum, the procedure described in Section C.3 of the appendixes can be followed. The photometric system defined here is also called a natural (or instrumental) photometric system for the Foveon camera, since magnitude measurements performed by this camera will be in that system, unless a transformation is applied to these magnitudes to transform them to another photometric system.

To obtain the camera response in the red, green and blue channels, we need to consider both the detector response and optics transmission. Figure 3.9 presents the quantum efficiencies in each band, plus the combined quantum efficiency of the image sensor of the Foveon camera.

3 Our b magnitudes should not be confused with the b magnitudes of the Strömgren uvby system, which are defined with a much narrower spectral band.

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Unfortunately, the transmission spectrum of the optics used in the test was unknown to the author and data provided for the image sensor used in the camera spanned only the 400nm-1000nm range. Not having the means to measure the transmittance of the optics in the range 350nm-1000nm and image sensor response in the near UV in time, the author of this thesis considered a flat transmission of the optics in the 400nm-1000nm range and no transmission outside this range. The spectral response and computed color indexes for black-bodies will not be exactly equal to those of the true photometric system of the camera used in the tests, but will be close enough for a preliminary analysis. In a series of computer simulations performed by the author, considering the transmission spectra of similar lenses to that used in the test (NAGASAWA et al., 2016) and assuming a set of probable values of image sensor response in the near-UV (obtained by extrapolation), he found that the computed color indexes for black-bodies would typically stay within 20% of the values derived here.

Figure 3.9 – Quantum efficiency of the Foveon X3 F13 image sensor.

Source: Drawn by the author using data provided by Gilblom (2017, personal communication).

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Multiplying the quantum efficiencies by the assumed optics transmission (flat in the 400nm-1000nm range and none outside this range) and by the wavelength (to convert from photon units to responsivity in energy units), somewhat rough approximations to the response functions of the Foveon camera were found and have been adopted in this work. These are plotted in Figure 3.10 as solid lines, together with the response functions of the standard B, V and I bands of the Johnson-Cousins system (BESSELL; MURPHY, 2012) plotted as dashed lines.

Figure 3.10 – Energy response functions adopted in this work

Source: Drawn by the author.

The author of this thesis could have adopted, instead, response functions that extends into the near UV (wavelengths shorter than 400 nm) and deeper in the IR (for wavelengths longer than 1000 nm) following a smooth extrapolation curve.

This would give curves closer to reality. However not having enough data to accurately predict how the detector response functions would extend into the near UV and considering that there might be significant differences in the transmission curve even among similar optics, it was considered to be safer to adopt an optics transmission that is zero outside the 400nm-1000nm range and flat within this

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range. At least this makes it very evident the limitations in the data used to derive the adopted response functions, since no real optical system presents abrupt, stepwise changes in their spectral response functions as the ones we have adopted. The unphysical assumption that there is no response for wavelengths shorter than 400 nm should affect mostly computations in the blue band for bluer (hotter) stars. The assumption of no response for wavelengths longer than 1000 nm is not so severe, as the response in all three bands is already very low at 1000 nm. The assumption of no response for wavelengths longer than 1000 nm also partially compensates the small drop in the infrared transmittance which was observed by Nagawasa et al. (2016) for some similar optics.

Using a procedure that will be described in more detail in Section 7.2.5, the relations between black-body temperatures with magnitudes and color indexes were found, as shown in Figure 3.11. In this figure, the horizontal axis gives the multiplicative inverse of the temperature, considering black-body temperatures in the range of 250,000 K (left edge of the plots) to 3000 K (right edges).

From Figure 3.11.a we can see that for black-bodies, the b magnitudes in the Foveon camera system are similar to the Johnson’s V magnitude, while the g and r magnitudes are intermediate between the V and I standard magnitudes. The magnitudes given are the apparent magnitudes of black-bodies for a hypothetical observer situated on its surface. Figure 3.11.b gives the relation between reciprocal of temperature and the color indexes. The higher the slope of these curves, the more sensitive is the color index to changes in the temperature (or more accurately, the reciprocal of the temperature) of the black-body. This can be seen better in Figure 3.12 which gives the derivative of the color indexes with the reciprocal of the temperature.

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Figure 3.11 – Magnitudes and color indexes for black-bodies versus reciprocal of the temperature

(a) (b)

Source: Drawn by the author.

A color index that varies little with temperature or other relevant physical characteristic of stars is not very useful for stellar identification, unless measurements in that color index can be made with a very low uncertainty.

Assuming that stars can be approximated as black-bodies, from Figure 3.12 we can see that the best color index for identifying different stars among those studied would probably be the V−I color index. In the photometric system of the Foveon camera, the best color index for stellar identification seems to be the b-r color index, which has about the same discrimination power of the standard B−V color index, except for cooler stars with temperature lower than 5000 K (1/T >

2∙10^-4 K^-1), where it is lower. Overall, for making comparisons between these two color indexes possible, we could say that the discrimination power of the b-r color index is roughly 0.9 times the discrimination power of the B−V color index.

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Figure 3.12 – Variation of color indexes for black-bodies with the reciprocal of the temperature

Source: Drawn by the author.

If we approximate stellar spectra with the spectra of black-bodies, only one color index is needed, since from one color index, the temperature of the black-body and any other color index can be obtained, as will be explained in Chapter 7.

Table 3.4 shows that the measurement error in the b-g and g-r color indexes tend to be smaller than the measurement error in the b-r color index. However, analyzing this data together with Figure 3.12 it can be seen that the b-r color index would be still a better choice for stellar identification than the b-g and g-r color indexes. The reason is that the b-r color index is twice as much sensitive for temperature variations among black-bodies than the b-g and g-r color indexes.

For the b-g and g-r color indexes to outperform the b-r color index, their measurement uncertainties should be less than half the measurement uncertainty of the b-r color index.

It is true that the spectra of stars are not exactly equal to black-bodies spectra, as exemplified in Section 7.3.2. However, the spectra of many stars can be

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approximated by the spectra of black-bodies to a good precision (HOLLOW, 2006), making the black-body model suitable for a preliminary analysis.