What Life would see around other Stars

[Widdekind]

Stable Planetary Systems -- those not possessing giant Hot Jupiters, which migrated all the way in towards their central stars, completely disrupting their Planetary Systems in the process -- apparently typically obey basic system-wide planetary profiles:

A typical calculation by Hughes (1992), in Fig. 2.5, shows the distances & types of planets that develop out of the Planetary Accretion Disks. Here, the big dots indicate Jovian Planets, and the small ones Terrestrial Planets & Kuiper Belt Objects. The generation of this sequence of planets w/ distance has also been found in simulations of Ida & Lin (2004, 2005). From Fig. 2.5, for example, it is seen that 5 Terrestrial Planets forum up to Jupiter distances [5.2 AU] around G-type stars [1 M_sol]. That such a result is typical has recently been concluded by Jacobsen (2005), in a review of the Planetesimal & Planetary Embryo Phase, as well as the main planet-building stage. He reported that computer simulations of the final stages of Accretion have succeeded in producing systems containing 2-4 Terrestrial Planets between 0-2 AU from the star.

Peter Ulmschneider. Intelligent Life in the Universe (2nd ed.), pg. 24.

 

[Widdekind]

Most Exo-Planetary Systems tend to be comparatively young (< ~6 Gyr; see above, post #22). For sake of comparison, then, most Exo-Planetary Systems may have formed after the early Universal Epoch of Quasars and AGNs. Attached is Figure 22 from Martin Rees' New Perspectives in Astrophysical Cosmology, pg. 77 (Rees uses a Hubble's Constant of 50 km/s/Mpc; current estimates imply H ~ 75 km/s/Mpc, corresponding to h₅₀ = 1.5 in the attached figure).

 

[Widdekind]

Stable Planetary Systems chart from Lewis Dartnell's Life in the Universe, pg. 149.

 

[Widdekind]

Stable Planetary Systems chart, merged from Ulmschneider & Dartnell (both above). Ulmschneider seemingly suggests, that the inner portions of Stellar Kuiper Belts typically produce Uranus- & Neptune-type planets, them being but "big Kuiper Belt Objects" (as it were). There seems to be some disagreement, as to the range of Terrestrial Planet Formation.

 

[Widdekind]

According to this amateur site, low-mass stars (< 0.2 M_sol) have purely convective cores, which affects their violent Flare Activities & Magnetic Dynamos.

Could this explain the change in slope, of the Habitable Zone's radius vs. star mass ?

 

[Widdekind]

Population I stars are the newest & youngest & most metal-rich stars around.

They formed from the "ashes" of burnt out (high mass) Population II stars, whose lower mass brethren are the oldest & most metal-poor stars still around.

Those Population II stars, in turn, formed from as-yet-unobservered Pop. III stars, back at the beginning of the Universe -- whose past presence is apparently necessary to explain the existence of what small amounts of metals there actually are in Pop. II stars.

Now, older & more metal-poor Pop. II stars possesses far fewer Planetary Systems -- for, Planetary Formation is strongly correlated w/ Metallicity. But, Pop. II stars tend, not only to be metal poor, but poor in particular types of metals -- especially Iron Peak Elements:

Theoretical galactic evolution models predict that early in the Universe there were more Alpha Elements relative to Fe. Type II supernovae [Core Collapse] mainly synthesize oxygen and the alpha-elements (Ne, Mg, Si, S, Ar, Ca and Ti) while Type Ia supernovae [White Dwarves] produce elements of the iron peak (V, Cr, Mn, Fe, Co and Ni).

http://en.wikipedia.org/wiki/Alpha_process

Thus, older Pop. II star systems may be especially "Iron poor". Even those few that possesses Planetary Systems, will possesses planets composed chiefly of C,N,O and the Alpha Elements (including Si), but w/o much Fe,Co,Ni. Thus, those "primitive planets", now many billions of years old, may be "all Mantle & Crust, no Core". They may not have either the (1) Iron-peak Elements necessary to produce Planetary Magnetic Fields; or (2) Heavy Radioactive Elements necessary to keep their Planetary Interiors warm, molten, & Geologically active.

W/ only tiny Iron Cores, weak Planetary Magnetic Fields, and comparatively rapid Geological cooling, primitive planets orbiting Pop. II stars may be uninhabitable[/color].

If so, this could affect the "search space" for SETI-type programs.

 

[marcus]

Widdekind I have to express congratulations and appreciation for this thread. You seem to me to be doing some very capable search-and-assembly of information. gathering from various sources and putting it together.
what I like is you seem to want to visualize carefully at the level of detail what kinds of star would have interesting kinds of exoplanets, and give the particulars about where life (and especially advanced life) might arise.

It is nice to see all these charts too. surely in future "exoplanet science" will be an active (possibly important) area of research.

 

[Widdekind]

Thanks !

 

[Widdekind]

ROUGH DRAFT​

Assuming some speculation, about Exoplanets, be permitted...

(1) G Y = C

Science-Fiction writer JT Bass, in his book The Godwhale, observes that, for all known Habitable Planets (N = 1),

G Y = C​

where G is the Surface Gravity, Y is Orbital Period, and C is the Speed of Light. For example, if you accelerated at one Earth Gravity, for one Earth Year, you would reach to the Speed of Light (ignoring Relativistic effects)*.

* This equation is sometimes called "Olga's Prayer" (OLGA was an intelligent robot starship).​

For example, compared to Earth, Venus scores about 0.56, while Mars — thought to be much more potentially habitable — scores about 0.71. Likewise, Gliese 581-c scores about 0.07, while Gliese 581-d — thought to be "a better candidate for habitability"* — scores about 0.50. Thus, for Terrestrial-type Planets, (Earth-normalized) scores increasingly close to one, do indeed seemingly coincide, w/ potential Habitability.

* http://en.wikipedia.org/wiki/Gliese_581_c​

For the following, we completely accept Bass' observation as valid.



(2) G scales as the Habitable Planet's Lifetime (for Planetological Activity)

Radioactive Decay, of heavy elements, dominates the Heat Budget of Terrestrial Planets. And, at least for a given Metallicty of the parent star, the mass of Radioactive elements -- and, hence, the Heat Budget of the Habitable Planet -- should scale as that planet's total Mass. But, that Heat Energy is radiated away, at a rate proportional to that planet's total Surface Area. Thus, the "Planetary Cooling Timescale" \tau_{p} should scale as that planet's Mass divided by its Surface Area, which is essentially its Surface Gravity:

\tau_{p} \propto \frac{M_{p}}{R_{p}^{2}} \propto G_{p}​

Based upon comparisons to Mars, Earth might be Geologically Active until it is 8-9 billion years old*. For our rough guestimations, we adopt the fiducial value, for the Lifetime of the Earth, of 10 Gyr -- or roughly 1 Gyr per m/s/s of Surface Gravity.

* http://www.astronomyforum.net/planetary-forums/82414-estimating-planetary-cooling-times.html​

(3) Y scales inversely as the central star's Stellar Lifetime (on the Main Sequence)

Stellar Luminosities scale roughly as the fourth-power of the stellar Mass (Bowers & Deeming. Stars, pg. 28). And, stars' Main Sequence Lifetimes scale roughly as their Masses (amount of fuel for fusion) divided by their Luminosities (fuel burn rate). Thus, Stellar Lifetimes scale as:

\tau_{*} \propto \frac{M_{*}}{L_{*}} \propto M_{*}^{-3}​

Meanwhile, those stars' Habitable Zones are defined as those regions, around the central star, where the received Light Flux is roughly that from the Sun, at the Earth:

\frac{L_{*}}{R_{HZ}^{2}} \approx constant​

Thus, adopting our previous approximations:

R_{HZ}^{2} \propto L_{*} \propto M_{*}^{4}
R_{HZ} \propto M_{*}^{2}​

And, Kepler's Laws inform us that:

Y_{HZ}^{2} \propto \frac{R_{HZ}^{3}}{M_{*}} \propto M_{*}^{5}
Y_{HZ} \propto M_{*}^{2.5}​

But, using only rough & round numbers:

Y_{HZ} \propto M_{*}^{\approx 3} \propto \frac{1}{\tau_{*}}​

For our rough guestimations, we adopt the fiducial value, for the Lifetime of the Sun, of 10 Gyr -- or roughly 10 Gyr per yr^-1 of Habitable Planet Orbital Period.



(4) G Y = C essentially equates the Lifetimes of the central star, and its Habitable Planet

We have seen that:

G \propto \tau_{p}
Y \propto \tau_{*}^{-1}​

Thus, the "G Y = C" equation can be converted to:

\frac{\tau_{p}}{\tau_{*}} \approx constant​

For our fiducial values, for planetary & stellar Lifetimes, as based upon the Sun & Earth (both being about 10 billion years):

\frac{\tau_{p}}{\tau_{*}} \approx 1​

This seems to suggest, that the Lifetimes, of the central star and its potentially Habitable Planet, must be matched*.

* Since Terrestrial-type Planets have a maximum Lifetime for Planetological Activity of about 30 billion years (see below), whereas small, dim, Dwarf Stars can remain on the Main Sequence for trillions of years, matching the Lifetimes amounts to imposing a minimum mass upon the central star of \approx 3^{-1/3} \; M_{\odot} \approx 0.7 \; M_{\odot}. This is not implausible, which claim Orange K-Dwarfs are the smallest stars that seem to produce Planetary Systems (see previous posts above). Furthermore, since Exoplanetologists argue that Terrestrial-type Planets must be massive enough to make at least about 1/3^rd Earth Gravities — in order to remain Planetologically Active long enough to support the development of Complex Life — such potentially Habitable Planets must have Lifetimes of at least about 3 billion years, which amounts to imposing a maximum mass upon the central star, of \approx 3^{1/3} \; M_{\odot} \approx 1.4 \; M_{\odot}. And, indeed, such Green F-Class stars are about as the biggest the apparently produce Planetary Systems (see previous posts above).​

(5) Probability analysis, & prediction, for observing first Habitable Planet

(A) IMF, & Probability Density distribution, for central stars

Saltpeter's standard Initial Mass Function (IMF) informs us of the approximate probabilities of observing stars of varying masses:

P(M_{*}) dM_{*} \propto M_{*}^{-\alpha} dM_{*}​

From the relation above, between star Mass and star Lifetime (\tau_{*} \propto \M_{*}^{-3}), we can convert this IMF from Masses to Lifetimes:

P(M_{*}) dM_{*} \equiv P(\tau_{*}) d\tau_{*}
P(\tau_{*}) \equiv P(M_{*}) \frac{dM_{*}}{d\tau_{*}} \propto -\frac{1}{3} \; \tau_{*}^{\frac{\alpha-4}{3}}​

This can be integrated to give the Normalization Constant, ensuring that the integral, over time, of the Probability Density Function, equals one. Quick application of Calculus reveals, that the resulting Normalized Probability Density Function is:

P(\tau_{*}) \equiv \frac{\alpha - 1}{3} \frac{1}{T_{max}} \left( \frac{\tau_{*}}{T_{max}}\right)^{\frac{\alpha - 4}{3}}​

where we adopt the standard Saltpeter exponent, of \alpha = 2.35.


(B) IMF, & Probability Density distribution, for potentially Habitable Planets

By essentially similar means, we may assume an IMF for Terrestrial-type Planets,

P(M_{p}) dM_{p} \propto M_{p}^{-\beta} dM_{p}​

Then, we can convert that planetary IMF to planetary Lifetimes (equivalent to planetary Surface Gravities):

P(M_{p}) dM_{p} \equiv P(G_{p}) dG_{p}​

To do so, we use a simple approximation, to the numerically computed relation, between Radius & Mass, for Terrestrial-type Planets. For Terrestrial-type Planets, of sufficiently small Planetary Mass (M M_{p} \leq 10 \; M_{\oplus}), this relation shows, that for Rocky Worlds, of Earth-like Composition, the Average Bulk Density (\rho) increases nearly linearly with Surface Gravity (G):

\rho = \rho_{0} + k \times G (eq. 1)​

where, using Earth-normalized units:

\rho_{0} \approx 0.536 [ 2.96 g cm^-3 ]
k \approx 0.464​

Note that \rho_{0} represents the natural "uncompressed density", of Terrestrial-type rocks, at zero Surface Gravity, consistent with the observed density of Asteroids*.

* http://www.scienceforums.net/forum/showthread.php?t=40699. For an even more detailed diagram, see: Caleb A. Scharf. Extrasolar Planets & Astrobiology, pg. 121.​

In particular, this simple approximation implies, that (in Earth-normalized units):

M_{p} = \rho_{p} \; R_{p}^{3}
G_{p} = \rho \; R_{p}​

so that:

M_{p} = \frac{G_{p}^{3}}{\rho_{p}^{2}} = \frac{G_{p}^{3}}{\left( \rho_{0} + k \times G_{p} \right)^{2}}​

Therefore:

P(G_{p}) = P(M_{p}) \frac{dM_{p}}{dG_{p}} = P(M_{p}) \; \left( \frac{G_{p}^2}}{\rho_{p}^3}} \right) \; \left( 2 \rho_{0} + \rho_{p} \right)​

Again, we assume a power-law IMF, for Terrestrial-type Planets. Note that this power-law's exponent, \beta, is surely smaller than that for Saltpeter's stellar IMF. For, numerical simulations of Planetary Formation, repeatedly produce Terrestrial-type Planets, relatively evenly distributed, across the whole range of masses, up to about 10 M_{\oplus}*. Below, we adopt exponents in the range 0 \leq \Beta &lt; 1.

* Caleb A. Scharf, ibid, pg. 123.​

(C) Joint Compound Probability Density distribution

Once we have the Normalized Probability Density Functions, for both parent stars, and their Terrestrial-type Planets, we can calculate the approximate probability of observing a parent star, and a Terrestrial-type Planet, both having roughly equivalent Lifetimes. If we accept a tolerance of T (10% say), so that the planetary Lifetime \tau_{p} need only lie within said tolerance of its central star's Lifetime, then we seek to integrate the Compound Probability Density Function:

P_{*}(t) \; P_{p}(t) \; (T \times t) \; dt​

from zero to the longest Lifetime allowed.

Now, stars of roughly 0.1 M_{\odot} are expected to live about 10 trillion years. But, Exoplanetologists argue, that the biggest possible Terrestrial-type Planet is about 10 M_{\otimes} -- which would have a Surface Gravity of about 3 Earth Gravities, and, hence, an expected Planetary Lifetime of about 30 billion years. Thus, the longest Lifetime allowed, in the integration of the above Compound Probability Density Function, is ~30 Gyr.

Numerical integration, using SciLab, reveals the following results, for varying IMF_p \beta values:

Beta      Probability
0         0.0081968
0.5       0.0035200
0.9       0.0008959

or, in rough & round numbers:

Beta      Probability
0         0.008
0.5       0.004
0.9       0.0009

These probabilities must be multiplied by at least two additional factors: first, the probability of a central star producing a Planetary System; and, second, the probability of that Planetary System putting a Terrestrial-type Planet in said central star's Habitable Zone. The former probability is about one-half to two-thirds*; the latter probability appears to be appreciable, as long as the Planetary System is not disrupted by Gas Giant Migrations of "Hot Jupiters"**.

* J.V. Narlikar. Star Factories, pg. ~30.
** Caleb A. Scharf, ibid., pg. 123. The plotted computer simulations produced a plethora of Terrestrial-type Planets, evenly distributed across the central star's Inner Star System. Indeed, the same can be said for our Solar System.​

Thus, this simply estimated probability, of observing a "G Y = C compliant" Star System, is of order 10^-3.

 

[Vanadium 50]

Science-Fiction writer JB Bass, in his book ____, observes that, for all known Habitable Planets (N = 1),

G Y = C​

First, this is PhysicsForums, not ScienceFictionForums.

Second, this equation is a numerical accident, no more and no less so than there are \pi \times 10^7 seconds in a year. Well, actually it's about seven times worse - the Bass coincidence is only good to about 3% and the one I posted is good to better than half a percent.

Third, it's not true that the number of habitable zone planets is one. There are dozens. If one picks, say Gleise 581c at random it's clear that this isn't even close to being true - it has something like double Earth's gravity and a year of only 5 days. And surely you aren't arguing that a planet in a similar orbit with only one-and-a-half percent of Earth's gravity would be habitable.

Fourth, your reference is terrible. Not only did you not give the book title, but there is no author named J.B. Bass on Amazon, and a search on Google turns up this thread. Pretty shoddy scholarship if you ask me.

Since the premise is nonsense - and unsupported by the data - so are any conclusions that stem from it.

 

[Widdekind]

ROUGH DRAFT​

Assuming some speculation, about Exoplanets, be permitted...

(1) G Y = C

Science-Fiction writer JT Bass, in his book The Godwhale, observes that, for all known Habitable Planets (N = 1),

G Y = C​

where G is the Surface Gravity, Y is Orbital Period, and C is the Speed of Light. For example, if you accelerated at one Earth Gravity, for one Earth Year, you would reach to the Speed of Light (ignoring Relativistic effects)*.

* This equation is sometimes called "Olga's Prayer" (OLGA was an intelligent robot starship).​

For example, compared to Earth, Venus scores about 0.56, while Mars — thought to be much more potentially habitable — scores about 0.71. Likewise, Gliese 581-c scores about 0.07, while Gliese 581-d — thought to be "a better candidate for habitability"* — scores about 0.50. Thus, for Terrestrial-type Planets, (Earth-normalized) scores increasingly close to one, do indeed seemingly coincide, w/ potential Habitability.

* http://en.wikipedia.org/wiki/Gliese_581_c​

For the following, we completely accept Bass' observation as valid.



(2) G scales as the Habitable Planet's Lifetime (for Planetological Activity)

Radioactive Decay, of heavy elements, dominates the Heat Budget of Terrestrial Planets. And, at least for a given Metallicty of the parent star, the mass of Radioactive elements -- and, hence, the Heat Budget of the Habitable Planet -- should scale as that planet's total Mass. But, that Heat Energy is radiated away, at a rate proportional to that planet's total Surface Area. Thus, the "Planetary Cooling Timescale" \tau_{p} should scale as that planet's Mass divided by its Surface Area, which is essentially its Surface Gravity:

\tau_{p} \propto \frac{M_{p}}{R_{p}^{2}} \propto G_{p}​

Based upon comparisons to Mars, Earth might be Geologically Active until it is 8-9 billion years old*. For our rough guestimations, we adopt the fiducial value, for the Lifetime of the Earth, of 10 Gyr -- or roughly 1 Gyr per m/s/s of Surface Gravity.

* http://www.astronomyforum.net/planetary-forums/82414-estimating-planetary-cooling-times.html​

(3) Y scales inversely as the central star's Stellar Lifetime (on the Main Sequence)

Stellar Luminosities scale roughly as the fourth-power of the stellar Mass (Bowers & Deeming. Stars, pg. 28). And, stars' Main Sequence Lifetimes scale roughly as their Masses (amount of fuel for fusion) divided by their Luminosities (fuel burn rate). Thus, Stellar Lifetimes scale as:

\tau_{*} \propto \frac{M_{*}}{L_{*}} \propto M_{*}^{-3}​

Meanwhile, those stars' Habitable Zones are defined as those regions, around the central star, where the received Light Flux is roughly that from the Sun, at the Earth:

\frac{L_{*}}{R_{HZ}^{2}} \approx constant​

Thus, adopting our previous approximations:

R_{HZ}^{2} \propto L_{*} \propto M_{*}^{4}
R_{HZ} \propto M_{*}^{2}​

And, Kepler's Laws inform us that:

Y_{HZ}^{2} \propto \frac{R_{HZ}^{3}}{M_{*}} \propto M_{*}^{5}
Y_{HZ} \propto M_{*}^{2.5}​

But, using only rough & round numbers:

Y_{HZ} \propto M_{*}^{\approx 3} \propto \frac{1}{\tau_{*}}​

For our rough guestimations, we adopt the fiducial value, for the Lifetime of the Sun, of 10 Gyr -- or roughly 10 Gyr per yr^-1 of Habitable Planet Orbital Period.



(4) G Y = C essentially equates the Lifetimes of the central star, and its Habitable Planet

We have seen that:

G \propto \tau_{p}
Y \propto \tau_{*}^{-1}​

Thus, the "G Y = C" equation can be converted to:

\frac{\tau_{p}}{\tau_{*}} \approx constant​

For our fiducial values, for planetary & stellar Lifetimes, as based upon the Sun & Earth (both being about 10 billion years):

\frac{\tau_{p}}{\tau_{*}} \approx 1​

This seems to suggest, that the Lifetimes, of the central star and its potentially Habitable Planet, must be matched*.

* Since Terrestrial-type Planets have a maximum Lifetime for Planetological Activity of about 30 billion years (see below), whereas small, dim, Dwarf Stars can remain on the Main Sequence for trillions of years, matching the Lifetimes amounts to imposing a minimum mass upon the central star of \approx 3^{-1/3} \; M_{\odot} \approx 0.7 \; M_{\odot}. This is not implausible, which claim Orange K-Dwarfs are the smallest stars that seem to produce Planetary Systems (see previous posts above). Furthermore, since Exoplanetologists argue that Terrestrial-type Planets must be massive enough to make at least about 1/3^rd Earth Gravities — in order to remain Planetologically Active long enough to support the development of Complex Life — such potentially Habitable Planets must have Lifetimes of at least about 3 billion years, which amounts to imposing a maximum mass upon the central star, of \approx 3^{1/3} \; M_{\odot} \approx 1.4 \; M_{\odot}. And, indeed, such Green F-Class stars are about as the biggest the apparently produce Planetary Systems (see previous posts above).​

(5) Probability analysis, & prediction, for observing first Habitable Planet

(A) IMF, & Probability Density distribution, for central stars

Saltpeter's standard Initial Mass Function (IMF) informs us of the approximate probabilities of observing stars of varying masses:

P(M_{*}) dM_{*} \propto M_{*}^{-\alpha} dM_{*}​

From the relation above, between star Mass and star Lifetime (\tau_{*} \propto \M_{*}^{-3}), we can convert this IMF from Masses to Lifetimes:

P(M_{*}) dM_{*} \equiv P(\tau_{*}) d\tau_{*}
P(\tau_{*}) \equiv P(M_{*}) \frac{dM_{*}}{d\tau_{*}} \propto -\frac{1}{3} \; \tau_{*}^{\frac{\alpha-4}{3}}​

This can be integrated to give the Normalization Constant, ensuring that the integral, over time, of the Probability Density Function, equals one. Quick application of Calculus reveals, that the resulting Normalized Probability Density Function is:

P(\tau_{*}) \equiv \frac{\alpha - 1}{3} \frac{1}{T_{max}} \left( \frac{\tau_{*}}{T_{max}}\right)^{\frac{\alpha - 4}{3}}​

where we adopt the standard Saltpeter exponent, of \alpha = 2.35.


(B) IMF, & Probability Density distribution, for potentially Habitable Planets

By essentially similar means, we may assume an IMF for Terrestrial-type Planets,

P(M_{p}) dM_{p} \propto M_{p}^{-\beta} dM_{p}​

Then, we can convert that planetary IMF to planetary Lifetimes (equivalent to planetary Surface Gravities):

P(M_{p}) dM_{p} \equiv P(G_{p}) dG_{p}​

To do so, we use a simple approximation, to the numerically computed relation, between Radius & Mass, for Terrestrial-type Planets. For Terrestrial-type Planets, of sufficiently small Planetary Mass (M M_{p} \leq 10 \; M_{\oplus}), this relation shows, that for Rocky Worlds, of Earth-like Composition, the Average Bulk Density (\rho) increases nearly linearly with Surface Gravity (G):

\rho_{p} = \rho_{0} + k \times G_{p} (eq. 1)​

where, using Earth-normalized units:

\rho_{0} \approx 0.536 [ 2.96 g cm^-3 ]
k \approx 0.464​

Note that \rho_{0} represents the natural "uncompressed density", of Terrestrial-type rocks, at zero Surface Gravity, consistent with the observed density of Asteroids*.

* http://www.scienceforums.net/forum/showthread.php?t=40699. For an even more detailed diagram, see: Caleb A. Scharf. Extrasolar Planets & Astrobiology, pg. 121.​

In particular, this simple approximation implies, that (in Earth-normalized units):

M_{p} = \rho_{p} \; R_{p}^{3}
G_{p} = \rho_{p} \; R_{p}​

so that:

M_{p} = \frac{G_{p}^{3}}{\rho_{p}^{2}} = \frac{G_{p}^{3}}{\left( \rho_{0} + k \times G_{p} \right)^{2}}​

Therefore:

P(G_{p}) = P(M_{p}) \frac{dM_{p}}{dG_{p}} = P(M_{p}) \; \left( \frac{G_{p}^2}{\rho_{p}^3} \right) \; \left( 2 \rho_{0} + \rho_{p} \right)​

Again, we assume a power-law IMF, for Terrestrial-type Planets. Note that this power-law's exponent, \beta, is surely smaller than that for Saltpeter's stellar IMF. For, numerical simulations of Planetary Formation, repeatedly produce Terrestrial-type Planets, relatively evenly distributed, across the whole range of masses, up to about 10 M_{\oplus}*. Below, we adopt exponents in the range 0 \leq \Beta &lt; 1.

* Caleb A. Scharf, ibid, pg. 123.​

(C) Joint Compound Probability Density distribution

Once we have the Normalized Probability Density Functions, for both parent stars, and their Terrestrial-type Planets, we can calculate the approximate probability of observing a parent star, and a Terrestrial-type Planet, both having roughly equivalent Lifetimes. If we accept a tolerance of T (10% say), so that the planetary Lifetime \tau_{p} need only lie within said tolerance of its central star's Lifetime, then we seek to integrate the Compound Probability Density Function:

P_{*}(t) \; P_{p}(t) \; (T \times t) \; dt​

from zero to the longest Lifetime allowed.

Now, stars of roughly 0.1 M_{\odot} are expected to live about 10 trillion years. But, Exoplanetologists argue, that the biggest possible Terrestrial-type Planet is about 10 M_{\otimes} -- which would have a Surface Gravity of about 3 Earth Gravities, and, hence, an expected Planetary Lifetime of about 30 billion years. Thus, the longest Lifetime allowed, in the integration of the above Compound Probability Density Function, is ~30 Gyr.

Numerical integration, using SciLab, reveals the following results, for varying IMF_p \beta values:

Beta      Probability
0         0.0081968
0.5       0.0035200
0.9       0.0008959

or, in rough & round numbers:

Beta      Probability
0         0.008
0.5       0.004
0.9       0.0009

These probabilities must be multiplied by at least two additional factors: first, the probability of a central star producing a Planetary System; and, second, the probability of that Planetary System putting a Terrestrial-type Planet in said central star's Habitable Zone. The former probability is about one-half to two-thirds*; the latter probability appears to be appreciable, as long as the Planetary System is not disrupted by Gas Giant Migrations of "Hot Jupiters"**.

* J.V. Narlikar. Star Factories, pg. ~30.
** Caleb A. Scharf, ibid., pg. 123. The plotted computer simulations produced a plethora of Terrestrial-type Planets, evenly distributed across the central star's Inner Star System. Indeed, the same can be said for our Solar System.​

Thus, this simply estimated probability, of observing a "G Y = C compliant" Star System, is of order 10^-3.

 

[cristo]

This thread is far too speculative to remain open.