Solar and proxy solar (stars of Table 5.6) results for r =1 AU are shown in Figures 5.11 and 5.12. At an age of 100 Myr the solar XUV and Lya emission flux ratios are 104.0 and 15.6, decreasing to 5.7 and 2.7, respectively, at an age of 1 Byr (3.56 Byr ago). However, beyond the present age, for the next several billion years the solar emission flux ratios decrease very slowly up to 8 Byr. Although the Rossby number increases over this time interval its effect is compensated by a corresponding increase in solar radius so that the ultraviolet
FlG. 5.10. Evolution of solar radius, effective temperature and B-V with age, based on the model of Girardi et al. (2000) for a star with 1 solar mass and solar metallicity. Also shown is the evolution of the rotation period, P from eqn (5.24), and Rossby number, K from eqn (5.23). (Vardavas 2005)
Fig. 5.11. Evolution of XUV(1-1200 A) emission flux ratios with age at 1 AU using eqn (5.37) for the sun (full squares) and proxy solar (empty squares) Sun in Time Programme stars (Table 5.6), with radius scaled to the solar value at the age of the star. The solid line is the power- law fit 6.4i~1'23. (Vardavas 2005)
FlG. 5.12. Evolution of Ly-a emission flux ratios with age at 1 AU using eqn (5.37) for the sun (full squares) and proxy solar (empty squares) Sun in Time Programme stars (Table 5.6) with radius scaled to the solar value at the age of the star. The solid line is the power- law fit 3.1i~0'72. (Vardavas 2005)
emission flux ratios decrease very slowly. Finally, we note that for a star like 3 Hyi whose actual radius is 1.89Rq at age 6.7 Byr, the emission flux ratios at 1 AU would be about 3 times those of the corresponding proxy solar star at the same age with radius of 1.1Rq (Fig. 5.10).
The stellar Rossby number increases with stellar age essentially according to t05 and the period of rotation according to t0 6, for G-type solar-like stars. Given the stellar rotation period and B-V we can compute the Rossby number and knowing the effective temperature compute the stellar surface XUV and Ly-a emission flux relative to the present sun. For stellar evolution studies, we can use the variation of the stellar B-V to derive the Rossby number with age. From the evolution of both the effective temperature and the stellar radius we can calculate the stellar XUV and Ly-a emission flux ratios with age, at any planetary distance from the star. These emission flux ratios can then be used in photochemical models (see Chapter 7) to examine the evolution of the composition of planetary atmospheres orbiting such stars.
From Fig. 5.10, we can see that the effective temperature of the Sun at zero main sequence age was about 0.97 of its present value of 5777 K and that its radius was 0.9 times its present value. These translate to a luminosity that was 0.72 its present value some 4.6 Byr ago. Thus, its effective temperature would
FlG. 5.13. Photospheric irradiance at 1 AU for the Sun from the library of Lejeune et al. (1997) is compared with the solar cycle mean solar irradiance of Gueymard (2004).
have been about 5600 K, which according to the Wien displacement law (§3.3.4) and the Planck function would have meant that the spectral irradiance in the visible and near-infra-red would have been slightly lower and its peak shifted from 5016 A to 5175 A. In contrast, the ultraviolet and XUV flux would have been appreciably higher, as we saw in the previous section. Further, if we assume that its mass has not changed appreciably (§5.3.1) since 4.6 Byr ago, then its gravity would have been 1.23 its present value with log(g) = 4.53, instead of its present value of 4.44.
Stellar photospheric irradiance F\ is tabulated in standard stellar libraries (see Buser and Kurucz 1992, Lejeune et al. 1997) given the stellar effective temperature, Teff, gravity, log(g) and metallicity, Z of the star. In Fig. 5.13 we compare the photospheric irradiance at 1 AU for the Sun taken from the Lejeune et al. (1997) standard stellar library with the solar cycle mean solar irradiance of Gueymard (2004). Down to 200 nm there is good agreement between the two profiles, given that the solar irradiance includes contributions from the chromosphere. The differences can be seen below 200 nm where the ultraviolet flux arises from the chromosphere and corona, and that varies with solar activity. As the Sun aged, its photospheric irradiance increased with time and solar activity decreased. The result was more visible and near-infra-red flux and less ultraviolet flux below 200 nm. The solar ultraviolet flux was probably enhanced 3.5 Byr ago (when life was well established on Earth), by a factor of about 3 at Lyman-a, as shown in Fig. 5.12, with little enhancement at wavelengths greater than 200 nm.
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