5
$\begingroup$

I am aware that similar questions have been asked but these variables are different. I have tried to calculate it myself or at least find an equation that could help find the forbidden zones for these planets. I found an answer on a previous post asking a similar question and went to this book pdf - https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf. On pages 48-52(slides 64 to 68) the author talks about forbidden zones, but the equation given only gives you the AU of a planet from a sun identical to Sol.

Is there an equation that can be used to find the forbidden zone around a planet? If so, what is it?

I was able to find an answer after a lot of digging. The equation is in the answers.

Below are variables that would probably be needed for such an equation. Please only answer the question above. I put the variables below so that I could see them clearly.

The Habitable Zone is 0.992AU to 1.429AU. This is because the star is 1.02 solar masses. It is a type G1.7V star. Sol's habitable zone is 0.9AU to 1.2AU. The Sun is a type G2V star.

The first planet is 1.7 times the mass of Earth with a radius of 1.138 Earths. The CMF(Core Mass Fraction) is 40% and the density is 6.354g/cm³(Earth's is around 5.51g/cm³). The gravity is 1.312 times that of Earth with an eccentricity is 0.003.

The second planet is the mass of Earth with a radius of 0.997 Earths. The CMF is 35% with a density of 5.568g/cm³. The gravity is 1.007 times that of Earth with an eccentricity of 0.02.

All numbers came from The Worldsmith by Artifexian - https://docs.google.com/spreadsheets/d/1AML0mIQcWDrrEHj-InXoYsV_QlhlFVuUalE3o-TwQco/copy

$\endgroup$
5
  • 1
    $\begingroup$ Does this change your thoughts on the subject? en.wikipedia.org/wiki/Klemperer_rosette $\endgroup$
    – Boba Fit
    May 8, 2023 at 1:04
  • $\begingroup$ @BobaFit Not really. I am asking how close these two planetary bodies can be to each other before their gravitational pull starts pulling them together. $\endgroup$
    – Martamo
    May 8, 2023 at 1:46
  • 1
    $\begingroup$ "Practically adjacent", if you'd consider a binary planet to be an acceptable answer ;-) The Pluto/Charon system has a mass ration of 1:8 between the two worlds, and the mass ratio between Earth and Mars is more like 1:10... $\endgroup$ May 8, 2023 at 8:39
  • $\begingroup$ To give you a fram of the scale of the size you are dealing with, Assuming that the star is identical in mass to Sol, Mars is just outside of the Habitable Zone (at ~1.5 AU) and Earth is exactly 1 AU from the sun (AU is defined as the distance from the Sun to Earth) Venus, is about 0.72 AU from the Sun, which is super close to the Earth... in space terms. $\endgroup$
    – hszmv
    May 8, 2023 at 18:53
  • $\begingroup$ Obviously, the type of star that the planets orbit will change where the habitable zone is. For example, when our sun turns to a Red Star (in about 5 billion years... earth is 4.6 billion years old for comparison of how immediate this is). The sun will expand in size that it will consume everything orbiting it as far as mars. $\endgroup$
    – hszmv
    May 8, 2023 at 18:58

2 Answers 2

5
$\begingroup$

NOTE: I'm not particularly happy with this answer. Notably because I've not used the equations myself and don't know if I'm representing them fairly. If another answer can improve on this, I'd be utterly delighted.

I believe you have misunderstood the relationships proposed in the book you link to. Here's a quote from the book I'll reference:

In general, the concept may be summarized as follows: We consider each planet separately, staarting with the largest (Jupiter). Each planet, together with the Sun, is considdered as a member of a two-body system, having a certain mass ratio $\mu$ (mass of the smaller divided by the sum of the masses of the two), a certain orbital eccentricity $e$, and a certain mean distance $\bar{D}$ (semimajor axis of its orbital ellipse). First, consider the pair Jupiter-Sun. In its orbit around the Sun, Jupiter creates a broad annular band about 250 million miles wide centered roughly on its orbit; within this band, no small third body can exist in a stable orbit, and one would not expect to find a major planet growing by accretion within this band. The dimensions of this "forbidden" region are functions of $\mu$, $e$, and $\bar{D}$.

Next, consider the the Saturn-Sun pair. The forbidden region created by Saturn is an an annular ring about 350 million miles wide wich does not overlap the forbidden region produced by Jupiter. It is as though each planet produces a standoff distance that is a function of its mass and its distance from the Sun, and within which other planets cannot orbit in a stable manner. This holds true for Neptune, Uranus, the Earth, Venus, Mars, and Mercury, as well as for Jupiter and Saturn. None of the forbidden regions of these planets overlaps any other. The only exception is Pluto. Ahtough Pluto's mean distance lies ourside Neptune's forbidden region, Pluto's orbit crosses into it; this may foreshadow an eventual catastrophic perturbation of Pluto's orbit by Neptune sometime in the distant future.

There are also wide gaps between certain adjacent forbidden regions within which small bodies can exist in stable orbits, notably the asteroid belt betwen Jupiter and Mars. Interestingly wide gaps are also present between the forbidden regions of Uranus and Saturn and between those of Neptune and Uranus, where small orbiting orjects (as yet undiscovered) may well exist in large numbers.

This provides a pretty good procedure that can be applied to any solar system.

  1. Begin with the largest planet in the solar system. Repeat for each planet largest mass to smallest.

  2. For each planet in turn, calculate the mass ratio.

$$\mu = \frac{planetary\ mass}{planetary\ mass + solar\ mass}$$

  1. Determine (perhaps arbitrarily) the eccentricity $\mu$ of the planet's orbit. It might be worth assuming $\mu=0$ (circular orbit) until you work out the repeatability of this procedure.

  2. Set the mean distance (semimajor axis) $\bar{D}$ for the target planet based on the forbidden zone calculations of all previous planets.

  3. Calculate the forbidden zone. Curiously, I don't find the equations for that in the book you linked to, nor have I found them with an admittedly trivial Google search. I did find this article that lists equations that may be useful (they're for calculating the forbidden zones for moons), but reading that article, they're non-trivial equations. Are we straining at a gnat?1

NOTE: As I said, the equations in the article I reference are for moons. Honestly, that might be a more than adequate simplification for worldbuilding purposes. However, even those aren't equations aren't trivially processed. The equation below can't be blindly used (note that double-prime mark). I'm growing fond of the idea of forbidden zones as I learn more about them, leading to me wondering if a simplified equation can be derived that could allow non-PhD worldbuilders to crank out adequate solar system designs.

$$R_{2,3}''+(1+\xi)* \left( \frac{\lambda m_1}{|R_{1,2}|^3} \right) *R_{2,3} = 0$$

where...

$$\xi= \left( \frac{(m_2+m_3)}{m_1} * \frac{|R_{1,2}|^3}{|R_{2,3}|^3} \right)$$

and...

  • m1 = mass of star
  • m2 = mass of planet
  • m3 = mass of moon
  • $R_{n,m} = \bar{D}$ = radius vector between two bodies

Note: I mentioned I wasn't particulary happy with this, right? If you assume $m_1$ is the mass of the galaxy's black hole and $R_{1,2}$ is the distance between that black hole and the sun, then you should be able to use the equation by treating the planets as moons. However, despite the article's enthusiastic conclusion that this produces results for forbidden zones, I don't see how that can happen, suggesting that the article is written for people trained in the art and not mere mortals like me. Ugh.

  1. Place the planet in relation to the other planets in the system. This will be an interative process because forbidden zones should not overlap (Pluto's exception should be treated as that). You may discover that, for example, your second planet calculated an overlapping forbidden zone because your initial assumption about $\bar{D}$ was too close to another planet, forcing you to change $\bar{D}$ and recalculate.

  2. It's natural for gaps to appear between planets.

  3. Rinse, wash, repeat....

Most of the planetary data you provide isn't relevant to this issue. A planet's radius, for example, isn't considered for calculation of forbidden zones. Neither is the planet's density. Just its mass.


1Let me clarify my statement. Back when I was a teenager I spent several weeks reading a book about the mathematics of rocketry and working through the equations to design a rocket. It was a TON of work, but I remember my parents and teachers being proud that I'd worked it out. Beaming with juvenile pride, I built the rocket I'd designed. It didn't work. Not even close. In fact it was a disaster. What I learned is that there's a HUGE difference between aquiring and learning how to use a hammer (tools) and actually knowing how to build a house (celestial mechanics). Please accept this wisdom for what it's worth: there's a reason why people spend years in education. If your audience can't appreciate when you're right and can't know when you're wrong, there's little value in getting every possible detail right. Thus, straining at a gnat.

$\endgroup$
3
  • $\begingroup$ I don't understand any of that. Thanks for attempting. I do want to ask. Where did you get D¯ from? $\endgroup$
    – Martamo
    May 10, 2023 at 1:58
  • $\begingroup$ D_BAR is the semimajor axis of the orbit. If the orbit is perfectly circular the major axis and semimajor axis are equal. Please keep in mind that you're hoping to find simple tools to answer complicated questions. Set your expectations accordingly. $\endgroup$
    – JBH
    May 10, 2023 at 3:06
  • $\begingroup$ I was able to find equations for the Roche Limit(the minimum orbital radius which is necessary for dust or particles to grow forming a moon, or necessary for an existing moon to remain internally stable) and the Hill Sphere/Radius(An astronomical body's Hill sphere is the region in which it dominates the attraction of satellites). In a separate article that author stated that planets should be 5 to 10 Hill Radii of the larger of the two planets you are using. I put it in an answer. $\endgroup$
    – Martamo
    May 10, 2023 at 3:49
0
$\begingroup$

I have done further research and wish to answer my own question.

Upon further investigation I found this thread from the Astronomy Stack Exchange talking about the minimum distance between planets. The accepted answer talked about a paper on creating a solar system. https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/. The author says that two planets should be 5 to 10 times their Hill Radius. More digging about the Hill Radius brought me to this Astronomy Stack Exchange talking about the Hill Sphere. Going past the accepted answer, because it was a bunch of math I didn't understand, brought me to an answer leading to this site. It had an equation for the Hill Sphere and the Roche Limit(the predecessor to the Hill Sphere). The Hill Radius can be calculated using this equation.

r = R [m/(3M)]^1/3

  • r = Hill Radius
  • R = Semi-major Axis
  • m = Mass of Planet
  • M = Mass of Star

This would make the Hill Sphere of my largest habitable world 1774199.41km

1.496x10^8((1.01524x10^25)/3(2.02878x10^30))^(1/3)

I then completed the Hill Radius for the second habitable planet and the minumum distance needed. I put them in a graph seen below. enter image description here

The Roche Limit calculation is R ≈ 2.5*rM

  • R = Roche Limit
  • r = Radius of Planet
  • M = Density of Planet

Will add equations at a later date.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .