### Calculation of solar elevation and azimuth angles versus time

For our numerical calculations, the solar elevation angle θ_{s}(*t*) from the horizon and the solar azimuth angle α_{s}(*t*) from south (axis *y*, Fig. 7A) were calculated as a function of time *t* with an algorithm based on a semi-analytical approximation (analytical Kepler’s orbits modified with astronomical perturbations) and the planetary theory VSOP 87 (Variations Séculaires des Orbites Planètaires) of Bretagnon and Francou^{30}. This method is valid for the 1950–2050 period with an accuracy of 0.01°. Using this algorithm, we calculated the geocentric ecliptical, then the geocentric equatorial, and finally the geocentric horizontal coordinates of the Sun, resulting in the values of θ_{s}(*t*) and α_{s}(*t*).

### Diurnal cloudiness

Total cloud cover (TCC) time series of high temporal resolution (1 h) were evaluated for the period 01.01.2009–31.12.2018 from the ERA5 reanalysis of the European Centre for Medium-Range Weather Forecasts^{31}. The geographic coverage is global with a native spatial resolution of 0.25° × 0.25° ≈ 27 km × 27 km. Climatological mean values of TCC were determined by averaging for each hour of each calendar day of every year in the vegetative period of sunflowers. Since TCC is a dimensionless relative parameter in the range 0–1 (0 is clear sky, 1 is overcast), the hourly climatological means are equivalent to the time-dependent probability 0 ≤ σ(*t*) ≤ 1 of cloudy situation. We determined the diurnal cloud probability function σ(*t*) in July, August and September in Boone County (Kentucky, USA, 39° N, − 84.75° E, Fig. 2A), central Italy (41.0° N, 15.0° E, Fig. 2B), central Hungary (47.0° N, 19.0° E, Fig. 2C), and south Sweden (58.0° North, 13.0° East, Fig. 2D). The cloudiness data used in our calculations correspond to the decade between 2009 and 2018. Because similar data are not readily available for the period when sunflowers were domesticated, we assume in this work that the data obtained in the last decade is historically representative. The validity of this assumption can be evaluated when paleo-climatological cloudiness data become available.

### Measurement of the elevation angle of mature sunflower heads versus time

In a sunflower plantation at Budaörs (near Budapest), we measured the elevation angle θ_{n} of the normal vector of the mature head of the same 100 sunflowers as a function of time *t*, approximately weakly from 6 July to 11 September 2020. The studied sunflowers were individuals in a given row of the plantation.

### Measurement of the absorption spectra of mature sunflower heads

The absorption spectra *A*(λ) of young (2 weeks after anthesis) and old (4 weeks after anthesis) inflorescence and back of mature sunflower heads were measured in the field with an Ocean Optics STS-VIS spectrometer (Ocean Insight, Largo, USA) in July 2020. Measurements were performed under total overcast conditions to ensure isotropic diffuse skylight illumination. At first, the reflection spectrum of the inflorescence/back was determined as follows: a spectrum was measured by directing the spectrometer’s head on the target at a distance of 5 cm, then another spectrum was registered by pointing the spectrometer to the overcast sky. In the laboratory these two spectra were divided by each other. Finally, assuming that all non-reflected light was absorbed, the absorption spectrum *A*(λ) = 1 − *R*(λ) was obtained by subtracting the reflection spectrum *R*(λ) from 1. Absorption spectra were measured for 3 sunflowers and then averaged.

### Calculation of sky irradiance absorbed by a sunflower inflorescence

In the *x*–*y*–*z* reference frame of Fig. 7A, let the normal vector of a mature sunflower inflorescence be

$$underline {text{n}} = , left( {{text{cos}}theta_{{text{n}}} cdot {text{sin}}alpha_{{text{n}}} ,{text{ cos}}theta_{{text{n}}} cdot {text{cos}}alpha_{{text{n}}} ,{text{ sin}}theta_{{text{n}}} } right),$$

(2)

where axes *x* and *y* point to west and south, axis *z* points vertically upward, the elevation angle − 90° ≤ θ_{n} ≤ + 90° is measured from the horizontal (θ_{n} > 0°: above the horizon, θ_{n} < 0°: below the horizon), and the azimuth angle α_{n} is measured clockwise from axis *y* (south). In our model, after sunflower anthesis angle α_{n} is constant, because the head does not follow the motion of the Sun in the sky, and angle θ_{n}(*t*) decreases monotonously with time *t* (due to the gradually increasing weight of the head) as measured in the field (Fig. 3). The unit vector pointing toward the Sun is:

$$underline {text{s}} = , left( {{text{cos}}theta_{{text{s}}} cdot {text{sin}}alpha_{{text{s}}} ,{text{ cos}}theta_{{text{s}}} cdot {text{cos}}alpha_{{text{s}}} ,{text{ sin}}theta_{{text{s}}} } right),$$

(3)

where θ_{s} and α_{s} are the solar elevation and azimuth angles (Fig. 7A). When θ_{n} = 0° and θ_{s} = 0° the inflorescence is vertical and the Sun is on the horizon, while for θ_{n} = 90° and θ_{s} = 74.5° the horizontal inflorescence looks at the zenith and the Sun culminates on 21 June at the 39° northern latitude of Boone County (Kentucky, USA, 39° N, − 84.75° E), the region from which the domesticated sunflower originates^{26}. In the case of α_{n} = − 90°, 0° and 90°, the inflorescence faces east, south, and west, respectively. Northeast, southeast, southwest and northwest facing inflorescences are defined as − 180° < α_{n} < − 90°, − 90° < α_{n} < 0°, 0° < α_{n} < 90° and 90° < α_{n} < 180°, respectively.

The global irradiance received from the sky-dome by a horizontal surface is the sum of the direct and diffuse irradiances:

$$I_{{{text{global}}}} = I_{{{text{direct}}}} + I_{{{text{diffuse}}}} ,$$

(4)

and

$$I_{{{text{diffuse}}}} = D cdot I_{{{text{global}}}} ,$$

(5)

where *D* is the diffuse fraction of global radiation. From (4) and (5) it follows:

$$I_{{{text{diffuse}}}} = I_{{{text{direct}}}} frac{D}{1 – D}.$$

(6)

There are two different meteorological situations: (i) In cloudy situation with time-dependent probability 0 ≤ σ(*t*) ≤ 1 clouds dominate which frequently occlude the Sun. (ii) In sunny situation with probability 1 − σ(*t*) direct sunlight dominates, because the Sun is occluded by clouds only rarely. The total light energy absorbed by a sunflower inflorescence from dawn to dusk on the i-th day is the sum of the energy of direct solar radiation (E_{{text{sun,i}}} ({uptheta }_{{text{n}}}, {{alpha }}_{{text{n}}} )), the energy (E_{{text{diffuse, i}}}^{{{text{cloud}}y}} ({uptheta }_{{text{n}}}, {{alpha }}_{{text{n}}} )) of diffuse radiation in cloudy weather, and the energy (E_{{text{diffuse, i}}}^{{{text{sunn}}y}} ({uptheta }_{{text{n}}}, {{alpha }}_{{text{n}}} )) of diffuse radiation in sunny weather:

$$E_{{text{i}}} = {E_{{text{diffuse,i}}}^{{{text{cloud}}y}}} ({uptheta }_{{text{n}}}, {{alpha }}_{{text{n}}} ) + {E_{{text{sun,i}}}} ({uptheta }_{{text{n}}}, {{alpha }}_{{text{n}}} ) + {E_{{text{diffuse,i}}}^{{{text{sunn}}y}}} ({uptheta }_{{text{n}}} {,alpha }_{{text{n}}} ).$$

(7)

Under cloudy conditions the diffuse energy component is:

$${E_{{text{diffuse,i}}}^{{{text{cloud}}y}}} ({uptheta }_{{text{n}}}, {{alpha }}_{{text{n}}} ) = Qfrac{{{uptheta }_{{text{n}}} + {{pi /2}}}}{{uppi }}intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {{sigma (}t{)}intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{inflor}}}} ({uplambda })I_{{{text{diffuse}}}}^{{{text{cloud}}y}} ({lambda ,}t){text{d}}lambda } } rightrangle {text{d}}t} ,$$

(8)

where *Q* is the surface area of the sunflower inflorescence, (I_{{{text{diffuse}}}}^{{{text{cloud}}y}} ({{lambda ,t}})) is the diffuse irradiance received by a horizontal surface in cloudy situation, λ_{min} = 400 nm ≤ λ ≤ λ_{max} = 700 nm is the wavelength interval of radiation in which the light absorption by sunflowers is physiologically relevant^{6}, (t_{{{text{rise}}}}^{{text{i}}}) and (t_{{{text{set}}}}^{{text{i}}}) denote the time of sunrise and sunset on the i-th day (after the first calendar day i = 1 the head does not follow the Sun), *A*_{inflor}(λ) is the absorption spectrum of the inflorescence, and the factor (θ_{n} + π/2)/π describes the proportion of the sky hemisphere from which an inflorescence with elevation angle θ_{n} receives diffuse skylight (Fig. 7B). Figure 2A illustrates the average absorption spectra *A*_{inflor}(λ) of young (2 weeks after anthesis) and old (4 weeks after anthesis) sunflower inflorescences. In our computations *A*_{inflor}(λ) was set to the absorption spectrum of young inflorescences in the first 3 weeks after anthesis, and then it was set to that of old inflorescences. In the spectral range λ_{min} = 400 nm ≤ λ ≤ λ_{max} = 700 nm, under cloudy conditions the average direct radiation is approximately 20% of the direct solar radiation (Fig. 8A) and this fraction has only a little variation with sky condition and time of year^{32}. Using (6) and the 0.2 factor, we obtain:

$$I_{{{text{diffuse}}}}^{{{text{cloud}}y}} ({{uplambda ,t}}) = frac{{D_{{{text{cloudy}}}} (t)}}{{1 – D_{{{text{cloudy}}}} (t)}}0.2I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]sin {uptheta }_{{text{s}}}^{{text{i}}} (t),$$

(9)

where *I*_{sun}(λ, θ_{s}) is the spectral distribution of irradiance of direct sunlight (Fig. 8B) incident on a unit surface perpendicular to the direct solar radiation, and *D*_{cloudy} is the diffuse fraction of global radiation under cloudy conditions (Fig. 8C). In meteorology, *I*_{global}, *I*_{direct} and *I*_{diffuse} are always measured on a horizontal detector (radiometer) surface, while the irradiance *I*_{Sun}(λ, θ_{s}) of sunlight in Fig. 8B was computed for a surface being perpendicular to the direct solar radiation. In order to take into consideration these two different orientations of the detector surface, *I*_{sun}(λ, θ_{s}) is multiplied by sinθ_{s} in (9).

**Figure 8**

Meteorological data used in the computations. (**A**) Spectral distribution of irradiance *I*_{sun}(λ, θ_{s}) for direct sunlight measured under cloudless sky conditions at solar noon with solar elevation angle θ_{s} = 53.5° and *I*_{cloud}(λ) for cloudlight measured on a completely overcast day at solar noon with solar elevation θ_{s} = 53.8° (after Fig. 2a of Dengel et al.^{32}). (**B**) Spectral distribution of irradiance *I*_{Sun}(λ, θ_{s}) of direct sunlight as a function of solar elevation angle θ_{s} ranging from 0° (horizon, lowermost curve) to 90° (zenith, uppermost curve) with an increment of 1° simulated by MODTRAN 3.7 (Ref.^{33}). (**C**) Diurnal variation of the annual mean diffuse fraction *D* (the proportion of diffuse sky radiation to global sky radiation) under sunny and cloudy conditions over northern China in April July 2017 (after Fig. 6 of Liu et al.^{39}).

Using MODTRAN (MODerate resolution TRANsmittance code)^{33,34,35,36} version 3.7 with a solar constant of 1362.12 W/m^{2} and the 1976 US Standard Atmosphere^{37}, we have computed the ground-level direct-normal spectral solar irradiance *I*_{sun}(λ, θ_{s}) as described by Egri et al.^{38}. *I*_{Sun}(λ, θ_{s}) is the energy of solar radiation per unit time, per unit area perpendicular to the radiation and per unit wavelength interval. Figure 8B shows the irradiance spectrum *I*_{Sun}(λ, θ_{s}) of direct sunlight for 91 different solar elevation angles θ_{s}.

Figure 8C shows the diurnal variation of the annual mean diffuse fraction *D* under cloudy and sunny conditions measured by Liu et al.^{39} over northern China. Since we did not find similar publicly available measurements for other regions of the Earth, we use these data further on in this work. From (8) and (9) we obtain the diffuse light energy per unit area received by a sunflower inflorescence under cloudy conditions:

$$e_{{text{diffuse,i}}}^{{{text{cloud}}y}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} ) = frac{{E_{{text{diffuse,i}}}^{{{text{cloud}}y}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} )}}{Q} = 0.2frac{{{uptheta }_{{text{n}}} + {{pi /2}}}}{{uppi }}intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {frac{{{sigma (}t{)}D_{{{text{cloudy}}}} (t)sin {uptheta }_{{text{s}}}^{{text{i}}} (t)}}{{1 – D_{{{text{cloudy}}}} (t)}}intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{inflor}}}} ({uplambda })I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]{{d}}lambda } } rightrangle {text{d}}t} .$$

(10)

If *I*_{direct}(λ, θ_{s}) is the direct irradiance illuminating the sunflower [Joule/(second·nanometer·meter^{2})], then the elementary direct light energy d*E* absorbed by the inflorescence in a time interval d*t* and in a wavelength bin dλ is:

$${text{d}}E = Q cdot {text{cos}}gamma cdot I_{{{text{direct}}}} left( {lambda , , theta_{{text{s}}} } right) cdot A_{{{text{inflor}}}} left( lambda right) cdot {text{d}}lambda cdot {text{d}}t,$$

(11)

where γ is the incidence angle between the unit vectors __n__ (n = 1) and __s__ (s = 1), the cosine of which can be expressed as:

$${text{cos}}gamma , = {underline{text{n}}} cdot {underline{text{s}}} = {text{ cos}}theta_{{text{n}}} cdot {text{sin}}alpha_{{text{n}}} cdot {text{cos}}theta_{{text{s}}} cdot {text{sin}}alpha_{{text{s}}} + {text{ cos}}theta_{{text{n}}} cdot {text{cos}}alpha_{{text{n}}} cdot {text{cos}}theta_{{text{s}}} cdot {text{cos}}alpha_{{text{s}}} + {text{ sin}}theta_{{text{n}}} cdot {text{sin}}theta_{{text{s}}} .$$

(12)

The inflorescence can absorb direct sunlight only if the following condition is satisfied:

− 90° < γ < + 90°, 0 < cosγ < 1, that is

$$0 < {text{cos}}theta_{{text{n}}} cdot {text{sin}}alpha_{{text{n}}} cdot {text{cos}}theta_{{text{s}}} cdot{text{sin}}alpha_{{text{s}}} + {text{ cos}}theta_{{text{n}}} cdot{text{cos}}alpha_{{text{n}}} cdot {text{cos}}theta_{{text{s}}} cdot {text{cos}}alpha_{{text{s}}} + {text{ sin}}theta_{{text{n}}} cdot {text{sin}}theta_{{text{s}}} < { 1}.$$

(13)

Using (11), under sunny conditions with probability 1 – σ(*t*), the direct solar energy per unit area received by the inflorescence is:

$$e_{{text{sun,i}}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} ) = frac{{E_{{text{sun,i}}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} )}}{Q} = intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {[1 – {sigma (}t{)]}cos {gamma (}t{)}intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{inflor}}}} ({uplambda })I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]{text{d}}lambda } } rightrangle {text{d}}t} ,$$

(14)

where the factor

$$cos {gamma (}t{)} = cos {uptheta }_{{text{n}}} sin {upalpha }_{{text{n}}} cos {uptheta }_{{text{s}}}^{{text{i}}} (t)sin {upalpha }_{{text{s}}}^{{text{i}}} (t) + cos {uptheta }_{{text{n}}} cos {upalpha }_{{text{n}}} cos {uptheta }_{{text{s}}}^{{text{i}}} (t)cos {upalpha }_{{text{s}}}^{{text{i}}} (t) + sin {uptheta }_{{text{n}}} sin {uptheta }_{{text{s}}}^{{text{i}}} (t),$$

(15)

is necessary, because the direct solar radiation is usually not perpendicular to the surface of the inflorescence. Using (6) under sunny conditions, the diffuse irradiance received by a horizontal surface is:

$$I_{{{text{diffuse}}}}^{{{text{sunny}}}} ({{lambda ,t}}) = frac{{D_{{{text{sunny}}}} (t)}}{{1 – D_{{{text{sunny}}}} (t)}}I_{{{text{sun}}}} [{uplambda },{{uptheta }_{{text{s}}}^{{text{i}}}} (t)]sin {uptheta }_{{text{s}}}^{{text{i}}} (t),$$

(16)

where the factor sinθ_{s} is again necessary for the reason mentioned when (9) was derived. Figure 8C shows the diurnal variation of the annual mean diffuse fraction *D*_{sunny} under sunny conditions^{39}. Using (16), under sunny conditions with probability 1 − σ(*t*), the diffuse light energy per unit area received by the inflorescence is:

$$e_{{text{diffuse,i}}}^{{{text{sunn}}y}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} ) = frac{{E_{{text{diffuse,i}}}^{{{text{sunn}}y}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} )}}{Q} = frac{{{uptheta }_{{text{n}}} + {{pi /2}}}}{{uppi }}intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {frac{{[1 – {sigma (}t{)]}D_{{{text{sunny}}}} (t)sin {uptheta }_{{text{s}}}^{{text{i}}} (t)}}{{1 – D_{{{text{sunny}}}} (t)}}intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{inflor}}}} ({uplambda })I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]{text{d}}lambda } } rightrangle {text{d}}t} ,$$

(17)

where the factor (θ_{n} + π/2)/π is again the proportion of the sky hemisphere from which the inflorescence receives diffuse skylight (Fig. 7B). Finally, the total light energy *e*_{total} per unit area absorbed by a mature sunflower inflorescence in the period between the stop of solar tracking and senescence is:

$$e_{{{text{total}}}} = sumlimits_{{{text{i}} = 1}}^{{{text{i}} = {text{m}}}} {e_{{text{i}}} } = sumlimits_{{{text{i}} = 1}}^{{{text{i}} = {text{m}}}} {left[ {e_{{text{diffuse,i}}}^{{{text{cloud}}y}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} ) + e_{{text{sun,i}}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} ) + e_{{text{diffuse,i}}}^{{{text{sunn}}y}} ({uptheta }_{{text{n}}} {{,alpha }}_{{text{n}}} )} right]} ,$$

(18)

where *m* is the last day of senescence, when the seeds are fully developed and ripe.

### Calculation of sky irradiance absorbed by the back of sunflower heads

The back side of sunflower heads is green (later yellowish green) due to its chlorophyll content, because it takes an important role in photosynthesis^{6}. Thus, we calculate here the total light energy *e*_{total,back} per unit area absorbed by the back of a mature sunflower head in the period between anthesis and senescence. The back receives the same three radiation components as the inflorescence. The only three differences are the following: (i) The normal vector __n___{back} of the back is opposite to the normal vector __n__ of the inflorescence (Fig. 7B): __n___{back} = −__n__. (ii) The back is illuminated by diffuse light originating from the sky region which is the complementary part of the sky-dome illuminating diffusely the inflorescence. This can be taken into consideration by factor 1 − [θ_{n}(*t*) + π/2]/π = [π-2θ_{n}(*t*)]/(2π), being the proportion of the sky from which the back of a sunflower head with elevation angle θ_{n}(*t*) receives diffuse skylight (Fig. 7B). (iii) The absorption spectrum *A*_{back}(λ) of the back of sunflower heads is different from the absorption spectrum *A*_{inflor}(λ) of the inflorescence (Fig. 2). Taking into account these differences, we obtain the total light energy *e*_{total,back} per unit area absorbed by the back of a mature sunflower head in the period between anthesis and senescence as follows:

$$e_{{text{total,back}}} = sumlimits_{{{text{i}} = 1}}^{{{text{i}} = {text{m}}}} {left{ {e_{{text{diffuse,i,back}}}^{{{text{sunn}}y}} left[ {{uptheta }_{{text{n}}} {(}t{{),alpha }}_{{text{n}}} } right] + e_{{text{diffuse,i,back}}}^{{{text{cloud}}y}} left[ {{uptheta }_{{text{n}}} {(}t{{),alpha }}_{{text{n}}} } right] + e_{{text{sun,i,back}}} left[ {{uptheta }_{{text{n}}} {(}t{{),alpha }}_{{text{n}}} } right]} right}} ,$$

(19)

where *m* is the last day of senescence, and the three components in sum (19) are:

$$e_{{text{diffuse,i,back}}}^{{{text{sunn}}y}} [{uptheta }_{{text{n}}} (t){{,alpha }}_{{text{n}}} ] = frac{{{uppi } – {{2uptheta }}_{{text{n}}} (t)}}{{{{2uppi }}}}intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {frac{{left[ {1 – {sigma (}t{)}} right]D_{{{text{sunny}}}} (t)sin {uptheta }_{{text{s}}}^{{text{i}}} (t)}}{{1 – D_{{{text{sunny}}}} (t)}}intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{back}}}} ({uplambda })I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]{text{d}}lambda } } rightrangle {text{d}}t} ,$$

(20)

$${e_{{text{diffuse,i,back}}}^{{{text{cloud}}y}}} [{uptheta }_{{text{n}}} (t){{,alpha }}_{{text{n}}} ] = 0.2frac{{{uppi } – {{2theta }}_{{text{n}}} (t)}}{{{{2pi }}}}intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {frac{{{sigma (}t{)}D_{{{text{cloudy}}}} (t)sin {uptheta }_{{text{s}}}^{{text{i}}} (t)}}{{1 – D_{{{text{cloudy}}}} (t)}}intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{back}}}} ({uplambda })I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]{text{d}}lambda } } rightrangle {text{d}}t} ,$$

(21)

$$e_{{text{sun,i,back}}} [{uptheta }_{{text{n}}} (t){{,alpha }}_{{text{n}}} ] = intlimits_{{t_{{{text{rise}}}}^{{text{i}}} }}^{{t_{{{text{set}}}}^{{text{i}}} }} {leftlangle {left[ {1 – {sigma (}t{)}} right] cdot left[ { – cos {gamma (}t{)}} right]intlimits_{{{uplambda }_{{{text{min}}}} }}^{{{uplambda }_{{{text{max}}}} }} {A_{{{text{back}}}} ({uplambda })I_{{{text{sun}}}} [{uplambda },{uptheta }_{{text{s}}}^{{text{i}}} (t)]{text{d}}lambda } } rightrangle {text{d}}t} .$$

(22)

The component *e*_{sun,i,back}[θ_{n}(*t*),α_{n}] expressed by (22) increases only when the back absorbs direct sunlight, which happens only, if the following condition is satisfied:

$$- 1 < cos {gamma (}t{)} = cos {uptheta }_{{text{n}}} sin {upalpha }_{{text{n}}} cos {uptheta }_{{text{s}}}^{{text{i}}} (t)sin {upalpha }_{{text{s}}}^{{text{i}}} (t) + cos {uptheta }_{{text{n}}} cos {upalpha }_{{text{n}}} cos {uptheta }_{{text{s}}}^{{text{i}}} (t)cos {upalpha }_{{text{s}}}^{{text{i}}} (t) + sin {uptheta }_{{text{n}}} sin {uptheta }_{{text{s}}}^{{text{i}}} (t) < 0.$$

(23)

Since in this case cosγ(*t*) < 0, the negative sign of cosγ(*t*) in (22) is necessary.

Figure 2B shows the average absorption spectra *A*_{back}(λ) of young (2 weeks after anthesis) and old (4 weeks after anthesis) backs of sunflower heads. In our computations *A*_{back}(λ) was set to the absorption spectrum of young backs in the first 3 weeks after anthesis, and then it was set to that of old backs.

### Ethical approval and informed consent

For our studies no permission, licence or approval was necessary.

Source: Ecology - nature.com