On the phenomenon of the blue Sun

. This study examines the cause of the blue colour of the Sun as observed after the eruption of Krakatoa in 1883 as well as other volcanic eruptions or massive forest ﬁre events. Aerosol particles, e.g., volcanic ash or products of biomass burning are believed to be able to modify the spectral distribution of transmitted solar radiation making it appear blue or green to a human observer. Previous studies already showed that narrow aerosol particle size distributions with radii on the order of about 500 nm can 5 lead to anomalous scattering, i.e., scattering cross sections increasing with increasing wavelength in the visible spectral range. In this work we treat the effect of Rayleigh scattering on the shape of the transmitted solar spectrum correctly employing radiative transfer (RT) simulations with the SCIATRAN RT-model. The colour associated with solar transmission spectra is determined based on the CIE colour matching functions and CIE chromaticity values. It is shown that a blue Sun can be simulated for aerosol optical depths (at 550 nm) of about τ = 0 . 5 (or higher) if Rayleigh scattering is taken into account. 10 Without considering Rayleigh scattering – as in most of the previous studies – a blue Sun is in principle produced with aerosol optical depths as low as about τ = 0 . 1 (at 550 nm), if the aerosol particle size distribution is chosen to maximize anomalous scattering in the visible spectral range. It is demonstrated that Rayleigh scattering – as expected – has a strong impact on the transmission spectrum, particularly at low solar elevation angles, and needs to be considered for a correct determination of the perceived colour of the Sun. We also test the hypothesis that the blue Sun after the eruption of Krakatoa was caused by 15 large abundances of water vapour in the atmosphere, as proposed in earlier studies. In addition, we present a case study on a particularly noteworthy blue-Sun-event in the past, i.e., the one related to the large Canadian forest ﬁres in September and observational parameters (SZA), Rayleigh scattering was found to have a signiﬁcant impact on the simulated colour of the Sun. Anomalous 295 extinction is more pronounced for narrow aerosol particle size distributions (given the right aerosol size), but the investigations based on the aerosol optical depth retrieved by Wilson (1951) demonstrated that a relatively weak increase in aerosol extinction with wavelength in the visible spectral range also allows simulating a blue Sun. This implies that blue Suns can also be produced for somewhat broader size distributions. Our results indicate that an aerosol optical depth of about 0 . 5 is sufﬁcient to explain a blue Sun, if the aerosol size parameters are chosen in an optimal way. If Rayleigh scattering is neglected – which 300 is unrealistic, but has been done in many earlier studies – an aerosol optical depth of only 0 . 1 is sufﬁcient to yield a blue Sun. Absorption by water vapour – also proposed as a potential explanation for the blue Sun phenomenon – was shown to be a very unlikely explanation. This study conﬁrms the result of several other studies, that anomalous extinction by aerosol particles of a speciﬁc size and size distribution allows explaining the occurrence of blue or green Suns. However, Rayleigh scattering should not be neglected.

We first describe the Mie simulations and the radiative transfer model employed, followed by a summary of the colour modelling approach used in this study.

Radiative transfer simulation: Mie scattering calculations and SCIATRAN
In order to model the transmitted solar radiation reaching the Earth's surface, first extinction coefficients of the aerosol particles were calculated using Mie theory and later implemented as an aerosol layer in the radiative transfer model SCIATRAN. In order to perform the Mie scattering calculations several assumptions have been made beforehand. We assume that the aerosol particle size distribution can be described by a mono-modal log-normal distribution: Here, N 0 is the total particle number density, r m the median radius, r the particle radius and S represents the geometric standard deviation of the distribution, a measure of the distribution width (Grainger, 2017). 70 Mie simulations were carried out for various combinations of distribution width S, refractive index n r and median radius r m , as described in more detail below. The Mie scattering calculations were carried out using routines from the Earth Observation Data Group (EODG) of the University of Oxford, available as downloads (for the software IDL) on their website (http://eodg. atm.ox.ac.uk/MIE/#intro, last checked: September 4, 2020). The choice of input values for the refractive indices and the median radii were guided by previous studies and literature (e.g., Ehlers et al., 2014;Kravitz et al., 2012;Penndorf, 1953). In this study 75 we examined real parts of the refractive index n r between 1.3 and 1.5 and particle radii ranging from 25 to 3000 nm. The imaginary part of the refractive index was set to zero (n i = 0).
Having calculated the extinction coefficients of the aerosol particles possibly leading to a blue Sun, the impact of these aerosols and of air molecules on the transmission of solar radiation through the atmosphere was determined using the radiative transfer model SCIATRAN (Rozanov et al., 2014). SCIATRAN was developed by the Institute of Environmental Physics (IUP) 80 of the University of Bremen for the interpretation of remote sensing measurements in the UV-visible-NIR-SWIR spectral range.
In the transmission modelling mode, SCIATRAN can be used to simulate the atmospheric transmission of sunlight reaching the Earth's surface. In addition to the light scattering and absorption due to atmospheric gases, an aerosol layer can be included in the calculations. This layer is represented by the calculated extinction coefficient spectra covering a certain altitude range.
The viewing geometry is set up with SCIATRAN in a way that the imaginary observer on the Earth surface is looking directly 85 into the Sun. The solar zenith angle (SZA) as well as the optical depth have to be considered as possible influencing factors.  (Simon, 2008). The colour matching functions x(λ), y(λ) and z(λ) are displayed in Fig. 1. Figure 1. CIE colour matching functions of the human eye for the CIE 1931 2 • -standard observer, after Judd (1951) and Vos (1978) (http://cvrl.ioo.ucl.ac.uk/cmfs.htm, last checked: September 4, 2020).
Combining these sensitivity curves with the incoming light's spectral distribution I(λ), the so-called XYZ tristimulus values can be calculated (see eqns. 2 -4), as described by the CIE (2004).
In the current study we employ the CIE chromaticity values (x and y) determined from the XYZ tristimulus values to characterize the colour associated with a given transmission spectrum. The chromaticity values are determined in the following way: and are displayed in a 2-D x-y plot.
In addition we convert the tristimulus values to sRGB (standard RGB), which can be directly used to display colours in IDL

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(Interactive Data Language), the software used to prepare all Figures shown in this study. The conversion was done according to the International Color Consortium (ICC) (see Stokes et al., 1996) with a linear matrix transformation and a non-linear gamma correction. The colours of the monochromatic colour arc -shown in the panels of the right column of Fig. 3, which will be discussed in detail below -as well as the transmitted solar spectra were determined this way. The procedure was verified using several external routines. Different websites offer colour value calculators and the display of colours according 115 to their sRGB-values (e.g., http://www.brucelindbloom.com, last checked: September 4, 2020); https://www.w3schools.com/ colors/colors_rgb.asp, last checked: September 4, 2020).

Results
As mentioned above, in many previous studies on the origin of a blue moon or Sun, enhanced scattering of red light is provided as an explanation of the phenomenon. Usually, only scattering by the aerosol particles is considered, and Rayleigh scattering 120 by air molecules is neglected. We start our considerations based on this assumption (Section 3.1) and then include a correct treatment of Rayleigh scattering based on SCIATRAN radiative transfer simulations in section 3.2.

Results without Rayleigh scattering and gaseous absorption
Previous studies, e.g., Penndorf (1953) suggest that the particle population leading to the blueing effect has a relatively narrow particle size distribution and a refractive index (real part n r ) of 1.3 to 1.5. Figure   As can be seen in Fig. 2, maximum anomalous extinction is obtained under the assumptions made for median radii in the 400

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-700 nm range, with the specific values depending on the assumed refractive index. Two features of the plots are particularly noteworthy. First, the broader the size distribution, the less pronounced the anomalous extinction is (compare top and bottom row of Fig. 2). Secondly, the median radius producing the maximum anomalous extinction depends on the refractive index, decreasing from about 700 nm to about 450 nm as the real part of the refractive index increases from 1.3 to 1.5. In other words, depending on the refractive index, particles of a slightly different size will more easily produce a blue Sun. It is also noticeable 135 that a lower refractive index leads to a larger particle radius range with stronger anomalous scattering, making the blue Sun phenomenon more probable. Unfortunately, no aerosol samples have been taken and examined from past events, therefore the true value of the refractive index of the particles producing a blue Sun is unknown. Penndorf (1953)  whitish-yellowish for small SZAs to orange for SZA = 90 • , as expected. This is easily explained by the λ −4 dependence of Rayleigh scattering (Strutt, 1871), i.e., radiation with shorter wavelength (blue light) is scattered more strongly than radiation with longer wavelengths. The pathlength of light through the atmosphere increases with increasing SZA, and the impact of Rayleigh scattering also increases, leading to an orange sunset which can be seen in the lowest panel of Fig. 3 where the Sun is at the horizon.

Simulation of a blue Sun
For the Sun to appear blue, the spectral irradiance values and therefore the transmission must be higher at shorter wavelengths relative to longer wavelengths in the visible spectral range. To reach this, in the following an aerosol layer was inserted into the model atmosphere and the radiative transfer calculations were repeated. Previous studies as well as the contour plot in Fig. 2 suggest that particles with a median radius of 400 -700 nm (depending on the refractive index) are able to cause a blue Sun. In 170 this study different particle sizes in the range from 25 to 3000 nm have been tested and Rayleigh scattering taken into account.   For the largest SZAs shown, the Sun assumes a deep blue colour. For increasing optical depth, the blueing effect is more pronounced for all SZAs, the transmission is further reduced and for an optical depth of, e.g., τ = 2 the Sun assumes an indigo colour for SZAs of 80 • and 90 • (not shown). Another aspect that must be considered is that colours may not be perceived if the transmitted solar radiation is too weak. According to Horvath et al. (1994), colour perception will not measurements are only available in the spectral range from 381 nm to 633 nm. Apparently, the optical depth increases from 450 to 650 nm, but the relative variation is with about 10% not very large. Using the optical depth spectrum we performed radiative transfer simulations for the solar zenith angle (SZA = 71 • ) of the blue Sun measurement by Wilson (1951) with SCIATRAN in order to test, whether this optical depth spectrum does indeed produce a blue Sun. The aerosol layer was assumed to occur 225 in 10 -12 km altitude, following Penndorf (1953). The vertical aerosol optical depth in this case was τ ≈ 3.2 at 550 nm based on the slant optical depth reported by Wilson (1951) and converted to vertical optical depth using SCIATRAN. The results are shown in Fig. 8 including Rayleigh scattering (top row) and without Rayleigh scattering (bottom row). A light blue colour of the Sun is already visible in the top row, i.e., including Rayleigh scattering, but the effect is more pronounced without Rayleigh the same or very similar systematic effects that largely cancelled out when taking the ratio of the two spectra. The results also demonstrate that a blue Sun can be reproduced even with a small relative spectral variation of the aerosol optical depth in the  13 https://doi.org/10.5194/cp-2020-117 Preprint. Discussion started: 22 September 2020 c Author(s) 2020. CC BY 4.0 License.

The role of absorption by H 2 O
Several observers took measurements of the spectral distribution of transmitted solar radiation during the green or blue Sun events in the aftermath of the 1883 eruption of Krakatoa (Symons et al., 1888). The red part of the spectrum was found to be markedly suppressed. The Fraunhofer A line (at 760 nm, caused by O 2 ) could not be observed during the events and the 245 Fraunhofer B line (at 687 nm, also caused by O 2 ) was difficult to observe and was not discernible for SZAs approaching 90 • (Symons et al., 1888, p. 212). The observers reported unusually strong absorption by "water" (or water vapour), and absorption by water vapour was also discussed as a potential explanation of the blue Suns after the Krakatoa eruption (Symons et al., 1888).
The effect of H 2 O vapor absorption on the transmission spectra and the resulting colour of the Sun was also tested using 250 SCIATRAN. For tropospheric H 2 O mixing ratios of 4% -which is often considered the upper limit in the troposphere -and an aerosol-free atmosphere, a blue colour of the Sun can not be reproduced (results not shown, but they are close to the results for the aerosol-free case shown in Fig. 3 in the atmosphere. One may expect that high H 2 O mixing ratios lead to smaller aerosol optical depths required to produce a blue colour of the Sun, because H 2 O absorbs strongly in the red part of the visible spectrum.

Discussion
In this study we investigated the influence of aerosol particles with different size distributions and refractive indices on the colour of the Sun, as perceived by a ground-based human observer. Different factors were examined that may be able to evoke 260 a shift in the transmitted solar spectral distribution towards a higher intensity at shorter wavelengths in the blue region of the visible spectrum.
It is obvious that a suppression of red light -relative to the shorter wavelengths -is the key prerequisite for producing a blue Sun or a blue moon. In principle this may be caused by molecular extinction or extinction by aerosol particles. Molecular extinction can be excluded as a cause (see also section 3.7). Extinction by aerosol particles may in principle be caused by Helens eruption in 1980. The imaginary part decreases with increasing wavelength in the 300 nm -700 nm spectral range, and so does the absorption cross section (or coefficient). Patterson et al. (1983) reported similar measurements of the ash emitted part of the refractive index of five different rock types. In some cases the imaginary part n i of the refractive index was found to be increasing with increasing wavelength in the visible spectral range. However, it has to be kept in mind that the absorption coefficient is proportional to the ratio of n i and the wavelength k ext ∝ ni λ . This again leads to k ext decreasing with increasing wavelength. To summarize, absorption by aerosol particles appears to be an unlikely cause of the blue Sun phenomenon.
The results presented so far were mostly for an aerosol layer in 6 -8 km altitude, which is certainly arbitrary. Simulations 280 were also carried out for aerosol layers in 2 -4 km and 10 -12 km altitude. We generally found little dependence of the results on layer altitude, except for SZAs approaching 90 • . At these large SZAs the blueing effect on the Sun gets more pronounced the lower the aerosol layer is (results not shown). It is beyond the scope of this study to present all cases in detail.
It should be pointed out that our results confirm the basic notion of previous studies, that anomalous scattering in the visible spectral range caused by a suitably sized aerosol population is able to explain the occurrence of a blue or green Sun. A narrow 285 aerosol particle size distribution with an appropriate mean or median radius will more easily lead to blue Suns compared to a broader size distribution. One general question is, how a narrow size distribution can be achieved in the real atmosphere. One possibility could be a size separation by sedimentation, perhaps accompanied by different wind directions at different altitudes, as already suggested by Gelbke (1951). The right combination of conditions for the occurrence of a blue Sun can be expected to arise only rarely, which provides an explanation for the infrequent observations of a blue Sun.

Conclusions
We investigated possible reasons for the occurrence of a blue Sun, including anomalous scattering by suitably sized aerosols, aerosol absorption and absorption by water vapour. This study included -for the first time to our best knowledge -a comprehensive treatment of Rayleigh scattering in simulations of a blue Sun. Depending on the specific aerosol and observational parameters (SZA), Rayleigh scattering was found to have a significant impact on the simulated colour of the Sun. Anomalous 295 extinction is more pronounced for narrow aerosol particle size distributions (given the right aerosol size), but the investigations based on the aerosol optical depth retrieved by Wilson (1951) demonstrated that a relatively weak increase in aerosol extinction with wavelength in the visible spectral range also allows simulating a blue Sun. This implies that blue Suns can also be produced for somewhat broader size distributions. Our results indicate that an aerosol optical depth of about 0.5 is sufficient to explain a blue Sun, if the aerosol size parameters are chosen in an optimal way. If Rayleigh scattering is neglected -which 300 is unrealistic, but has been done in many earlier studies -an aerosol optical depth of only 0.1 is sufficient to yield a blue Sun.
Absorption by water vapour -also proposed as a potential explanation for the blue Sun phenomenon -was shown to be a very unlikely explanation. This study confirms the result of several other studies, that anomalous extinction by aerosol particles of a specific size and size distribution allows explaining the occurrence of blue or green Suns. However, Rayleigh scattering should not be neglected.