Water Transport to Habitable Planets

[Widdekind]

A star's Habitable Zone (HZ) is the region, around said star, where water exists as a liquid. In essence, the Habitable Zone is the Water Zone (Venus, Earth, Mars). Within this Water Zone, closer in towards the central star, water cannot condense. In essence, this is the Steam Zone (Mercury). But, beyond the Habitable Zone, farther from the central star, where water freezes, is the Ice Zone (Jupiter, Saturn)*. Roughly speaking the Habitable Zone is bounded between Rain Line (where water condenses as a liquid) and the Snow Line (where water freezes out as snow).

*

Carroll & Ostlie.

Introduction to Modern Astrophysics

, pg. ~893. Even farther from the central star,

Methane

also freezes, making a

Methane Ice Zone

(Uranus, Neptune)

.

Now, by virtue of forming farther from their central stars, out in the Ice Zone, colder planets can keep more of their water, through their Planet Formation process. For example, Jupiter's small moon Europa has about ten times (10x) more water than the Earth*. It probably possesses a Global Ocean about 100 km deep, making the moon the Solar System's largest reservoir of liquid water**. Indeed, ice's condensation dramatically increases the density, in early circum-stellar disks, of solid material. This causes the accumulation of significantly larger Proto-Planetary Cores (~10 M_earth), which can then retain their primordial Gaseous Envelopes, thereby accounting for the Gas Giants' huge masses***.

*

Prof. Woodruff T. Sullivan.

Athena Lecture Series -- Astrobiology

(Lecture #4)

,

8 Mar. 2009 AD

.

**

National Geographic Channel

Naked Science -- Journey to Jupiter

(TV)

***

Caleb Scharf.

Astrobiology & Extra-Solar Planets

.

Conversely, closer in towards the central star, ever hotter temperatures boil away increasing quantities of water, making such inner planets progressively drier (cf. Mercury). Thus, most water on such planets is probably brought back in by impacts, from Planetesimals, Comets, & Asteroids, scattered into the inner system, by the pull of Gas Giants' gravities (eg. Jupiter) from further out*.

*

National Geographic Channel

Naked Science -- Birth of the Oceans

(TV)

 

 

 

CONCLUSIONS: For Habitable Zones, Primordial Water retention, through the Planet Forming process, is inversely proportional to Temperature, which increases inwards, towards the central star. Thus, all else being equal, potentially habitable planets' quantities of water should tend to decrease inwards, towards the central star. In particular, potentially habitable planets born on the outer edge of their parent stars' Habitable Zones could frequently be created possessing comparatively bigger planetary Oceans.

 

Furthermore, potentially habitable planets orbiting smaller, dimmer stars (M,K-Class) could get greater quantities of water, by being physically closer to their system's Snow Line and water-bearing Icy Bodies. This is because the physical width (in AU) of a star's Habitable Zone is proportional to that star's mass. To see this, first consider an orbiting planet's expected Black Body Temperature, at Thermal Equilibrium:

[math]\frac{L_{*}}{4 \pi r^{2}} \; ( \pi R_{p}^{2}) = \sigma T_{p}^{4} \; (4 \pi R_{p}^{2})[/math]

Now, Planetary Global Temperatures (T_p) must be essentially the same, in all Habitable Zones (~300 K), by definition of liquid water being present. So, for a given star (L_*), the Orbital Radius ® of the Habitable Zone is:

[math]r^{2} = \frac{L_{*}}{16 \pi \sigma T_{p}^{4}}[/math]

Thus, the width of that star's Habitable Zone is roughly:

[math]\delta r = \sqrt{\frac{L_{*}}{16 \pi \sigma}} \left( \frac{-2}{T_{p}^{3}}\right) \delta T_{p} = -2 r \frac{\delta T_{p}}{T_{p}}[/math]

Since, by definition, every Habitable Zone has essentially the same requisite Planetary Temperature (T_p ~ 300 K) and permissible temperature drop (dT_p ~-100 K), the width of Habitable Zones scales as their radii.

 

In addition, stars' Luminosities (L_*) roughly scale as the fourth power of their Masses (M_*)*. So:

[math]\delta r \propto r \propto M_{*}^{2}[/math]

Bigger stars become so much brighter, that their Habitable Zones are pushed profoundly further outwards, where the Temperature Gradient (~r^-1/2) is significantly lower, which widens the Habitable Zone.

*

Bowers & Deeming.

Astrophysics I -- Stars

, pg. ~26.

Thus, the distance between a potentially habitable planet, orbiting in its parent star's Habitable Zone, and that star system's Snow Line and water-bearing Icy Bodies beyond, is dramatically reduced around smaller & dimmer stars. This could conceivably make Water Transport, by those water-bearing bodies, more common, creating larger Oceans, on habitable planets in such systems.

 

 

 

ADDENDUM: It can be shown, that an orbiting planet's Total Energy (E_p) and Angular Momentum (L_p) are roughly:

[math]E_{p,tot} = - \frac{G M_{*}}{2 r}[/math]

[math]L_{p} = \sqrt{G M_{*} r (1-e^{2})}[/math]

Restricting our attention to Habitable Zones, where dr ~ r ~ M_*², we therefore have that (as above):

[math]\delta E_{p,tot} = - E_{p,tot} \frac{\delta r}{r} \propto M_{*}^{-1}[/math]

[math]\delta L_{p} = \frac{L_{p}}{2} \frac{\delta r}{r} \propto M_{*}^{3/2}[/math]

Thus, compared to Habitable Zone planets orbiting bigger & brighter stars, those orbiting around smaller & dimmer stars are:

• Physically closer, in terms of Orbital Radius, to their star system's Snow Line & Icy Bodies
  
• More similar, in terms of Angular Momentum, to those Icy Bodies
  
• More different, in terms of Total Energy, from those Icy Bodies
  

Thus, if differences in Total Energy dramatically reduce the gravitational scatterings, of Icy Bodies into Inner Systems, Habitable Zone planets orbiting smaller & dimmer stars could conceivably tend to be comparatively Water Poor. But, if differences in Angular Momentum, or Orbital Distance, are the biggest barriers to Water Transport, Habitable Zone planets orbiting smaller & dimmer stars could conceivably tend to be comparatively Water Rich.

[GDG]

An interesting exposition. Of course, Venus is not believed to have any liquid water at the surface, due to a runaway greenhouse effect. Mars is generally too cold (an low pressure) to have liquid water. And the equation does not take into account heating by other means, such as radioactive decay and tidal stress (from orbiting a nearby massive planet).

 

And, of course, not every system resembles our solar system (in fact, AFAIK no other system closely resembles our system). There are several examples of systems with "hot Jupiters", or gas giants that orbit close to the parent star.

 

Your formula may also need some correction factors for albedo, as this affects how much solar radiation is absorbed.