Calculation of Goldilocks Zone for a given planet and a given star

Distance from a star where water can exist in liquid state (between 273 K and 373 K), calculated with the star’s luminosity and the planet’s albedo
by student Mauro Benjamin Mistretta

Definitions:

Variables:
​
Planet’s orbit: orbit
temperature on planet’s surface: temperature
star’s luminosity: luminosity
albedo: albedo
​
Constants (input in the kernel, within the formulas):
​
σ = 5.67*
-8
10
W m
-2
​
-4
K
​
AU = 1.4986*
11
10
m

Planet’s effective temperature (considering albedo)

A planet’s effective temperature is calculated by the means of the formula
L=4π
2
R
σ
4
T
(1)Where L is the luminosity of the star.The temperature T , for the luminosity and the orbital radius, is:​T =
4
L
4π
2
R
σ
​​ENERGY (the flux) received by the planet’s surface exposed to the star’s light is calculated with the formula:​Flux = σ
4
T
= σ
L
4π
2
R
σ
=
L
4π
2
R
(2)​The planet’s irradiated surface = π
2
r
​where r is the radius of the planet.​The total energy received by the planet’s irradiated surface is:​E = Flux · Irradiated surface =
L
4π
2
R
π
2
r
=
L
2
r
4
2
R
​​By rearranging the formula (1) , we can calculate the planet’s effective temperature:​Effective temperature =
4
E
4πσ
2
r
=
4
L
16πσ
2
R
(3)
Manipulate[(((((100-albedo)/100)*3.862*10^26*luminosity)/(16π*5.67*10^-8*(1.4986*10^11*orbit)^2))^(1/4)),{luminosity,0.001,100000},{orbit,0.001,200},{albedo,0,100}]
Out[]=
​
luminosity
orbit
albedo
254.015
for example: Earth’s albedo 31 , luminosity 1 L⊙ and orbit 1 (AU) yields an effective T = 254.015 K

Orbital radius (in AU) for a planet to have a given effective temperature, albedo and star’s luminosity

This calculation is achieved by rearranging equation (3):​R =
L
16πσ
4
T
​​
Manipulate[((((((100-albedo)/100)*3.862*10^26*luminosity)/(16π*5.67*10^-8*temperature^4)))^(1/2))/(1.4986*10^11),{albedo,0,100},{luminosity,0.1,100000},{temperature,0.1,1000}]
Out[]=
​
albedo
luminosity
temperature
102.968
For example, a planet around Tau Ceti (0.5 L⊙ ) should orbit at 0.7 AU in order to receive the same Earth’s solar insolation level (Earth’s effective temperature is 254 K, with an albedo of 31).

Find goldilocks zone with inputs of albedo, star’s luminosity and planet’s surface temperature (boundaries 273 K - 373 K) - output in AU

calculation:

Formulainvolved:L=4π
2
R
σ
4
T
(1)​where R is the orbital radius of the planet, and σ is the Stefan-Boltzmann constant, defined above.​In order to find a simple formula for this task, I have considered the following magnitudes:​Flux from the star, given the freezing point and boiling point of water, assuming an albedo: σ
4
T
;The above flux if there were no albedo:
flux
noalbedo
=
σ
4
T
(100/(100-albedo));​The irradiated planet’s surface can be considered as the area of the circle with radius r (radius of the planet) which is exposed to the star’s light.​​Energy received on the irradiated planet’s surface:
flux
noalbedo
· planet’s surface (π
2
r
)
;​Then, the temperature T of the planet’s surface is calculated by the rearranging the Boltzmann law:​T =
4
Energyreceived
σ
(4)​We can now calculate the distance at a specific temperature (273 K and 373 K) rearranging the formula 1.1 :​Distance (AU) =
L
4πσ
4
T
(5)​Rearranging (3) with (1) , we have:​​Distance (in AU) =
L(insolarluminosities)
4πσ
4
T
·100/(100-albedo)
​by plugging in the calculation the two boundary temperatures we obtain the distances. The program let you change the albedo and the star’s luminosity:
{Manipulate[((((luminosity*3.862*10^26)/(4π*5.67*10^-8*273^4*(100/(100-albedo)))))^(1/2))/(1.4986*10^11),{luminosity,0.00001,100000},{albedo,0,100}],Manipulate[((((luminosity*3.862*10^26)/(4π*5.67*10^-8*373^4*(100/(100-albedo)))))^(1/2))/(1.4986*10^11),{luminosity,0.00001,100000},{albedo,0,100}]}
Out[]=

​
luminosity
albedo
1.7315
,
​
luminosity
albedo
0.927536

examples (with Earth’s albedo of 31):
​
Sun : Outer 1.731 AU / Inner 0.927 AU
Vega : 40 L⊙ → Outer 10.951 AU / Inner 5.86625 AU
Proxima Centauri: 0.0017 L⊙ ⟶ Outer 0.07 AU / Inner 0.038 AU
Shaula: 36,300 L⊙ ⟶ Outer 329.89 AU / Inner 176.719 AU