5 Radiation

$Q_{rad}$ in equation (5) is given by convergence of net radiative heat flux which is calculated by using radiative transfer equation. We consider following radiation processes in this model; absorption of near infrared solar radiation (NIR), absorption and emission of infrared radiation associated with atmospheric CO${}_{2}$, absorption and scattering of solar radiation, and absorption and emission of infrared radiation associated with dust.

$Q_{rad}$ is represented as follows.

$$Q_{rad}= Q_{rad,IR} + Q_{rad,NIR} + Q_{rad,dust,SR} + Q_{rad,dust,IR}.$$ (22)

$Q_{rad,IR}$ and $Q_{rad,NIR}$ are the infrared and near infrared radiative heating rate associated with CO${}_{2}$ $Q_{rad,dust,SR}$ and $Q_{rad,dust,IR}$ are the solar and infrared radiative heating rate associated with dust. The governing equations to calculate these heating rate are described in following sections.

5.1 Radiative transfer of atmospheric CO${}_{2}$

Both infrared and near infrared radiative flux associated with CO${}_{2}$ are calculated by Goody narrow band model (c.f., Goody and Young, 1989). In calculating infrared radiative flux, CO${}_{2}$ 15 $\mu$m band is only considered. The upward and downward infrared radiative flux $F_{IR}^{\uparrow}, F_{IR}^{\downarrow}$ and the infrared radiative heating rate per unit mass $Q_{rad,IR}$ are calculated as follows.

$$F_{IR}^{\uparrow}(z) = \sum _{i}\Delta \nu _{i}\left\{ \pi B_{\nu _{i},T}(z=0){\cal T}_{... ...\int _{0}^{z}\pi B_{\nu _{i},T}(z')\DD{{\cal T }_{i}(z,z')}{z'}\Dd z' \right\},$$ (23)

$$F_{IR}^{\downarrow}(z) = \sum _{i}\Delta \nu _{i}\left\{ \int _{z}^{\infty }\pi B_{\nu _{i},T}(z')\DD{{\cal T}_{i}(z,z')}{z'}\Dd z' \right\},$$ (24)

$$Q_{rad,IR} = -\frac{1}{\rho _{0}c_{p}}\DP{}{z}(F_{IR}^{\uparrow}(z) - F_{IR}^{\downarrow}(z) ).$$ (25)

$\Delta \nu _{i}$ is the $i$th narrow band width and $B_{\nu _{i},T}$ is the Plank function which is represented as follows.

$$B_{\nu _{i},T} = \frac{2hc^{2}\nu _{i}^{3}}{e^{hc\nu_{i}/kT... ... \frac{1.19\times 10^{-8}\nu _{i}^{3}}{e^{1.4387\nu_{i}/T}-1},$$ (26)

where $h$ is the Plank constant, $c$ is speed of light, $k$ is the Boltzmann constant, and $T$ is temperature. ${\cal T}_{i}(z,z')$ is the transmission function averaged over $\Delta \nu _{i}$ around $\nu _{i}$.

$${\cal T}_{i}(z,z') = \exp ( - W_{i}/\Delta \nu _{i}), \qua... ...ac{s_{i} u(z,z')} {\sqrt{ 1 + s_{i}u(z,z')/\alpha ^{*}_{i}}},$$

$$u(z,z') = \int _{z}^{z'}1.67\rho _{0}\Dd z, \quad \alpha ... .../p_{0}, \quad \overline{p} = \int _{z}^{z'}P_{0} \Dd u / u.$$

$s_{i}$ is line strength, $\alpha ^{*}_{i}$ is square root of the product of line strength and line width and $\alpha _{i}$ is the reference value of $\alpha ^{*}_{i}$, $u$ is effective path length, and $p_{0}$ is reference pressure (= 1013 hPa).

In calculating near infrared solar radiative flux, CO${}_{2}$ 4.3 $\mu$m, 2.7 $\mu$m, and 2.0 $\mu$m band are considered. The near infrared solar radiative flux $F_{NIR}^{\downarrow}$ and $Q_{rad,NIR}$ are calculated as follows.

$$F_{NIR}^{\downarrow}(z) = \sum _{i}\Delta \nu _{i}\left\{ S_{\nu _{i}}{\cal T}_{i}(\infty,z)\mu_{0} \right\},$$ (27)

$$Q_{rad,NIR} = \frac{1}{\rho _{0}}\DP{F_{IR}^{\downarrow}(z)}{z},$$ (28)

where $\mu _{0}=\cos \zeta$, $\zeta$ is the solar zenith angle, and $S_{\nu _{i}}$ is the solar radiative flux per unit wave length at the top of atmosphere which is represented as follows.

$$S_{\nu _{i}} = B_{\nu _{i},T_{sol}} \left(\frac{F_{s}}{\sigma T_{sol}^{4}}\right),$$ (29)

$$F_{s} = I_{0}\left(\frac{r_{0}}{r}\right)^{2}\mu_{0},$$ (30)

where $T_{sol}$ is the surface temperature of the sun (= 5760 K), $\sigma$ is the Stefan-Boltzmann constant (= 5.67 $\times 10{}^{-8}$ Wm${}^{-2}$K${}^{-4}$), $I_{0}$ is solar constant on the mean radius of Mars orbit (= 591 Wm${}^{-2}$), $r$ and $r_{0}$ is the radius of Mars orbit and its mean value, $F_{s}$ is solar radiative flux at the top of atmosphere. $F_{s}$ is depend on season, latitude and local time. Detail descriptions of $F_{s}$ and $\cos \zeta$ are shown in 第5.3節.

The transmission function averaged over $\Delta \nu _{i}$ in near infrared wavelength region is similar to that in infrared wavelength region except for the effective path length $u$.

$$u(z,z') = \int _{z}^{z'}1.67\rho _{0}\Dd z / \mu_{0},$$

Parameters

The number of narrow band and its band width are similar to those of Savijärvi (1991a). The line strength and the square root of the product of line strength and line width are quoted from those at 220 K listed by Houghton (1986). These vaues are listed in Table 4 $\sim$ Table 7.

CO${}_{2}$ 15 $\mu$m band ranges from 500 cm${}^{-1}$ to 900 cm${}^{-1}$ and 4.3 $\mu$m band ranges from 2200 cm${}^{-1}$ to 2450 cm${}^{-1}$, where $\Delta \nu _{i}$ is equal to 25 cm${}^{-1}$. CO${}_{2}$ 2.7 $\mu$m band ranges from 3150 cm${}^{-1}$ to 4100 cm${}^{-1}$ and 4.0 $\mu$m band ranges from 4600 cm${}^{-1}$ to 5400 cm${}^{-1}$ , where $\Delta \nu$ is equal to 100 cm${}^{-1}$.

表 4: Parameters of CO${}_{2}$ 15 $\mu$m band

$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$	$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$
512.5	1.952 $\times 10{}^{-2}$	2.870 $\times 10{}^{-1}$	712.5	1.232 $\times 10{}^{3}$	8.387 $\times 10{}^{1}$
537.5	2.785 $\times 10{}^{-1}$	1.215 $\times 10{}^{0}$	737.5	2.042 $\times 10{}^{2}$	2.852 $\times 10{}^{1}$
562.5	5.495 $\times 10{}^{-1}$	2.404 $\times 10{}^{0}$	762.5	7.278 $\times 10{}^{0}$	6.239 $\times 10{}^{0}$
587.5	5.331 $\times 10{}^{0}$	1.958 $\times 10{}^{1}$	787.5	1.337 $\times 10{}^{0}$	2.765 $\times 10{}^{0}$
612.5	5.196 $\times 10{}^{2}$	5.804 $\times 10{}^{1}$	812.5	3.974 $\times 10{}^{-1}$	8.897 $\times 10{}^{-1}$
637.5	7.778 $\times 10{}^{3}$	2.084 $\times 10{}^{2}$	837.5	1.280 $\times 10{}^{-2}$	3.198 $\times 10{}^{-1}$
662.5	8.746 $\times 10{}^{4}$	7.594 $\times 10{}^{2}$	862.5	2.501 $\times 10{}^{-3}$	1.506 $\times 10{}^{-1}$
687.5	2.600 $\times 10{}^{4}$	2.635 $\times 10{}^{2}$	887.5	3.937 $\times 10{}^{-3}$	1.446 $\times 10{}^{-1}$

表 5: Parameters of CO${}_{2}$ 4.3 $\mu$m band

$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$	$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$
2212.5	9.504 $\times 10{}^{-1}$	2.866 $\times 10{}^{0}$	2337.5	5.587 $\times 10{}^{5}$	1.206 $\times 10{}^{3}$
2237.5	2.217 $\times 10{}^{2}$	3.000 $\times 10{}^{1}$	2362.5	6.819 $\times 10{}^{5}$	1.182 $\times 10{}^{3}$
2262.5	4.566 $\times 10{}^{3}$	1.134 $\times 10{}^{2}$	2387.5	1.256 $\times 10{}^{4}$	8.873 $\times 10{}^{1}$
2287.5	7.965 $\times 10{}^{3}$	2.011 $\times 10{}^{2}$	2412.5	7.065 $\times 10{}^{-1}$	3.404 $\times 10{}^{-1}$
2312.5	1.055 $\times 10{}^{5}$	5.880 $\times 10{}^{2}$	2437.5	8.522 $\times 10{}^{-2}$	4.236 $\times 10{}^{-1}$

表 6: Parameters of CO${}_{2}$ 2.7 $\mu$m band

$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$	$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$
3150	1.324 $\times 10{}^{-1}$	9.836 $\times 10{}^{-1}$	3650	1.543 $\times 10{}^{4}$	3.245 $\times 10{}^{2}$
3250	7.731 $\times 10{}^{-2}$	4.900 $\times 10{}^{-1}$	3750	1.649 $\times 10{}^{4}$	2.722 $\times 10{}^{2}$
3350	1.232 $\times 10{}^{0}$	2.952 $\times 10{}^{0}$	3850	1.180 $\times 10{}^{-1}$	9.535 $\times 10{}^{-1}$
3450	5.159 $\times 10{}^{0}$	7.639 $\times 10{}^{0}$	3950	1.464 $\times 10{}^{-2}$	2.601 $\times 10{}^{-1}$
3550	4.299 $\times 10{}^{3}$	1.914 $\times 10{}^{2}$	4050	1.251 $\times 10{}^{-2}$	2.021 $\times 10{}^{-1}$

表 7: Parameters of CO${}_{2}$ 2.0 $\mu$m band

$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$	$\nu _{i}$(cm${}^{-1}$)	$s_{i}$	$\alpha _{i}$
4650	2.185 $\times 10{}^{-1}$	1.916 $\times 10{}^{0}$	5050	8.778 $\times 10{}^{1}$	2.012 $\times 10{}^{1}$
4750	2.040 $\times 10{}^{0}$	6.475 $\times 10{}^{0}$	5150	8.346 $\times 10{}^{1}$	1.804 $\times 10{}^{1}$
4850	1.197 $\times 10{}^{2}$	3.112 $\times 10{}^{1}$	5250	8.518 $\times 10{}^{-2}$	8.474 $\times 10{}^{-1}$
4950	4.829 $\times 10{}^{2}$	5.759 $\times 10{}^{1}$	5350	4.951 $\times 10{}^{-1}$	1.597 $\times 10{}^{0}$

5.2 Radiative transfer of dust

The solar and infrared radiative flux associated with dust are calculated by using the $\delta$-Eddington approximation (c.f., Liou, 1980). The $\delta$-Eddington approximation is well used in calculating radiative transfer with anisotropic scattering. The asymmetry factor of dust for solar and infrared radiation are between 0 and 1 which means forward scattering occurs.

The upward and downward diffuse solar radiative flux per unit wave length associated with dust $F_{dif,\nu_{i}}^{\uparrow}$, $F_{dif,\nu _{i}}^{\downarrow}$ are obtained as solutions of following equations.

$$\DD{F_{dif,\nu _{i}}^{\uparrow}}{\tau _{\nu _{i}}^{*}} = \gamma _{1,\nu _{i}}F_{dif,\nu _{i}}^{\uparrow} - \gamma _{2,\nu ... ...i}} \tilde{\omega}_{\nu _{i}}^{*}S_{\nu _{i}}e^{-\tau _{\nu _{i}}^{*}/\mu_{0}},$$ (31)

$$\DD{F_{dif,\nu _{i}}^{\downarrow}}{\tau _{\nu _{i}}^{*}} = \gamma _{2,\nu _{i}}F_{dif,\nu _{i}}^{\uparrow} - \gamma _{1,\nu ... ...}}) \tilde{\omega}_{\nu _{i}}^{*}S_{\nu _{i}}e^{-\tau _{\nu _{i}}^{*}/\mu_{0}}.$$ (32)

The boundary condition of (31) and (32) are that $F_{dif,\nu_{i}}^{\downarrow}=0$ at the top of atmosphere and $F_{dif,\nu_{i}}^{\uparrow}= F_{dif,\nu}^{\downarrow}\times A$ at the surface, where $A$ is the surface albedo. $\gamma _{1,\nu _{i}},\gamma _{2,\nu _{i}},\gamma _{3,\nu _{i}}$ are expressed as follows.

$$\gamma _{1,\nu _{i}} = \frac{1}{4}[7-(4+3g_{\nu _{i}}^{*}... ...amma _{3,\nu _{i}} = \frac{1}{4}(2-3g_{\nu _{i}}^{*}\mu_{0}),$$

where $\tau _{\nu _{i}}^{*}, \tilde{\omega}_{\nu _{i}}^{*}, g_{\nu _{i}}^{*}$ are optical depth, single scattering albedo and asymmetry factor scaled by $\delta$-Eddington approximation, which are given as follows.

$$\tau _{\nu _{i}}^{*}=(1-\tilde{\omega}_{\nu _{i}}g_{\nu _{i... ...quad g_{\nu _{i}}^{*} = \frac{g_{\nu _{i}}}{1+g_{\nu _{i}}},$$

where $\tau _{\nu _{i}}, \tilde{\omega}_{\nu _{i}}, g_{\nu _{i}}$ are optical depth, single scattering albedo and asymmetry factor, respectively.

The upward and downward infrared radiative flux per unit wave length associated with dust are obtained as solutions of similar equations used for calculation of diffuse solar flux ((31), (32)) except for the last term in right hand side of each equation.

$$\DD{F_{IR,\nu _{i}}^{\uparrow}}{\tau _{\nu _{i}}^{*}} = \gamma _{1,\nu _{i}}F_{IR,\nu _{i}}^{\uparrow} - \gamma _{2,\nu _... ...} -2\pi (1-\tilde{\omega}_{\nu _{i}}^{*}) B_{\nu _{i},T}(\tau _{\nu _{i}}^{*}),$$ (33)

$$\DD{F_{IR,\nu _{i}}^{\downarrow}}{\tau _{\nu _{i}}^{*}} = \gamma _{2,\nu _{i}}F_{IR,\nu _{i}}^{\uparrow} - \gamma _{1,\nu _... ...w} +2\pi (1-\tilde{\omega}_{\nu _{i}}^{*})B_{\nu _{i},T}(\tau _{\nu _{i}}^{*}).$$ (34)

The boundary condition of (33) and (34) is that $F_{IR,\nu_{i}}^{\downarrow}=0$ at the top of atmosphere and $F_{IR,\nu_{i}}^{\uparrow}$ is equal to $\pi B_{\nu, T_{sfc}}$ at the surface. The Plank function $B_{\nu _{i},T}$ in (33) and (34) is averaged over the band width.

$$B_{\nu_{i},T} = \frac{1}{\nu _{2}-\nu _{1}} \int _{\nu _{1}}^{\nu _{2}} B_{\nu ,T}\Dd \nu.$$

$\nu_{1},\nu_{2}$ are the lower and upper wave length of the band.

The radiative heating rate associated with dust is calculated as follows.

$$Q_{rad,dust,SR} = - \frac{1}{\rho _{a}c_{p}}\DD{}{z} \left[\sum _{\nu _{i} } \Delta... ...w}-F_{dif,\nu _{i}}^{\downarrow}- F_{dir,\nu _{i}}^{\downarrow}\right) \right],$$ (35)

$$Q_{rad,dust,IR} = -\frac{1}{\rho _{a}c_{p}}\DD{}{z} \left[\sum _{\nu _{i} } \Delta ... ...} \left(F_{IR,\nu _{i}}^{\uparrow}-F_{IR,\nu _{i}}^{\downarrow}\right) \right].$$ (36)

$F_{dir,\nu _{i}}^{\downarrow}$ is the direct solar radiative flux per unit wave length,

$$F_{dir,\nu _{i}}^{\downarrow} = \mu _{0}S_{\nu _{i}} e^{-\tau_{\nu _{i}}/\mu _{0}}$$ (37)

The dust opacity is calculated by using the mass mixing ratio and effective radius of dust. In this model, we suppose that the size distribution of dust particle is the modified gamma distribution (Toon et al., 1977).

$$\DD{n(r)}{r} = n_{0}r ^{\alpha} \exp \left[ - \left(\frac{... ...gamma }\right) \left(\frac{r}{r_{m}}\right)^{\gamma }\right].$$ (38)

Dust opacity

The monoclomatic optical depth $\tau _{\nu}$ is represented by using the extinction coefficient per unit volume $\beta _{e,\nu}$ as follows.

$$\tau _{\nu}(z) = - \int _{z_{t}}^{z} \beta _{e,\nu}(r) \Dd z$$ (39)

where $z_{t}$ is altitude at the top of atmosphere. $\beta _{\nu,e}$ is given as follows.

$$\beta _{e,\nu} = \int _{0}^{\infty} \sigma_{e,\nu}(r)\DD{n(r)}{r}\Dd r$$ (40)

where $\sigma_{e,\nu}$ is the extinction cross section, $dn(r)/dr$ is the size distribution of scattering particle (cf. Liou, 1980; Shibata, 1999). By using extinction coefficient per unit mass $k_{e}$, (40) is rewritten as follows.

$$\rho _{a}q_{s}k_{e,\nu} = \int _{0}^{\infty} \sigma_{e,\nu}(r)\DD{n(r)}{r}\Dd r$$ (41)

where $\rho _{a}$ is atmospheric density, and $q_{s}$ is mass mixing ratio of scattering particle. Similarly, the scattering and absorption coefficient per unit volume are represented by using the scattering cross section $\sigma_{s,\nu}$ and the absorption cross section $\sigma_{a,\nu}$ as follows.

$$\beta _{s,\nu} = \int _{0}^{\infty} \sigma_{s,\nu}(r)\DD{n(r)}{r}\Dd r,$$ (42)

$$\beta _{a,\nu} = \int _{0}^{\infty} \sigma_{a,\nu}(r)\DD{n(r)}{r}\Dd r,$$ (43)

and the single scattering albedo $\tilde{\omega}_{\nu}$ is given as follows.

$$\tilde{\omega}_{\nu} = \frac{\beta _{s,\nu}}{\beta _{a,\nu}}$$ (44)

The extinction efficiency $Q_{e,\nu}$ is defined as the ration of extinction cross section to geometric cross section.

$$Q_{e,\nu} = \frac{\sigma_{e,\nu}}{\pi r^{2}},$$ (45)

Similarly, the scattering efficiency $Q_{s,\nu}$ and absorption efficiency $Q_{a,\nu}$ is defined as follows.

$$Q_{s,\nu} = \frac{\sigma _{s,\nu}}{\pi r^{2}},$$ (46)

$$Q_{a,\nu} = \frac{\sigma _{a,\nu}}{\pi r^{2}}.$$ (47)

In this model, the dust opacity is derived from the mass mixing ratio of atmospheric dust. Given parameters are the cross section weighted mean extinction efficiency $\overline{Q}_{e,\nu}$, the single scattering albedo $\tilde{\omega}_{\nu}$, the size distribution function of dust $dn(r)/dr$, the mode radius $r_{m}$, the effective (or, cross section weighted mean) radius $r_{eff}$, and the density of dust particle $\rho _{d}$. $\overline{Q}_{e,\nu}$ and $r_{eff}$ are defined as follows, respectively.

$$\overline{Q}_{e,\nu} \equiv \frac{\int _{0}^{\infty}Q_{e,\nu}\pi r ^{2}\DD{n(r)}{r}\Dd r} {\int _{0}^{\infty}\pi r ^{2}\DD{n(r)}{r} \Dd r},$$ (48)

$$r_{eff} \equiv \frac{\int _{0}^{\infty}r ^{3}\DD{n(r)}{r}\Dd r} {\int _{0}^{\infty}r ^{2}\DD{n(r)}{r} \Dd r},$$ (49)

Supposing that the shape of scattering particle is sphere, the extinction coefficient per unit mass is given as follows.

$$\beta _{e,\nu} = \overline{Q}_{e,\nu} \int _{0}^{\infty} \pi r^{2}\DD{n(r)}{r}\Dd r,$$

$$= \frac{\overline{Q}_{e,\nu}}{r_{eff}} \int _{0}^{\infty} \pi r^{3}\DD{n(r)}{r}\Dd r,$$

$$= \frac{\overline{Q}_{e,\nu}}{r_{eff}} \frac{3\rho _{a}q_{s}}{4\pi \rho _{d}},$$ (50)

where $\rho _{a}$ is the atmospheric density. Therefore, the optical depth can be represented by using the mass mixing ratio as follows.

$$\tau _{\nu} = -\int _{z_{t}}^{z} \frac{\overline{Q}_{e,\nu}}{r_{eff}} \frac{3\rho _{a}q_{s}}{4\pi \rho _{s}} \Dd z,$$ (51)

Parameters

The values of band width and optical parameters of dust (extinction efficiency, single scattering albedo, asymmetry factor) considered in this model are following to those of Forget et al. (1999) except for 11.6-20 $\mu$m band of dust. The overlap between visible band of dust and CO${}_{2}$ near infrared band is omitted.

The 5-11.6 $\mu$m infrared dust opacity $\tau _{5-1.6 \mu m}$ is obtained by dividing the visible dust opacity by the visible to infrared opacity ratio $\tau _{0.67 \mu m}/\tau _{9 \mu m}$, which is set to be 2. (Forget, 1998). The 20-200 $\mu$m infrared dust opacity is calculated by using $\tau _{5-11.6 \mu m}$ and the value of $Q_{e,\nu_{i}}/Q_{e,0.67\mu m}$ shown in Table 8.

表 8: Band width and optical parameters of dust

Band($\mu$m)	Band(cm${}^{-2}$)	$Q_{e,\nu_{i}}/Q_{e,0.67\mu m}$	$\tilde{\omega}_{\nu _{i}}$	$g_{\nu_{i}}$
0.1-5 $\mu$m	2000-10${}^{5}$	1.0	0.920	0.55
5-11.6 $\mu$m	870-2000	0.253	0.470	0.528
20-200 $\mu$m	50-500	0.166	0.370	0.362

ダストの光学パラメータ

表 9: Other parameters

Parameters	Standard values	Note
$Q_{e,0.67 \mu m}$	3.04	Ockert-Bell, et al. (1997)
$\tau _{0.67 \mu m}/\tau _{9 \mu m}$	2	Forget (1998)
$r_{eff}$	2.5 $\mu$m	Pollack et al. (1979)
$r_{m}$	0.4 $\mu$m	Pollack et al. (1979)

その他のダストに関するパラメータ

5.3 Solar flux and zenith angle

The solar flux at the top of atmosphere $F_{s}$ is depend on season, latitude and local time. In this section, we show $F_{s}$ as a function of local time at a specified season and latitude.

Suppose that $I_{0}$ (Wm${}^{-2}$) is solar constant on the mean orbital radius of Planet, $r$ and $r_{0}$ is the radius of orbit and its mean value, $\zeta$ is solar zenith angle, $\phi$ is latitude, $\delta$ is the solar inclination, $h$ is the hour angle ( $= 2\pi t/T -\pi$, $T$ is length of day ). $F_{s}$ is represented by using these variables as follows.

$$F_{s} = I_{0}\left(\frac{r_{0}}{r}\right)^{2}\cos \zeta,$$ (52)

$$\cos \zeta = \sin \phi \sin \delta + \cos \phi \cos \delta \cos h,$$ (53)

(c.f., Ogura, 1999). $r$ and $\delta$ are given as follows.

$$\begin{eqnarray*} r &=& \frac{a(1-e^{2})}{1+e\cos \omega}, \\ \sin \delta &=& \sin \alpha \sin (\omega - \omega _{0}) \end{eqnarray*}$$

where $\theta$ is the longitude relative to the perihelion, $a$ is the semimajor axis of orbit, $e$ is the eccentricity, $\alpha$ is the declination, $\omega$; is the true anomaly, and $\omega _{0}$ is the longitude of vernal equinox relative to the perihelion. By introducing the areocentric longitude of the sun $L_{s}\equiv \omega - \omega _{0}$, $F_{s}$ is rewritten as follows.

$$F_{s} = I_{0}\left(\frac{1+e\cos (L_{s}+\omega_{0})}{1+e^{2... ...s h \cos \phi \sqrt{1-\sin ^{2}\alpha \sin ^{2}L_{s}} \right]$$ (54)

Parameters

表 10: Parameters for solar flux and zenith angle

Parameters	Standard values	Note
$\phi$	20${}^{\circ}$N	Pollack et al. (1979)
$L_{s}$	100 ${}^{\circ}$	〃
$e$	0.093
$\alpha$	25.2${}^{\circ}$
$\omega _{0}$	110${}^{\circ}$	Carr (1996), Fig. 1
$I_{0}$	591 Wm${}^{-2}$