Geoscience Reference
In-Depth Information
Fig. 13.11
Left panel
: Schematic presentation of an S-type system. The two stars of the binary,
primary and secondary, revolve around their centre of mass (
CoM
) whilst the planet orbits only
one of the stars (
top panel
). It is, however, customary to neglect the motion of the binary around
its CoM and consider the motion of the secondary around a stationary primary (
bottom panel
)
In this equation, F
Pl
is the total flux received by the planet, L
i
and T
i
(i=Pr, Sec)
represent the luminosity and effective temperature of the primary and secondary
stars, f is the cloud fraction of the planet's atmosphere and W
i
.f;T
i
/ is the binary
stars' spectral weight factor. The quantities r
PlPr
and r
PlSec
in Eq. (
13.1
)represent
the distances between the planet and the primary and secondary stars, respectively
(Fig.
13.11
). In using Eq. (
13.1
), we normalise the weighting factor to the flux of
the Sun.
From Eq. (
13.1
), the boundaries of the HZ of the binary can be defined as
distances where the total flux received by the planet is equal to the flux that Earth
receives from the Sun at the inner and outer edges of its HZ. Since in an S-type
system, the planet revolves around one star of the binary, we determine the inner and
outer edges of the HZ with respect to the planet-hosting star. As shown in Fig.
13.11
,
it is customary to consider the primary of the system to be stationary and calculate
the orbital elements with respect to the stationary primary star. In the rest of this
section, we will follow this convention and consider the planet-hosting star to be the
primary star as well. In that case, the range of the HZ of the binary can be obtained
from
W
Pr
.f;T
Pr
/
L
Pr
.T
Pr
/
l
xBin
C
W
Sec
.f;T
Sec
/
L
Sec
.T
Sec
/
r
PlSec
L
Sun
l
xSun
D
:
(13.2)
In Eq. (
13.2
), the quantity l
x
represents the inner and outer edges of the HZ with
x=(in,out). As mentioned earlier, the values of l
in
and l
out
are model dependent and
change for different values of cloud fraction, f , and atmosphere composition.
13.7.3
Calculation of Spectral Weight Factors
To calculate the spectral weight factor W.f;T/ for each star of the binary and in
terms of their SEDs, we calculate the stellar flux at the top of the atmosphere of an
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