To model the sunlight scattered by NLC particles and transmitted to the Earth's surface, the Mie code implemented into the radiative transfer software SCIATRAN was used. This allows for the calculation of aerosol optical parameters by SCIATRAN and the simultaneous implementation as an aerosol layer at a certain height. The NLC particle size distribution was assumed to be a mono-modal log-normal distribution: 1n(r)=N02π⋅ln⁡(S)⋅r⋅exp⁡-(ln⁡r-ln⁡rm)22ln⁡2(S), where N0 is the total particle number density, rm the median radius, r the particle radius and S the geometric standard deviation of the distribution . The calculations were carried out for median radii ranging from 10 to 1000 nm and constant values for S=1.4 and N0=100 cm-3. Note that the vertical optical depth of the cloud layer is additionally specified, which leads to an adjustment of the value of N0. The input values were guided by previous studies and literature on this topic e.g.. In order to simulate the solar radiation scattered by aerosols and air molecules in a spherical atmosphere, while considering refraction effects for the direct solar beam and the scattered light, the “spher_scat” mode was used in SCIATRAN . SCIATRAN was developed by the Institute of Environmental Physics (IUP) of the University of Bremen as a forward model for the retrieval of atmospheric parameters from measurements with the SCIAMACHY instrument on ESA's Envisat spacecraft. More information on SCIATRAN can be found at https://www.iup.uni-bremen.de/sciatran/ (last access: 29 April 2022). The model output contains radiance values at different wavelengths. These data were multiplied by the solar spectrum incident on the Earth's atmosphere (SORCE data (Solar Radiation and Climate Experiment)) to obtain the resulting spectral distribution of the solar radiation scattered to an observer at the Earth's surface.

The colour corresponding to a given scattered solar spectrum was determined and displayed using a standard approach based on the CIE XYZ colour space established in 1931 e.g.. Using the CIE colour-matching functions x‾(λ), y‾(λ) and z‾(λ) after and , which quantify the spectral sensitivity of the three cone cells of the human eye, the CIE tristimulus values X, Y and Z are determined : 2X=k∫380 nm800 nmI(λ)x‾(λ)dλ,3Y=k∫380 nm800 nmI(λ)y‾(λ)dλ,4Z=k∫380 nm800 nmI(λ)z‾(λ)dλ, where I(λ) is the given radiance spectrum, and the normalising factor k is defined as 5k=100∫380 nm800 nmIachromatic(λ)y‾(λ)dλ, with Iachromatic(λ) as a reference spectrum with the colour impression of white. In the case of self-luminaries, k remains indeterminate. Based on the XYZ tristimulus values, the CIE chromaticity values x and y are calculated using 6x=XX+Y+Zy=YX+Y+Z. These chromaticity values characterise the colour independently of the brightness and are displayed in a 2-D plot, the so-called CIE chromaticity diagram or “gamut”. Furthermore, the XYZ tristimulus values were converted to sRGB (standard RGB), which can be used to display the colours in the programming software IDL (Interactive Data Language). More detailed information can be found in a previous paper by .

Figure  shows scattered solar spectra determined by multiplying the SORCE solar spectrum by the scattered radiance spectra simulated with SCIATRAN. The left panel of Fig.  includes an NLC with the following characteristics: rm=50 nm, S=1.4, vertical optical depth of τNLC=10-4, 1 km vertical extent and a centre altitude of zNLC=82 km. The right panel shows the background spectrum without NLCs. Both spectra correspond to a solar zenith angle (SZA) of 98∘, a solar azimuth angle (SAA) of 0∘ and a viewing zenith angle (VZA) of 65∘. Note that the radiances in the case with NLCs are about an order of magnitude larger than in the background case.

At twilight – i.e. the sun being below the horizon – NLCs appear to an observer on the Earth's surface with a bluish colour. This is determined by the absorption of solar radiation in the O3 Chappuis bands with maxima at 575 and 603 nm, filtering the longer wavelengths and the NLC particle size distribution parameters (here: rm=50 nm and S=1.4). So the scattered solar radiation including NLCs appears blue (left panel of Fig. ). Without NLCs (right panel of Fig. ), the spectrum also exhibits a peak at about 750 nm, resulting in a slightly different blue hue (Fig. ). Figure  shows a CIE chromaticity diagram with the chromaticity values x and y on the axes. The arc with the filled colour circles represents the positions of the spectral colours with the corresponding wavelengths in the x–y plane. The connecting line at the bottom of the arc cannot be represented by pure spectral colours and is called the ”line of purples”. The colours in the diagram are based on the conversion of the chromaticity values to sRGB as described in Sect. . The small “×” corresponds to the chromaticity values of the unattenuated solar spectrum. Simulations for other optical depths (in the range of 10-3 to 10-6) show only minor differences in the resulting colours, which is why for the sake of clarity a separate presentation is omitted.

Figures  and show solar scattering spectra (left column) with the resulting colour impression (right column) for an observer on the Earth's surface. The simulations were carried out for SZA=98∘; viewing zenith angles of 10, 20, 40, 60, and 80∘ (from top to bottom); and a solar azimuth angle of 0∘. The simulations differ in the assumed vertical optical depth of the NLCs: τNLC=10-4 (Fig. ) and τNLC=10-5 (Fig. ). Both sets of plots show the Chappuis bands of ozone, whose visibility decreases with increasing VZA (from top to bottom). Note that for the studies of the conditions of NLC illumination by the Sun and the effects of ozone, the local solar zenith angle (SZAL), i.e. the solar zenith angle at the location of the cloud, can also be used (as discussed further in Sect. 4). Furthermore, these simulations also show that the Chappuis bands are more clearly visible with decreasing NLC optical depth (compare Figs. –), and the spectral maximum at about 750 nm has no noticeable effect on the colour of NLCs. Considering that single scattering is a valid approximation as discussed in Sect. , these observations can be explained by the following geometrical considerations. With NLCs at an altitude of 82 km, the scattered radiation comes primarily from this altitude and is hardly affected by stratospheric ozone due to the slight atmospheric penetration of solar radiation on its path to the NLC, at least for the SZA considered here. For SZA=98∘ and VZA=65∘, the tangent height of the solar beam is about 40 km. Without NLCs the scattered light is purely Rayleigh scattered, and since the density in the atmosphere increases exponentially with decreasing altitude, the scattered radiation comes from lower altitudes and is more affected by the stratospheric ozone. Accordingly, the O3 Chappuis bands are more clearly visible. Therefore, with a sufficiently large NLC optical depth and a certain viewing geometry, the NLC signal dominate over the background signal, and the Chappuis absorption is no longer visible. However, the NLCs still appear blueish. The finding that absorption in the Chappuis bands of O3 is not required to explain the blue colour of NLCs in some situations is a new result compared to earlier work e.g..

As evident from the spectra in Fig. , ozone absorption may also affect the colour of noctilucent clouds. The effect of ozone was already investigated by , who made measurements of the spectral radiance of NLCs with a photoelectric spectropolarimeter. Figure  shows a CIE chromaticity diagram including NLCs and different ozone column densities.

For a vertical ozone column density of 300 DU, the colour changes with the VZAs (45, 65 and 85∘) from dark blue to light blue. This corresponds to a natural colour gradient of NLCs during twilight. With more ozone (600 DU), a shift of colours to smaller x values, i.e. a blue shift, can be observed. With a lower ozone column density (100 DU), a shift to larger x values occurs. This can be explained by the effect of ozone absorption in the Chappuis bands. Due to the long light path through the atmosphere, green, yellow, orange and short-wave red light is effectively absorbed by ozone, so blue light predominates. With more ozone, this effect is intensified and leads to a more saturated blue colour (compare 600 DU). However, a noticeable colour change only occurs at a low ozone column density (100 DU). This is due to the lower attenuation of the long-wave light by ozone absorption, resulting in a shift to the reddish region of the gamut (see VZA=85∘). In addition, the effect of ozone absorption increases for larger VZAs, due to the longer path through the ozone layer from the observer to the NLC. Since the changes in colour and the positions in the CIE chromaticity diagram corresponding to the different ozone column densities are closer together, it requires an unrealistically small amount of ozone for the colour of the NLCs to change. Nevertheless, ozone absorption must be taken into account to explain the colour of noctilucent clouds.

The typical particle radii of visible NLCs are in the range of 10–80 nm (see Sect. ). In order to test the effect of the NLC particle size on the colour of the clouds, we performed SCIATRAN simulations for different median radii of the assumed mono-modal log-normal particle size distribution, i.e. 10, 50, 200, 600 and 1000 nm. The width parameter is kept constant at S=1.4, and the optical depth was assumed to be τNLC=10-4 in all cases. Figure  shows a chromaticity diagram with simulated colours for increasing particle sizes.

As expected, solar radiation scattered by particles with typical radii (10 to 50 nm) is perceived as blue by a ground-based observer. In addition, these radii also show the colour change from dark blue to white blue or light blue with the VZAs (45, 65 and 85∘). Assuming significantly larger particles, the scattered light becomes more reddish (compare 600 and 1000 nm). That means, when the particles become larger and scatter spectrally more neutral, the reddish colouring is also visible from the Earth's surface. Most of the calculations by , who have summarised various NLC measurements, are in agreement with our results, only in one case do their calculations show smaller particles in the yellowish/orange region of the CIE chromaticity diagram. The simulations displayed in Fig.  show that the typical colour of NLCs is only present for certain particle sizes. Furthermore, it can be shown that the light scattered by NLCs in the visible spectral range contains important information on the size of NLC particles. Thus, very large particles can be excluded by the resulting colour.

During sunset, a reddening appears on the horizon (see Fig. ), which accompanies most NLC observations (depending on the SZA). Figures  and show simulated solar scattering spectra (left column) with the resulting colour impression (right column) for a ground-based observer with SZA=98∘; VZA=84, 87, and 90∘ (from top to bottom); and SAA=0∘. Figure  shows the calculated results for NLCs with the following parameters: rm=50 nm, S=1.4, vertical optical depth of τNLC=10-4 and an altitude of zNLC=82 km. In comparison, Fig.  illustrates the background without NLCs.

Both plots depict the colour change from blue to orange near the horizon. Note here that the radiance values of the maximum in the short-wave blue spectral range at about 470 nm are larger for the simulations with NLCs than for the background case, especially for VZA=84∘ (upper panel). This results in different positions in the CIE chromaticity diagram. In contrast, the spectra for VZA=90∘ (lower panel) show no significant differences. Due to the very small radiance values for the case with NLCs in the range of about 400 nm, the positions in the CIE chromaticity diagram differ but not the resulting colour impression of orange. Since the simulated spectra near the horizon barely deviate in intensity and spectral shape, it can be concluded, as expected, that NLCs play no decisive role in the red colouring of the horizon. It should be noted that the tropospheric aerosol loading is highly variable, and the colours of the twilight sky may differ. But the main point here is the effect of NLCs on the reddish colour of the horizon.

In earlier studies on simulations of the spectral distribution of solar radiation scattered by NLC particles, the contribution of multiply scattered radiation has been neglected e.g.. Using SCIATRAN, the contribution of multiple scattering to the NLC spectra as seen by a ground-based observer can be easily simulated. For calculations with a more accurate consideration of multiple scattering, which is referred to as the fully spherical solution, i.e. “number of iterations” >1 (see Sect. ), SCIATRAN version 4.5.5 is now used. Figure  shows the difference between both methods for scattered solar spectra with VZA=65∘ (left panel) and the ratio for different VZAs (right panel).

The differences are mainly in the short-wave blue spectral range (maximum factor of 1.34). Overall, they are not crucial in the context of the current study. This is especially the case for the NLC-viewing geometry relevant VZAs here (45, 65 and 85∘). Furthermore, Fig.  shows that the more accurate consideration of multiple scattering has no visible effect on the CIE chromaticity values and the resulting colour, which is due to the blue CIE colour-matching function z‾(λ) having its maximum at 450 nm. Therefore, the approximate multiple scattering treatment method is sufficient for the simulations performed here.

In comparison, Fig.  shows CIE chromaticity values for different viewing geometries (VZAs of 45, 65, 85∘ and SZA of 98∘) and for single and multiple scattering simulations. For the calculations considering multiple scattering, the fully spherical solution was used here. The simulations show that for multiple scattering the colour is slightly bluer than for single scattering. This is due to Rayleigh scattering and the resulting preference for the short-wave blue light. However, no significant differences are observed.

The spectra in Fig.  (left panel) show the simulated spectral distribution with and without the multiple scattering contribution for VZA=65∘. The right panel compares the ratio of multiple scattering and single scattering for different VZAs.

The scattered solar spectra show that for the single scattering the simulated values are smaller than for the multiple scattering (the maximum relative difference between the two spectra is 23 %). Especially in the short-wave range, the influence of multiple scattering is significant. Overall, the differences for the VZAs of the NLC-viewing geometry used here are not significant with respect to the focus of this study and confirm the results of Fig. . As above, the weak influence of the large differences at shorter wavelengths (compare Fig. ) is due to the blue CIE colour-matching function z‾(λ) with a maximum at 450 nm. However, the differences depend highly on the VZA. Near the zenith, multiple scattering has a large influence and results in a maximum factor of about 3.

Single scattering is a valid approximation for the NLC-viewing geometry used here, but it should be noted that, depending on the relevant VZA range, the effect of multiple scattering may dominate and lead to different conclusions.