In this Appendix, we present all scattering calculations and results used in Section 5. Scattering in the atmosphere is a combination of Rayleigh scattering by molecules and Mie scattering by particulates. The scattering due to these two components can be dealt with individually. We begin by describing a general model for the scattering in a spherical atmosphere and then discuss the specific parameters needed to calculate Rayleigh and Mie scattering affecting observations from Las Campanas Observatory. Finally, we present the predicted contribution of scattered light to the program observations analyzed in this paper. We address zodiacal light and the integrated starlight as scattering sources separately.
The first calculations of radiative transfer in a Rayleigh scattering atmosphere were published by Chandresekar in 1950. Since then, a number of authors have published radiative transfer calculations addressing Rayleigh and Mie scattering in a curved atmosphere (e.g. Sekera 1952, Sekera & Ashburn 1953, and Ashburn 1954), and the effects of multiple-scattering (Dave 1964, de Bary & Bullrich 1964, and de Bary 1964). Careful measurements of zodiacal light over the sky and intensity distributions of the daytime sky have empirically demonstrated the accuracy of those calculations (e.g. Elterman 1966; Green, Deepak, & Lipofsky 1971; Weinberg 1964, Dumont 1965).
To calculate the atmospheric scattering affecting the observations described in this Paper, we begin by adopting the scattering geometry and coordinate system definitions used by Wolstencroft & van Breda (1967, hereafter WvB67), illustrated in Figure A1: (A, )_{O} and (A', )_{X} are azimuth/zenith-distance coordinate systems centered on the observer at O and a generic point, X, along the line of sight, respectively. The problem is then to calculate the brightness observed at O, along the line of sight (A, )_{O} = (A_{0}, z_{0})_{O}.
Following WvB67, scattering occurs at the point X for radiation which entered the atmosphere at the point N from the direction N', given by (A, )_{O} or (A', )_{X}. The light arriving at X from N' can be expressed as
(A1) |
in which the above-the-atmosphere source has flux L(A', ), light is attenuated by e^{-Cext() h1(,)} as it travels along NX, and s is the distance along that path. Attenuation is a function of C_{ext}(), the extinction cross section of the scattering particles in cm^{2}, and of h_{1}(, ), the effective column density of particles along the line of sight. The effective column density is defined by the local zenith angle, , and the distance, , which defines the point X relative to the center of the Earth (see Figure A1):
(A2) |
in which s_{max}(, ) is the distance from X to the top of the atmosphere at N, and n(') is the atmospheric number density of molecules in cm^{-3} as a function of distance from the center of the Earth, ', and as a function of the distance s' along the line XN.
The light scattered towards the observer from X is then
(A3) |
in which P() is the scattering phase function and I_{X}, the flux arriving at point X is given in Equation A1. Finally, the scattered light is further attenuated by the factor e^{-Cext()h2(z0,)}, in which
(A4) |
The total flux scattered into the line of sight (z_{0}, A_{0}) from sources distributed over entire visible hemisphere of the sky is then
(A5) |
The visible sky at the point X dips below the observer's horizon at large values of s. Hence, the limit of the integral over , is greater than /2 by the value f () = cos^{-1}(R / ) 14°, where R is the radius of the Earth (6371 km). The equations needed to change variables between (A',) and (z_{0}, A_{0}) are given WvB67.
The phase function for Rayleigh scattering is P() = 1 + cos^{2}(). The atmospheric density is given by the standard barometric formula n() = n_{0}e^{-H/H0}, where H is the altitude above sea level (H = - R), the scale height is H_{0} = 7.99km, the density at sea level is n_{0} = 2.67 × 10^{19} cm^{-3} , and the effective scattering cross-section for air is C_{scat} = 7.78 × 10^{-27}( / 4600Å)^{-4}cm^{2} (see Schubert & Walterscheid 1999 and van de Hulst 1952). For molecules in the atmosphere, extinction is entirely due to scattering, so that C_{ext} = C_{scat}. Atmospheric extinction due to Rayleigh scattering is then ^{R}() = C_{ext}() _{R}^{} n() d. For the duPont telescope at Las Campanas, which is at an altitude of 2.28 km, the expected Rayleigh extinction is ^{R}(4600Å) = 0.12.
The phase function for Mie scattering by particulates in the atmosphere depends on the distribution of particle sizes, and must be empirically determined. We adopt the phase function measured by Green, Deepak & Lipofsky (1971) from their complete analysis of the Mie (particulate) scattering and Rayleigh (molecular) scattering components of the atmosphere based on the scattering of sunlight. Their results are in good agreement with theoretical scattering models and other estimates of the size-distribution of particles in the atmosphere and have a negligible dependence on wavelength for our purposes (see Elterman 1966, and Deepak & Green 1970). The scattering and extinction coefficients for particulate scattering are a function of the size distribution of particles and vary with time and geography. The extinction due to Mie scattering can, however, be inferred from the observed extinction for a point source and the calculated Rayleigh extinction coefficient: ^{M} = _{obs} - ^{R} ~ 0.05 at 4500Å (see Figure A2). This value is in good agreement with estimates for Tenerife by Dumont (1965) and for Haleakala by Weinberg (1964). This is not surprising as our observed _{obs}() is consistent with the CTIO curves (Baldwin & Stone 1984, Stone & Baldwin 1983), and ^{R}() is simply a function of the atmospheric density.
Unlike the case for molecules, the attenuation caused by particulates is not entirely due to scattering. Staude (1975) adopts values of C_{scat} = 4.47 × 10^{-11} cm^{2} and C_{ext} = 7.53 × 10^{-11} cm^{2} for dry air at 4200Å. With a sea level density of n_{0} = 1.11 × 10^{4}cm^{3}, and a distribution scale height of only h_{0} = 1.2 km, these parameters give ^{M} ~ 0.01 at 2 km. We have scaled C_{scat} and C_{ext} to give values consistent with our observed value of ^{M}. Scaling H_{0} or assuming a different value of H would have the same effect on I^{M}_{scat}(, ZL).
The scattering model discussed above describes a single scattering event. However, multiple scattering events become significant for scattering angles 30° (de Bary 1964, de Bary & Bullrich 1964). Consequently, we apply a multiple scattering correction for Rayleigh scattering which is adopted from Dave (1964) and plotted in Figure A3. The correction factor plotted in Figure A3 is simply the factor by which the intensity of scattered light increases over the single-scattering case. Multiple scattering does not occur due to particulates (Mie case) because of the small scattering angles which dominate that process and very small values of ^{M}().
Figure A3. Correction factor for multiple scattering, F_{MS}, as a function of the Rayleigh extinction. |
To confirm the accuracy of our calculations, we checked our scattering model against published results of Staude (1975), WvB67, and Ashburn (1954) for a uniform, sky-filling source of unit flux. We find that our results are consistent to within 4% for zenith angles z 80° before the multiple scattering correction is applied. (WvB67 predates evidence for the effects of multiple scattering, and Staude adopts the same corrections used here.) The uncertainty in the multiple scattering correction is roughly 4-7%, increasing with larger values of .
Using the expressions above, we calculate the scattered light flux, I_{scat}(), resulting from Mie scattering by particulates and Rayleigh scattering by molecules throughout the nights of our observations. The results depend explicitly on the absolute position of the Sun and the Galactic center relative to the observatory and relative to the target field. In the following sections, we consider the cases of zodiacal light (ZL) and integrated starlight (ISL) as the extra-terrestrial source of flux separately.
To calculate the scattered zodiacal light along the line of sight of our observations, we adopt values for the zodiacal flux given in Leinert et al. (1998), which are taken from Levasseur-Regourd & Dumont (1980) with values at elongations < 30° added from space-based observations. These ZL values are above-the-atmosphere fluxes and are in excellent agreement with later space-based results (see Leinert et al. 1981). To obtain a smooth flux distribution of ZL on the sky (see Figure A4), we use the spherical interpolation method developed by Renka (1997).
In Figure A5, we show the integration pattern in (zenith angle) and A' (azimuth) used to calculate I_{scat}(, ZL) (Equation A6) at the indicated times during the nights of our observations. Actual calculations were done with twice the number of integration points shown in the figures. The visible part of the sky (shown by the integration pattern) is at least 30 degrees from the Sun during our observations.
As a technical detail, we have made the simplifying assumption that the spectral shape of the ZL over the visible hemisphere is uniform. That is, only the mean flux of the ZL changes. Although there are variations in the color (defined in Equation 1) of the ZL over the range 3900-5100Å from = 30° to = 180°, the total change is empirically less than ~ 8% and our target is in the center of the expected color range (e.g., Frey et al. 1974, Leinert et al. 1981). We have run trail scattering models in which we change the flux with over the sky by ± 4%, and we find that the effect on the predicted scattered flux is changed by 0.4% at airmasses higher than 1.6, and 0.2% at the lowest airmass. In other words, by ignoring the color variation in ZL over the sky, the scattered light model will be wrong by 0.2% at 3900Å relative to the value at 5100Å, or ± 0.1% over the range 3900-5100Å for our observing situation (positions of the Sun relative to the target and the horizon).
Figure A7. Same as Figure A6, but here we plot the contribution of Mie scattered zodiacal light along the line of sight. Again, the scattered light flux is plotted as a fraction of the above-the-atmosphere zodiacal light flux in our target field. |
Figure A8. Same as Figure A6, but here each line indicates I^{R}_{scat}(, ZL) as a function of wavelength at discrete times. Scattering is maximized at high airmass (far west of zenith), and minimized slightly west of zenith, when the airmass is still relatively low and the Sun has had time to set far enough that highest flux regions of zodiacal light are no longer in the visible hemisphere of the sky. |
Figure A9. Same as Figure A8, but showing the Mie scattered zodiacal light. |
The predicted Rayleigh and Mie scattering flux of ZL, I^{R}_{scat} and I^{M}_{scat}, respectively, along the line of sight to our target field throughout our observations is shown in Figures A6 - A9. In those Figures, we show the scattered light as a function of the above-the-atmosphere ZL flux in target field at ( = 3.00 h, = - 20.18 d), I_{(3h, - 20d)}(ZL). This removes the spectrum of the ZL and highlights the wavelength dependence of I_{scat}. The predicted scattered flux is not symmetric about the zenith because the distribution of ZL over the sky is not symmetric: the scattered light will be smaller at the same airmass if the Sun is further below the horizon, i.e. the middle of the night. The scattered flux is therefore minimized near UT ~ 4, when the field is still at low airmass and the brightest regions of the ZL are below the horizon. In Figure A10, we show the total combined effect of the atmosphere on the ZL flux received from the target field:
(A6) |
Figure A10. The net effect of the atmosphere on the observed spectrum of zodiacal light. I_{net}(ZL) is the flux received from the target field plus the scattered light coming from the entire visible hemisphere of the sky: I_{net}(, ZL) = I_{scat}(, ZL) - I_{(3h, - 20d)}(ZL) (1 - e^{- (,obs)}). I_{net}(, ZL) is plotted as a function of wavelength at discrete times, as labeled. Flux units are the same as in Figures A6 - A9. |
Finally, from I_{net}(, ZL) we can calculate an effective extinction for the ZL from our target field at the specific times at which our observations occurred. The effective extinction is defined by the equation
(A7) |
The effective extinction is plotted in Figure A11, and is specific to our target field, times of observation, observed extinction, geographic latitude and longitude, and altitude.
Figure A11. Each line shows the effective extinction for the zodiacal light as a function of wavelength for our observations at the indicated UT. The effective extinction corresponds to the net loss of light relative to the above-the-atmosphere flux of the zodiacal light in our field of view (see Figure A10). For comparison with the total observed extinction derived from standard stars see Figure 4. |
The result which is applied to our ZL measurement from the modeling discussed here is _{eff}(, t), which corresponds to I_{net}(, t,, ZL) rather than I_{scat}(, t,, ZL). The virtue of this approach is that the absolute flux accuracy of the adopted ZL over the sky (Fig. A4) does not affect our results; only the accuracy of the relative flux distribution over the sky matters. In the regions of the sky which dominate the scattering for our observations (solar elongations of > 30°), the relative flux errors for the ZL over the visible hemisphere of the sky are 5% over large areas (> 30°), and better on small scales. Such errors will propagate into final measurement of the ZL at the level of < 1% at high airmass, and < 0.4% at low airmass. Nevertheless, we note that the above-the-atmosphere value of the ZL from Levasseur-Regourd & Dumont (1980) agrees with our measurement in our target field to within 2%.
To evaluate the accuracy of our calculated values of I_{net}(ZL) (see Equation A7), we estimate that the uncertainty in our scattering calculations is 8% at the low zenith angles (< 30°) where the bulk of our observations occur. This estimate is based on the comparisons between scattering models and atmospheric measurements presented in Green et al. (1971), Dave (1964), and Staude (1975), and is consistent with the uncertainties discussed in WvB67, Ashburn (1954), and Sekera & Ashburn (1953). The time-weighted average of I_{scat}(ZL) over the course of our observations is ~ 0.15 × I_{(3h, - 20d)}(, ZL), so I_{scat}(ZL) has an uncertainty of 1.2% of the ZL flux in our target field. The uncertainty in the observed extinction is much less than 1% and adds negligibly to this error. See Section 6 for further discussion of the accuracies of the zodiacal light measurement.
We can also assess the accuracy of _{eff}() independently from our own data, as discussed in Section 5. Notice that I_{net}(ZL) changes with time in a way which is only weakly dependent on wavelength. A consistent solution for the ZL with both wavelength and airmass will be strong confirmation of the accuracy of the values for _{eff}(, t) calculated here.
Unlike the scattered ZL, the scattering which results from integrated starlight (ISL) must be incorporated into our analysis of the observed night sky spectrum as an absolute flux value. We must therefore first derive a spectrum for the ISL as a function of position over the sky. To do so, we have followed the method suggested by Mattila (1980a, 1980b), which we briefly summarize here.
The spatial and flux distribution of stars of all spectral types can be described by exponential distributions perpendicular to the Galactic plane (in the z direction) and narrow Gaussian distributions in intrinsic magnitude. The mean emission per pc^{3} from stars of type i as a function of distance from the Galactic plane, z, can be written as
(A8) |
in which D_{i}(0) is the number density of stars per cubic parsec in the plane, h_{i} is the scale height of the vertical distribution, and M_{i} is the mean absolute magnitude of the spectral type i. The observed flux is also attenuated by interstellar extinction, which can be expressed by a two-component extinction law characterized by a total extinction a_{0}() = a_{1}() + a_{2}(), with a_{1}(): a_{2}() in the ratio 1.84 : 0.62 (Neckel 1965). The z-dependence of extinction can be written as
(A9) |
for z given in parsecs (Neckel 1965, Neckel 1980). We find a good fit to the observed ISL by adopting standard values for a_{0}() (Zombeck 1990), scaled to a_{0}(V) = 1.5 mag kpc^{-1}.
In cylindrical coordinates, the flux per unit solid angle (ergs s^{-1} cm^{-2} sr^{-1} Å^{-1}) from stars fainter than m_{0} along the line of sight at Galactic latitude b can be expressed by the volume integral
(A10) |
in which r is the distance along the line of sight from the observer in parsecs and f_{i} is the spectral energy density of a star of type i in ergs s^{-1} cm^{-2} Å^{-1}. In the derivation of the above integral, the 1/4r^{2} loss of flux from each star along the line of sight has canceled with the r^{2} d in the volume integral, and we have changed variables using the relation r = z/sinb. The lower limit of integration is simply the distance modulus for stars of each type corresponding to the bright magnitude cut-off, m_{0}, so that z_{0} = 10^{0.2(m0 - Mi + 5)} in parsecs. Finally, the extinction from z to the observer is
(A11) |
Using 33 individual stellar types described by the parameters M_{i}, D_{i}, and h_{i} from Wainscoat et al. (1992), we obtain integrated spectra which agree with the observed star counts at V and B (Roach & Megill 1961, see also Allen 1973) to m_{0} = 6 V mag at |b| > 5° to better than 10%, which is more than adequate for our purposes. The spectral energy densities for each stellar type, f_{i}() were obtained from the STScI archive (Jacoby, Hunter, & Christian 1984) and have a resolution of roughly 4Å. We include stars by type with m_{0} < 6 V mag from the SAO star catalog by hand. We felt this was necessary as the statistical variation in stellar density on small scales around the solar neighborhood can have a significant impact on the accuracy of the model, while variations are apparently averaged out in stellar populations at large distances. In Figure A12, we plot the total integrated starlight (ISL) with no bright magnitude cut off at 0° < |b| < 90°. The total flux at |b| = 90° is roughly 20 × 10^{-9} ergs s^{-1} cm^{-2} sr^{-1} Å^{-1}, while the flux near the plane is as high as 300 × 10^{-9} ergs s^{-1} cm^{-2} sr^{-1} Å^{-1}. Interstellar extinction limits the ISL flux in the plane in our model, probably more than is appropriate. However the flux rises rapidly even 1° degree above the plane to more realistic values. The limited sky area at b = 0° precludes this from impacting the accuracy of the models.
Figure A12. The mean integrated starlight (ISL) spectra as a function of Galactic latitude (as labeled) produced by the model described Section A.3. |
Figure A13. Integrated starlight over the sky from star counts at V. Flux units are 1 × 10^{-9} ergs s^{-1} cm^{-2} sr^{-1} Å^{-1}. |
Figure A14. Integration pattern for the calculation of scattered ISL flux. Each plot is an Aitoff projection of the sky in Galactic coordinates. The center of the Galaxy is at the center of each plot (l = 0°, b = 0°). The Galactic plane within 30 degrees of the Galactic center is marked by the thick line. Our target field is indicated by the square. The Galactic coordinates of the zenith can be seen as the "bullseye" center of the integration pattern. The coordinates of the local horizon are shown by the edge of the integration pattern. The Galactic plane is running along the horizon at the start of the night (UT = 2.0 hr), and is perpendicular to the horizon at the end of the night (UT = 7.5 hr). The Galactic center is never above the horizon. See Figure A5 for further discussion. |
Figure A15 shows the total scattered ISL flux due to Rayleigh and Mie scattering which contributes to observations of our target field at the beginning of the observing night (UT = 2.0 hr). The total flux (~ 12 × 10^{-9} ergs s^{-1} cm^{-2} sr^{-1} Å^{-1}) is roughly 12% of the ZL flux above the atmosphere in our target field. The ISL flux in the observed sky spectrum will impact our measurement of the ZL flux only if the ISL spectrum has the same features as the solar (zodiacal light) spectrum. In the lower plot of Figure A15, we therefore plot ratio of the I_{scat}(, t, , ISL) to the solar spectrum, normalized at 4600Å. It is clear from this plot that the scattered ISL and solar spectra differ by 5-10% at > 4500Å, but by a factor of 3-5 in the strength of spectral features at less than 4500Å. In Figure A16, we plot the ratio of I_{scat}(, t, , ISL) throughout the night to I_{scat}(, t, , ISL) at UT = 2.0 hr. From this plot it is clear that strength of spectral features changes only very weakly throughout the night, by < 1% over the majority of the spectrum and by < 4% at 3900-4000Å (CaI H & K). The consistency of our ZL measurement (Section 6) over the full wavelength range 3900-5100Å is, therefore, a strong test of the accuracy of the predicted contribution of scattered ISL. As discussed in Section 5, the predicted scattered ISL flux is entirely consistent with our observations. See Section 5 for further discussion.
Obviously, the model we describe above makes no allowance for variation in the ISL with Galactic longitude. For comparison, we show in Figure A13 an Aitoff projection of the ISL from star counts over the sky, which shows that the ISL has only minor dependence on longitude at b > 20°. At lower latitudes where the variation is greater with longitude, the spectroscopic model we employ does give a good approximation to the average ISL. Because the contribution to the scattering comes from a wide spread in longitude (compare Figures A14 and A13), the mean value is adequate for our purposes. To test this, we ran simulations in which we maintained the mean ISL flux with latitude, but varied the ISL flux with longitude by ± 25%. We find that the total scattered ISL is affected by less than 9% throughout the night due to longitudinal variations around the mean.
The mean flux in our models for the ISL is consistent with the star counts of Roach & Megill (1961) to within ± 10% at both V and B. As in the previous section, we estimate that the uncertainty in our scattering calculations is 8% at the low zenith angles (< 30°) where the bulk of our observations occur. Combining these, we estimate the uncertainty in I_{scat}(, t,, ISL) to be 13%. As the relative mean strength of spectral features in starlight is 0.6-4% of the total ZL flux in our target field at the 4000 - 5200Å, this corresponds to an uncertainty of < 0.5%.
Any significant errors in our model, either in mean flux as might be caused by longitudinal variations in the ISL, or in the spectral energy distribution, would show up as inconsistencies in the solution for the ZL flux as a function of wavelength. Furthermore, errors would be worst at higher airmass, where Figure A14 shows that the low-galactic-latitude sky has a greater impact on the scattered ISL, the mean flux is greater, and the stellar-type mix is more sensitive. No such variations with wavelength are found, as we have discussed in Section 5. See Section 6 for further discussion of the accuracies of the zodiacal light measurement.