It has been usual to define the habitable zone as the zone where there can be fluid water for billions of years on the surface of a planet, i.e. where the surface temperature of the planet is between T
_{1}=373K and T
_{2}=273K. This may, however, be an unrealistically large temperature interval for the development of complex life. If the average surface temperature of the Earth, which is 288 K, decreases by for example 10 degrees, then the conditions may be as in the Huronian glaciation where most of the Earth was covered by ice. If the temperature of the surface is higher than about , there would also be difficulties for complex life, and the rate of loss of hydrogen would be dangerously high over a billion year time scale. Hence is seems more realistic to choose, say
T_{1}=373K and
T_{1}=278K for the zone permitting evolution of complex life.
Usually one talks about one planetary zone of a star, indicating that it depends upon the properties of the star only. This has been a useful simplification in order to obtain conceptual simplicity, but now we need a more encompassing and physically realistic definition.
We shall here consider how changing luminosity, L, of a star changes the position of its habitable zone, and how albedo, A, and greenhouse effect (G) affect a planet’s position relative to the habitable zone of a planetary system. If we consider a planet moving along a circular path around a star, it will have a fixed distance from the star. The habitable zone will move radially when
L changes and the surface temperature of the planet will change due to changes of the albedo and the greenhouse effect. This may be described by introducing a fictitious radial displacement of the planet relatively to the habitable zone. If for example an increasing greenhouse effect causes a temperature increase the planet would “move” inwards in the radial zone.
However, the actual physical situation is that the planet has no radial motion. Hence it is the radial zone that gets an outwards displacement due to the increased greenhouse effect. This means that the habitable zone is not a property only of the star. It depends also upon the physical properties of a planet. In the Solar system, for example the Earth and Venus have different habitable zones due to the great difference of the greenhouse effect of these planets. The habitable zone of Venus is farther out in the Solar system than the habitable zone of the Earth.
For these reasons we suggest introducing the concept “planetary habitable zone” (PHZ), defined by taking into account the planet’s albedo and greenhouse effect when calculating at what distance from the star the planet’s surface temperature is
T_{1} (a maximal temperature compatible with evolution of complex life) and
T_{2} (a corresponding minimum temperature). Also an agreement should be obtained on realistic values of
T_{1} and
T_{2} to be used in the definition of the PHZ.
Radiation energy balance gives for the average surface temperature of the planet
,(A1)
where σ=5,67
^{.}10
^{-8}W/
m^{2}k
^{4} is the Stefan-Boltzmann constant, A is the planet’s albedo, S the star constant corresponding to the Solar constant S=1,37kW/m
^{2} at the Earth, i.e. S is the incoming radiation outside the atmosphere per unit surface area normal to the direction of the incoming radiation, and is the absorptivity of the planet’s atmosphere. For the Earth it is usual to put
A=0,26 and
ε=0. The absorptivity of the atmosphere represents the greenhouse effect. With no greenhouse effect we have
ε=0, and with maximal greenhouse effect where the atmosphere acts as a black body, we have
ε=1.
The relationship between the luminosity of a star and the “star constant”, i.e. received radiation per unit area at a distance r from the center of the star, is S=L/4πr
^{2}. Let S
_{1} be the value of the star constant at the inner boundary of the PHZ and S
_{2} the value at the outer boundary. Then
.(A2)
Hence
.(A3)
Let the width of the PHZ be
.(A4)
This gives
,(A5)
or
. (A6)
Inserting the relationship (A3) gives
. (A7)
With
T_{1}=373K and
T_{2}=273K we get Δ
r=0.87
r. On the other hand
T_{1}=343K and
T_{2}=278K gives Δr=0.52r. The width of the PHZ is approximately half the distance of its inner boundary to the star.
Equation (A7) shows that the width of the PHZ is proportional to the distance of its inner boundary from the star. As the star gets warmer and the PHZ moves outwards, the PHZ will broaden.
We shall now find the velocity of PHZ due to a change of the luminosity of the star, and of the albedo and greenhouse effect of the planet. We shall represent the greenhouse effect by only one parameter, the absorptivity, ε, of its atmosphere. From
(A8)
we get
.(A9)
Inserting this into equation (A7) we find how the width of the PHZ depends upon the luminosity of the mother star and the albedo and greenhouse effect of the planet,
,(A10)
Where
L_{e} is the luminosity of the Sun, and
ε_{⊗} and
A_{⊗} are the absorptivity of the Earth’s atmosphere and the albedo of the Earth, respectively. The formula shows that decreasing luminosity, increasing albedo and decreasing greenhouse effect (decreasing value of
ε) makes the PHZ thinner, which decreases the probability of finding a planet within the PHZ.
Differentiating equation (A9) we find
.(A11)
The physical interpretation of this equation is that the PHZ has a velocity
, (A12)
where
(A13)
is the contribution to the velocity of the PHZ due to a changing luminosity of the star,
(A14)
is the contribution due to a changing albedo, and
(A15)
is contribution due to a changing greenhouse effect.
The time taken for the PHZ to move so that a planet without radial motion changes its position relative to it from the outer boundary to the inner boundary is Δ
t=Δ
r/
V_{PHZ} . It follows from equations (A7) and (A11) that
.(A16)
This is a characteristic time telling how long a planet with circular path can remain inside the PHZ when the luminosity of the mother star and the albedo and greenhouse effect of the planet change. The characteristic time due to the change of the mother star’s luminosity, only, is
. (A17)
The rate of change of the luminosity of a main sequence star is with good accuracy given by [
16]
,(A18)
where
L_{e}=3,85
^{.}10
^{26}W is the present day solar luminosity, and
t_{e}=4.47Gyr is the age of the Sun. Differentiating and inserting the present rate of change into equation (A17) gives
. (A19)
Inserting
T_{1}=333K and
T_{2}=278K gives Δ
t_{L}=2.17
t_{e}=9.8
Gyr. The characteristic times associated with changes of the greenhouse effect and the albedo are much shorter.
Furthermore equations (A10) and (A18) give the time evolution of the width of the habitable zone in the Solar system as
, (A20)
where Δ
r_{e} is the present width of the habitable zone in the Solar system.
Small stars emit less radiation than large stars. The luminosity of main sequence stars with mass 0.43
M_{e}<
M<2
M_{e} is
(A21)
where
M_{e} is the mass of the Sun. Inserting this into equation (A9) we find that the distance of the inner boundary of the habitable zone has a distance from the center of the star,
, (A22)
where
r_{e} is the corresponding distance in the Solar system, i.e.
r_{e}=1AU . Inserting equation (A18) into equation (A7) we find how the extension of the habitable zone depends upon the mass of the mother star in this mass interval,
.(A23)
For a star with
M=0.6
M_{e}, for example, this gives Δ
r=0.36Δ
r_{e} or about the third of the width of the habitable zone in the Solar system. Hence the probability that a planet moving around such a small star is positioned within the habitable zone of the star is less than that of a planet in the Solar system. It may be noted that by inserting (A17) into equation (A10) the width of the PHZ is given as
(A24)