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I've been wanting to create a simulator to run tests based on properties of a star to find patterns in how planets form. Are there any already made equations for any of these countless relationships? Here are a few examples to further explain what sort of information I am looking for.

For example, how does the mass of a star relate to the distance of the frost line, and is there an already existing equation to estimate this?

More general example; If I wanted to find out how to estimate, say, a planets surface temperature based on a stars temperature and distance, what is the best way to do this?

Another example; To find the most likely average mass of a potential planet, what information would I need? Distance to star? Stars mass? Distance to nearest planets?

Another; How would I go about finding likely measurements of compositions of any hypothetical planets (surface/atmosphere/etc)

Absolutely any information is very welcome. I do not require any form of proven or precise measurements. If there is no equations, then all I need is what factors are involved in specific attribute development.

I'd like to emphasize my understanding that this is an extremely complex subject to go into detail. All I am asking for is any information or recommendations.

Here is a related question to my dilemma, in addition to this information, I am looking for more. Absolutely anything that may help, probably will. I am attempting to make as highly detailed of a model as possible.

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  • $\begingroup$ A planet's surface temperature will depend heavily on the planet and not just the star and distance. The earth's surface would be approximately as frigid as the moon if it weren't for our atmosphere, for example. $\endgroup$ – zibadawa timmy May 31 '15 at 22:32
  • $\begingroup$ Understood; I was just giving a few examples to explain what I was looking for, but I am looking for precise formulas to accurately model this with. $\endgroup$ – ChaosOverlord Jun 4 '15 at 7:52
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I may have to answer this in parts, a bit at a time, so I apologize if I'm not able to answer everything at the moment.

If I wanted to find out how to estimate, say, a planets surface temperature based on a stars temperature and distance, what is the best way to do this?

Ah, I just studied this about a week ago! What you're looking for is the effective temperature of the planet, which can be calculated as $$T=\left(\frac{L(1-A)}{16 \pi \sigma D^2} \right)^{\frac{1}{4}}$$ where $L$ is the star's luminosity, $A$ is the planet's albedo, $\sigma$ is the Stefan-Boltzmann constant, and $D$ is the distance from the star.

how does the mass of a star relate to the distance of the frost line, and is there an already existing equation to estimate this?

I wasn't able to find an explicit equation for this, but it's calculable. The frost line is a certain distance from a star in a stellar nebula such that the temperature is about 150 Kelvin.

If you know the mass of the star, you can use the mass-luminosity relation: $$\left(\frac{L}{L_{\odot}} \right)=\left(\frac{M}{M_{\odot}} \right)^a$$ Re-arrange that to solve for the luminosity. The inverse-square law says that the intensity of power drops off as the distance from the source increases: $$I=\frac{P}{4 \pi r^2}$$ You can solve for $r$ if you can figure out the required intensity: $$r=\frac{1}{2}\sqrt{\frac{P}{\pi I}}$$ Here, $P \approx L$. Figure out the necessary intensity to heat the nebula to 150 K and you've found the frost line.

To find the most likely average mass of a potential planet, what information would I need?

I can't help you much there, but I can give you a starter. The density $\rho$ of a circumstellar disk at a distance $r$ (as given by Michael Woolfson in On the Origin of Planets) is $$\rho (r)=C \exp \left[-\frac{(r-r_{\text{peak}})^2}{2 \sigma ^2} \right]$$ where $\sigma$ is one standard deviation and $r_{\text{peak}}$ is the peak density, and $C$ is a constant.

As promised, here are some more, set forth by Woolfson.

The disk decays over time; its density at time $t$ is approximated by $$\rho_t=\rho e^{- \gamma t}$$ where $\gamma$ is a decay constant. If you've taken certain math course, you've probably seen this equation many times before, just in with $\rho=y$ and $\gamma=k$.

Another equation - really, a couple equations - deal with accretion by a small dust particle in a circumstellar disk. The variables involved are: $m$ (mass of the particle), $t$ (time), $s$ (the radius of the particle), $\rho_s$ (the density of the particle), $\rho$ (the density of the disk), $f$ (the fraction of the medium that is dust), $T$ (the temperature), and $k$ (Boltzmann's constant, not be confused with the Stefan-Boltzmann constant).

The relevant pair of equations are: $$\frac{dm}{dt}=4 \pi s^2 \rho_s \frac{ds}{dt}$$ $$\frac{ds}{dt}=\frac{3f \rho}{4} \left( \frac{kT}{4 \pi \rho_s^3} \right)^{1/2}s^{-3/2}$$ Using these, you can find the size and mass of a given dust particle at any given time.

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    $\begingroup$ Absolutely awesome information so far Michael Woolfsons equation was particularly interesting $\endgroup$ – ChaosOverlord Jan 12 '15 at 1:30
  • $\begingroup$ This is great info, though not quite enough for me to develop a model. Did you have any more, or any sources I can look at myself? $\endgroup$ – ChaosOverlord Jan 15 '15 at 2:57
  • $\begingroup$ @ChaosOverlord There are some, but not any that will fully address your task. $\endgroup$ – HDE 226868 Jan 15 '15 at 3:28
  • $\begingroup$ If your willing to post it, please do. Like I said in the question any help is greatly appreciated. $\endgroup$ – ChaosOverlord Jan 17 '15 at 4:56
  • $\begingroup$ @ChaosOverlord Okay. I'm a bit busy this weekend (prepping for exams), but I'll put some up when I have the time. $\endgroup$ – HDE 226868 Jan 17 '15 at 14:27
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This link has a lot of simulators for each individual property. I do not know if it has everything, but I hope it helps. Each simulator includes the equation it uses. http://astro.unl.edu/animationsLinks.html

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  • $\begingroup$ Link-only answers are generally frowned upon; can you explain it in more detail? $\endgroup$ – HDE 226868 May 31 '15 at 18:25
  • $\begingroup$ Excellent resources, thanks greatly. Atmospheric Retention...HR Diagram Explorer...Celestial distance methods...planet formation temperatures...solar system properties...spectroscopic parallax...it even comes with source code...habitable zones...with these I can make leaps forward in my modeling of accretion, and even take it further to simulate development throughout a bodies lifespan...absolute gold The only thing I'm missing now for this part of my model is the exact correlations between a planets mass/temp/density (exact composition)/magnetic fields/etc $\endgroup$ – ChaosOverlord Jun 4 '15 at 7:50

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