Architect of Worlds – Step Ten: Place Dominant Gas Giant

July 20, 2018 Sharrukin Comments 0 Comment

Step Ten: Place Dominant Gas Giant

The evolution of the protoplanetary disk, and the formation of planets, will be dominated by the presence of the first gas giant planet to form. This planet may or not be the most massive, but it is usually the gas giant planet that forms closest to the primary star, and its movement through the disk will tend to affect the formation of other planets. In this step, we determine where the dominant gas giant planet (if any) forms, and how it migrates across the protoplanetary disk. This will, in turn, tell us how many gas giants may form.

Procedure

Begin by checking whether the dominant giant forms in a “hot” or “cold” region of the protoplanetary disk, or whether a dominant gas giant will form at all. Then determine how many gas giants can form in the planetary system, and how the dominant gas giant migrates to its final position.

First Case: Hot Dominant Gas Giant

If the protoplanetary disk has very high density of dust, or if the primary star is very bright and so has a distant snow line, the dominant gas giant may form in the warm, dry region inside the snow line. To check for this possibility, compute the following. If M is the mass of the star in solar masses, K is the system’s metallicity, and D is the disk mass factor:

$R=\frac{16}{{(M\times K\times D)}^2}$

R is measured in AU. If R is less than the disk inner radius, set R to be the disk inner radius instead. The result is the radius at which the dominant gas giant will form, based solely on the accretion of stony planetesimals. This will occur only if:

R is less than the snow line radius;
R is less than the slow-accretion radius; and
R is less than the radius of any forbidden zone.

If all three conditions hold, make a note that the dominant gas giant forms at this radius. Otherwise, check the second case.

Second Case: Cold Dominant Gas Giant

In most cases, the dominant gas giant will form outside the snow line, where ice as well as dust is available for the formation of planetesimals. To check for this possibility, compute the following. If M is the mass of the star in solar masses, K is the system’s metallicity, and D is the disk mass factor:

$R=\frac{1}{{(M\times K\times D)}^2}$

For a quick check, if the radius for the first case above was computed, simply divide it by 16 to get the radius for this case. If R is less than the radius of the snow line, set R to be the radius of the snow line instead. The result is the radius at which the dominant gas giant will form, based on the accretion of icy planetesimals. This will occur only if:

R is less than the slow-accretion radius; and
R is less than the radius of any forbidden zone.

If both conditions hold, make a note that the dominant gas giant forms at this radius. Otherwise, no gas giant will form in this planetary system; skip ahead to Step Eleven.

Number of Possible Gas Giants

Depending on the size and mass of the protoplanetary disk, more gas giants may form at larger orbital radii. To determine the further evolution of the planetary system, we need to estimate how many gas giants are possible.

To make this estimate, compute the following. Let R be the radius at which the dominant gas giant forms, as determined in the two cases above. Let R_max be the slow-accretion radius or the radius of any forbidden zone, whichever is less. Then:

$N=1+(6\times\log_{10}{\frac{R_{max}}{R}})$

Round N down to the nearest integer. The result is the estimated number of possible gas giants in the planetary system. Note that N must be at least 1, otherwise no gas giant can form in the system and we should already have skipped ahead to Step Eleven.

Disk Migration

Once the dominant gas giant begins to form, it is likely to migrate through the protoplanetary disk. At first, it will migrate inward due to interactions with the gas of the disk. As it migrates inward, its gravity will disrupt the orbits of any planetesimals it approaches or passes, affecting the later evolution of inner planets. To estimate the extent of the dominant gas giant’s inward migration, roll 3d6 on the Planetary Migration Table. Modify this roll by -3 if the disk mass factor is 4 or greater, or by +3 if the disk mass factor is less than 1.

Planetary Migration Table
Roll (3d6)	Status After Inward Migration
3-6	Epistellar Gas Giant – Dominant gas giant migrates inward to the disk inner edge radius.
7-9	Strong migration – Dominant gas giant migrates inward to about 0.25 times its initial orbital radius, or to the disk inner edge radius, whichever is greater.
10-12	Moderate migration – Dominant gas giant migrates inward to about 0.5 times its initial orbital radius, or to the disk inner edge radius, whichever is greater.
13-15	Weak migration – Dominant gas giant migrates inward to 0.75 times its initial orbital radius, or to the disk inner edge radius, whichever is greater.
16-18	No migration – Dominant gas giant fails to migrate inward at all.

The table indicates how to estimate the orbital radius to which the dominant gas giant migrates during this phase of its evolution. For the strong migration, moderate migration, or weak migration cases, feel free to adjust the multiplying factor by up to 0.1 in either direction. Make a note of the resulting orbital radius.

The Grand Tack

At some point in its formation, the dominant gas giant may fall into a strong resonance interaction with one or more gas giants forming further away from the primary star. This is likely to halt inward migration through the protoplanetary disk, and may lead to outward migration back away from the star.

A Grand Tack will take place only if there are at least two possible gas giants in the planetary system, as estimated above. If this is the case, roll 3d6. A Grand Tack takes place if the result is 13 or higher.

If a Grand Tack takes place, estimate the final orbital radius of the dominant gas giant by rolling 3d6 and applying the following:

$R=(1+\frac{3d6}{10})\times R_M$

Here, R_M is the planet’s orbital radius after any inward migration is applied, and R is its orbital radius after the Grand Tack is finished. If R is greater than half the radius of any forbidden zone, then set R to that value. Otherwise, feel free to adjust the final orbital radius by up to 5% in either direction.

Selecting for an Earthlike world: The critical orbital radius for an Earthlike world depends on the current (rather than initial) luminosity of its primary star. To estimate this orbital radius, if L is the star’s current luminosity, then:

$R=\sqrt L$

Here, R is the most likely orbital radius for an Earthlike world. A terrestrial planet with enough mass to support an Earthlike environment is only likely to form in one of three cases:

The dominant gas giant migrated inward completely past this radius, and no Grand Tack event took place to pull it back outward;
The dominant gas giant migrated inward but entered a Grand Tack event before reaching about 1.5 times this radius; or
The dominant gas giant did not migrate inward at all.

Of these three cases, the second (moderate inward migration followed by a Grand Tack) is the most likely to give rise to an Earthlike world.

Examples

Arcadia: Alice suspects that the primary star of the Arcadia system is too dim to promote the formation of a hot dominant gas giant, but she checks anyway. The star’s mass is 0.82 solar masses, its metallicity is 0.44, and the disk mass factor she selected earlier is 2. She computes:

$\frac{16}{{(0.82\times0.44\times2.0)}^2}\approx30.7$

This is, as Alice expected, well past the slow-accretion line. Moving on to the case of a cold dominant gas giant, she divides the above result by 16 and gets a radius of about 1.9 AU. This is inside the snow line, so Alice resets the radius to be equal to the snow line at 2.2 AU; this is where the dominant gas giant will form.

Alice now needs to estimate how many gas giants could form in the Arcadia planetary system. She computes:

$1+\left(6\times\log_{10}{\frac{14}{2.2}}\right)\approx5.8$

Rounding down to the nearest integer, she finds that the Arcadia system could have as many as five gas giants in it. Depending on subsequent results, it may have fewer than this number, but it cannot have more.

Rather than generate the dominant gas giant’s evolution at random, Alice wants to maximize the probability of an Earthlike planet forming at the critical orbital radius, which she computes from the primary star’s current luminosity of 0.34 solar units:

$\sqrt{0.34}\approx0.58$

She therefore decides that the Arcadia system’s primary gas giant exhibited weak inward migration, moving from its initial orbital radius of 2.2 AU to about 1.7 AU, inside the snow line but nowhere near the eventual Earthlike world’s position. Then she decides that the planet underwent a Grand Tack event, plausible since there are at least two possible gas giants in the system. She decides that the dominant gas giant migrated back outward to an orbital radius of 4.4 AU. She makes note of all three radii for future reference.

Beta Nine: Bob knows that a red dwarf star will almost certainly not develop a hot gas giant, so he moves directly to the second case, computing the radius at which a cold gas giant will form:

$\frac{1}{{(0.18\times2.5\times0.5)}^2}\approx19.8$

This is far beyond the inner edge of the forbidden zone created by the star’s brown-dwarf companion. The primary star of the Beta Nine system will not have any gas giant planets at all. Bob moves on to the next step in the design sequence.

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