Investigation of Extra-Solar Planetry Disks

[Widdekind]

Minimum Mass Solar Nebula (MMSN) model

 

Caleb A. Scharf (Extrasolar Planets & Astrobiology, pg. 101) mentions the MMSN model, for the (vertically integrated) surface density, of the proto-Solar proto-Planetary disk:

[math]\sigma® \equiv \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2}[/math]

where [math]\sigma_{0} = 1700 \; g \; cm^{-2}[/math] and [math]r_{0} = 1 AU[/math]. The total disk mass is estimated to have been about [math]M_{disk} \approx 0.018 \; M_{\odot} \approx 6000 \; M_{\oplus}[/math].

 

 

Now, for the general case, of extra-Solar proto-Planetary disks, we have, for the disks' masses & angular momentums, the following mathematical definitions:

[math]M_{disk} = \int 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2}[/math]

[math] = 2 \pi \sigma_{0} r_{0}^{2} \int dx \; x^{-1/2}[/math]

[math] \equiv 2 \pi \sigma_{0} r_{0}^{2} \; I_{1}[/math]

and:

[math]L_{disk} = \int 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} \sqrt{G \; M_{*} \; r}[/math]

[math] = 2 \pi \sigma_{0} r_{0}^{2} \sqrt{G \; M_{*} \; r_{0}} \int dx [/math]

[math] \equiv \frac{M_{disk}}{I_{1}} \sqrt{G \; M_{*} \; r_{0}} \; I_{2}[/math]

Knowing the total disk mass (above), and angular momentum (Carroll & Ostlie. Intro. to Mod. Astrophys., pg. 893), we can calibrate these equations. We find, w/ approximate self consistency, that the best fit values for the integrals are:

[math]I_{1} \approx 15[/math]

[math]I_{2} \approx 3[/math]

 

 

Now, we know (Carroll & Ostlie, ibid.) that for stars [math]\leq 2 M_{\odot}[/math]:

[math]\frac{L_{tot}}{M} \propto M^{2/3}[/math]

[math]\frac{L_{*}}{M} \propto M^{16/3}[/math]

The residual difference between these values has probably been deposited into (inferred) Planetary systems (Carroll & Ostlie, ibid.). Therefore, we define [math]L_{disk} \equiv L_{tot} - L_{*}[/math].

 

 

Furthermore, we know that the Planetary disk mass is zero for [math]2 M_{\odot}[/math] stars, and [math]0.018 M_{\odot}[/math] for [math]1 M_{\odot}[/math] stars (e.g. Sun). In addition, we doubt that the Planetary disks of minimum-mass stars [math]\left( M \approx 0.08 M_{\odot} \approx 27,000 M_{\oplus} \right)[/math] can exceed the mass of those central stars. With these two data points, plus the (presumed) upper bound, we can confidently allege a linear increase of disk mass, with decreasing stellar mass, as:

[math]M_{disk} \approx 6000 M_{\oplus} \times \left( 2 - \frac{M_{*}}{M_{\odot}} \right)[/math]

Having an empirical formula for disk angular momentum [math]\left( L_{disk}\right)[/math], and a plausible estimate for disk mass [math]\left( M_{disk} \right)[/math], as functions of stellar mass [math]\left( M_{*} \right)[/math], we can, in principal, use the aforementioned mathematical definitions (along w/ their best fit values for [math]I_{1}[/math] and [math]I_{2}[/math]) to deduce the best fit values for the parameters [math]\sigma_{0}[/math] ("central" density) and [math]r_{0}[/math] (radial scale length), as:

[math]r_{0} \equiv \frac{1}{G M_{*}} \left( \frac{L_{disk}}{M_{disk}} \frac{I_{1}}{I_{2}} \right)^{2}[/math]

and:

[math]\sigma_{0} \equiv \frac{M_{disk}}{2 \pi I_{1}} \frac{1}{r_{0}^{2}} = \frac{\left( G M_{*}\right)^{2}}{2 \pi} \left( \frac{M_{disk}}{I_{1}}\right)^{5} \left( \frac{L_{disk}}{I_{2}}\right)^{-4}[/math]

Units can be verified by inspection.

 

 

SciLab was used to numerically calculate the Disk Scale Radii & Disk Central Densities, as functions of stellar mass:

Fig. 1

-- Disk Scale Radius vs. Stellar Mass

 

Fig. 2

-- Disk Central Density vs. Stellar Mass

It was numerically verified, that these values do, indeed, reproduce the original estimate for total disk mass (above).

 

Now, the MMSN model is defined as having [math]Z = 0.01[/math] abundance of rocky materials (Caleb A. Scharf, ibid.). For our Solar System, then, we may define [math]\Delta[/math] to be the (relative) distance, to either side of the Earth's orbit, which must be swept up to accumulate [math]1 M_{\oplus}[/math] of rocky material:

[math]1 M_{\oplus} = Z \int_{1 AU \times (1-\Delta)}^{1 AU \times (1+\Delta)} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} = Z \left( 2 \pi \sigma_{0} r_{0}^{2} \right) \int_{1-\Delta}^{1+\Delta} dx \; x^{-1/2}[/math]

Substituting from the formula for total disk mass (above), we have that:

[math]1 M_{\oplus} = Z \frac{M_{disk,\odot}}{I_{1}} \int_{1-\Delta}^{1+\Delta} dx \; x^{-1/2}[/math]

The solution is straightforward, and yields [math]\Delta \approx 1/8[/math].

 

 

Armed with this definition for [math]\Delta[/math], we can calculate how much mass lies within the equivalent regions, of the Habitable Zones, around other stars. The only additional information required is the evolution, of the distance to said Habitable Zones, as a function of stellar mass. Now, the definition of the Habitable Zone is that:

[math]\frac{L_{*}}{D_{HZ}^{2}} = constant[/math]

And, to close approximation (Bowers & Deeming. Stars, pg. 28), [math]L_{*} \propto M_{*}^{4}[/math]. So, [math]D_{HZ} \propto M_{*}^{2}[/math]. Therefore, following the foregoing, we define:

[math]M_{HZ,rock} \equiv Z \int_{D_{HZ} \times (1-\Delta)}^{D_{HZ} \times (1+\Delta)} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2}[/math]

[math] = Z \frac{M_{disk}}{I_{1}} \int_{\frac{D_{HZ}}{r_{0}} \left( 1-\Delta \right) }^{\frac{D_{HZ}}{r_{0}} \left( 1+\Delta \right) } dx \; x^{-1/2}[/math]

[math]= Z \frac{M_{disk}}{I_{1}} \sqrt{\frac{D_{HZ}}{r_{0}}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2}[/math]

[math]= 1 M_{\oplus} \times \frac{M_{disk}}{M_{disk,\odot}} \times \sqrt{\frac{D_{HZ}}{r_{0}}} [/math]

This solution is also straightforward, and SciLab was used again, to numerically calculate these estimated Habitable Zone masses of rocky material, as a function of stellar mass:

Fig. 3

-- Habitable Zone Rock Mass vs. Stellar Mass

Apparently plausibly, the amount of rocky mass, in the central star's Habitable Zone, increases inexorably, with decreasing stellar mass [math]\leq 2 M_{\odot}[/math].

 

 

Finally, we can calculate quantities associated with the Angular Momentum, of that rocky mass, inside said stars' Habitable Zones. First, we can define the total Angular Momentum, of all that rock, revolving around in the disk:

[math]L_{HZ,rock} \equiv Z \int_{D_{HZ} \times (1-\Delta)}^{D_{HZ} \times (1+\Delta)} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} \sqrt{G M_{*} r}[/math]

[math] = Z \frac{M_{disk}}{I_{1}} \sqrt{G M_{*} r_{0}} \int_{\frac{D_{HZ}}{r_{0}} \left( 1-\Delta \right) }^{\frac{D_{HZ}}{r_{0}} \left( 1+\Delta \right) } dx[/math]

[math] = Z \frac{L_{disk}}{I_{2}} \frac{D_{HZ}}{r_{0}} \int_{1-\Delta}^{1+\Delta} dy[/math]

Next, we can define the "Keplerian" Angular Momentum, of all that rock, once collapsed into a single Planetesimal, orbiting at that star system's scaled equivalent of 1 AU:

[math]L_{Kepler} \equiv M_{HZ,rock} \times \sqrt{G M_{*} D_{HZ}}[/math]

[math]= Z \frac{M_{disk}}{I_{1}} \sqrt{\frac{D_{HZ}}{r_{0}}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2} \times \sqrt{G M_{*} D_{HZ}}[/math]

[math] = Z \frac{L_{disk}}{I_{2}} \frac{D_{HZ}}{r_{0}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2}[/math]

Now, presumably, any excess of [math]L_{HZ,rock}[/math] over [math]L_{Keplerian}[/math] will set the Planetesimal spinning on its axis. We therefore define [math]L_{spin} \equiv L_{HZ,rock} - L_{Kepler}[/math], which we divide by the Planetesimal's mass [math]M_{HZ,rock}[/math] as per normal normalization. Noting that:

[math]M_{HZ,rock} = Z \frac{L_{disk}}{I_{2}} \frac{D_{HZ}}{r_{0}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2} \times \frac{1}{\sqrt{G M_{*} D_{HZ}}}[/math]

this yields:

[math]\left( \frac{L}{M} \right)_{planet} = \sqrt{G M_{*} D_{HZ}} \times \left( \frac{\int_{1-\Delta}^{1+\Delta} dy}{\int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2}} -1 \right)[/math]

[math]= \sqrt{G M_{*} D_{HZ}} \times \left( \frac{\Delta}{\sqrt{1+\Delta} - \sqrt{1-\Delta} } -1 \right)[/math]

[math]\approx \sqrt{G M_{*} D_{HZ}} \times \left( 0.998 - 1 \right)[/math]

[math]\approx -0.002 \sqrt{G M_{*} D_{HZ}} [/math]

[math]\propto - M_{*}^{3/2}[/math]

Assuming planets typically rotate prograde, this analysis seemingly suggests that planets preferentially accumulate material from beyond their orbit [math]\left( r > r_{p} \right)[/math], rather than from within it [math]\left( r < r_{p} \right)[/math]. It is remotely possible, that these results imply, that bigger & brighter stars probably possess planets that tend to spin (rotate) more rapidly.

 

 

Lastly, note that the change in Specific Angular Momentum (SAM), across the star's Habitable Zone, increases strongly with stellar mass:

[math]\delta \left( \frac{L}{M} \right) = \delta \sqrt{G M_{*} D_{HZ} } = \frac{1}{2} \sqrt{\frac{G M_{*}}{D_{HZ}}} \delta D_{HZ} = \sqrt{G M_{*} D_{HZ} } \times \frac{\Delta}{2} [/math]

[math]\propto M_{*}^{3/2}[/math]

Thus, in-so-far as SAM poses the biggest obstacle to the accretion of Planetesimals, smaller & dimmer stars may be more likely to have fewer but (much) bigger planets, whereas bigger & brighter stars might be more likely to have more but (much) smaller planets. For example, an M-Class red dwarf might have one Super-Earth in its inner system, and one or two Super-Jupiters in its outer system. Conversely, an F-Class green dwarf might have many Mars- & Mercury-sized planets in its inner system, and many Uranus- & Neptune-sized gas giants in its outer system.

 

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Figure 21.16 from Carroll & Ostlie (ibid.)

 

http://www.freeimagehosting.net/image.php?fbaed847c5.jpg

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Disk Zone Masses

 

It is straightforward to define Disk Zone Masses:

[math]M_{zone} = \int_{r_{i}}^{r_{o}} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} = 2 \pi \sigma_{0} r_{0}^{2} \int_{x_{i}}^{x_{o}} dx \; x^{-1/2}[/math]

[math][/math]

[math] = \frac{M_{disk}}{I_{1}} \int_{x_{i}}^{x_{o}} dx \; x^{-1/2} [/math]

Now, stellar proto-Planetary Disks are divided into four successive zones (Carroll & Ostlie, ibid., pg. ~898):

1. Metal Zone -- only metals solidify
   
2. Rock Zone -- metals & rocks solidify
   
3. Ice Zone -- beyond Snow Line, water ice also condenses
   
4. Methane Ice Zone -- beyond Methane Snow Line, methane ices also condense
   

From the information of our Solar System, the "Metal Line" (0.5 AU), Snow Line (5 AU), and Methane Snow Line (30 AU) can be scaled for other star systems, precisely as per [math]D_{HZ}[/math] above.

 

 

SciLab was used to numerically calculate these quantities, and plot said Zone Masses, as functions of stellar mass:

Fig. 4

-- Zone Masses vs. Stellar Mass

It is found, that a star-system-wide scaling factor f_loss ~ 0.10 cleanly & consistently accounts for the observed amounts of mass remaining in our Solar System. (For example, out of the inferred initial Solar proto-Planetary Disk mass of [math]6000 M_{\oplus}[/math], over [math]20 M_{\oplus}[/math] of rock material remained, according to calculations, in the Rock Zone, while over [math]4000 M_{\oplus}[/math] is calculated to have resided in the Ice Zone (Jupiter, Saturn, Uranus), and over [math]200 M_{\oplus}[/math] is calculated to have resided in the Methane Ice Zone (Neptune+). Across the Solar System, all these values strongly suggest a "retention factor" of roughly 10%.) These reduced "residual masses", for the Ice- & Methane-Ice Zones, appear in the above plot, alongside the unscaled Rock Zone masses (effectively x10) for purposes of visualization.

 

 

It is immediately apparent, that, b/c of the inexorably increasing comparative coreward concentration of disk mass, with decreasing star mass, the most massive planet-bearing stars (F- & A-Class) actually boast the biggest Outer Star Systems (Ice- & Methane-Ice-Zones). In particular, the A-Class stars [math]\left( M > 1.5 M_{\odot} \right)[/math] boast the biggest Neptune-like Ice Giants, some surely exceeding [math]100 M_{\oplus}[/math]. This is (practically) completely consistent, with the observations from Fomalhaut System, which boasts a (single) big Ice Giant, (Fomalhaut b) (~115 AU) and Kuiper Belt (~150 AU), out in its Methane Ice Zone (beyond ~125 AU).

 

 

Conversely, the smallest stars support the most massive Inner Star Systems, housing [math]5-8 M_{\oplus}[/math] of rocky planetary material. However, with so much mass, crammed into such confined regions, it surely seems somewhat certain that such planetary systems could become incredibly unstable, and "torque themselves to shreds", perhaps explaining why small stars seemingly mostly lack Planetary Systems:

Fig. 5

--

Hertzsprung-Russell Diagram

in

Luminosity - Mass

space, also indicating known

Exoplanet

-bearing systems

(W.T. Sullivan III & J.A. Baross.,

Planets & Life

, pg. 445;

cf

.

link

)

.

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Estimating Number & Mass of HZ Exoplanets

 

Consider two Planetesimals , each of mass [math]m[/math], orbiting their central star. Their (assumed circular) orbits are separated by a distance [math]\delta D[/math]. From Kepler's Laws, we know that the difference in Angular Momentum, between both orbits, is:

[math]\Delta L = m \sqrt{G M_{*}} \frac{\delta D}{2 sqrt{D}}[/math]

Now, for both Planetesimals to merge, they must match their Angular Momentums, which requires Torques. We (crudely) guestimate the magnitude of said torques, with a simple model, of periodic pulses of strong gravitational influence, at every planetary alignment:

[math]\tau = F \times \Delta D \approx \frac{G m^{2}}{\Delta D^{2}} \times \Delta D[/math]

[math][/math]

[math]\Delta t \propto \Delta \omega^{-1} = \left( \frac{3}{2} \sqrt{\frac{G M_{*}}{D^{5}}} \Delta D \right)^{-1}[/math]

Setting [math]\Delta L \equiv \tau \times \Delta t[/math], we have, after rearranging:

[math]\left( \frac{\Delta D}{D} \right)^{3} \propto \frac{4}{3} \left( \frac{m}{M_{*}} \right)[/math]

[math][/math]

[math]\frac{\Delta D}{D} \propto \left( \frac{m}{M_{*}} \right)^{1/3}[/math]

Strictly speaking, the characteristic Planetesimal mass [math]m[/math] is most probably proportional to the retained Rock Zone mass [math]\left( M_{rock} \times Z \times f_{loss} \right)[/math]. But since most material resides in the Rock Zone (i.e., Inner Star System), we can closely approximate the right result w/ [math]M_{rock} \approx M_{disk}[/math]. Then we have that:

[math]\frac{m}{M_{*}} \approx Z \; f_{loss} \frac{M_{disk}}{M{*}} = Z \; f_{loss} \frac{0.018 M_{\odot}}{M_{*}} \left(2 - \frac{M_{*}}{M_{\odot}} \right)[/math]

[math][/math]

[math] = Z \; f_{loss} \; 0.018 \times \left( \frac{2}{\mu} - 1 \right) [/math]

where [math]\mu \equiv M_{*} / M_{\odot}[/math]. SciLab was used to calculate the (slightly) more accurate estimates, as originally defined:

Fig. 6

-- Estimated Number & Mass of HZ

Exoplanets

vs. Star Mass

For this figure, it was assumed that our Solar System supports 2.5 Habitable Zone (HZ) planets (Venus, Earth, Mars), so that a relative [math]\Delta D / D = 2.5[/math] 'should' sweep out its entire HZ, amassing all the (retained rocky) material, into a single planet. Thus, the estimated number of HZ planets is merely the inverse of the relative [math]\Delta D / D[/math], and the estimated mass per HZ planet is merely the (retained rocky) material mass divided by that above value. This analysis seemingly strongly suggests, that bigger brighter stars support many more Mars-, Mercury-, and Moon-mass worlds (scattered across their vast HZs), whereas smaller stars support single Super-Earths.

Edited December 29, 2009 by Widdekind
Consecutive posts merged.