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Given that the mass of a planet at a given orbital distance r is dependent on the protoplanetary material in the neighborhood of r at the time of formation (assuming no planetary migration), what functional form do we expect the masses of planets around a star in theory?

My initial guess was $M(r) \propto r e ^{-r} $, which is roughly similar to our own solar system (with a minimum near the origin, a peak at Jupiter, and then decreasing as you move out again). This is from taking a guess to the density of of protoplanetary material around a young star as $e ^{-r} $, and then multiplying this by the volume of a ring at distance r from the star, $ 2 \pi r \cdot dA$.

Is my guess correct? Do we expect planet masses to be distributed in such a way? Or is there another functional form that is more accurate?

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At zeros order such assumption might be made, but a powerlaw relation is more common and accepted.

Also a protoplanetary disk is more complex as is the planet formation process which may include radial migration of the protoplanets. So the mass $m(r)$ available at a distance $r$ might not be exactly representative for the planetary mass found at that distance later on.

Weidenschilling's reference model from 1977 for the minimum mass solar nebular gives a density variation of the nebula as $$ \varrho(r) \propto r^{-2} $$ and the column density (thus all matter vertically integrated or considered compressed to the mid-plane) as $$ \sigma(r) \propto r^{-1}. $$ That is gas and solids combined, but given good mixing, this describes also the (initial) radial distribution of solids; typically a solid to gas mass ratio of 1:100 (minimum) to 3:100 (dense) is considered.

More modern disk models, taking into account further effects like viscosity lead to a surface density variation of $\sigma(r) \propto r^{-p}$ with $p\lessapprox 1$, thus possibly even shallower.

Beyond that, models get more complex, if you start to take into account density variations throughout the disk imposed by the different condensation temperatures of different materials. The most notable of these is the snowline at around 3...5 astronomical units which increases the available mass of solids for planetesimals significantly. So you have to introduce something like $$ \Sigma^{MMSN}_{solid}(r) = 7.1F_{ice}\left(\frac{r}{1AU}\right)^{3/2} \qquad \mathrm{with}\qquad F_{ice} = \begin{cases}1, r < r_{ice}\\ 4.2, r \ge r_{ice}\end{cases} $$ (see Min et al (2011), who cite Thommes and Duncan (2006) for this equation.

Now, once the protoplanets formed this way, the larger ones will start interacting with the disk and through that also with eachother. Planetary migration WILL occur at least to some degree. That's an area which is still in ongoing research and a lot of ideas and processes have been brought forward in the last decade or two on this topic - and IMHO no "final decision" can be made there yet.

Thus it might actually be worthwhile to analyse the known exoplanet systems to give an idea of the radial planet mass distribution (a thing done in this area of research anyway to benchmark which types of system a disk and / or migration model can explain). The type impossible to explain without migration are the hot jupiters, thus giant planets in close-in orbits which orbit their host star within days, even closer than Mercury in our own solar system. Given this, there is no general formula available which gives the planetary mass as a function of radial distance. Even when the observation bias acts in favour of detecting giant planets close-in, this is some process which does happen often (plot from exoplanet.eu): Planetary mass as function of orbital period

Looking at the systems with multiple known planets seems like the way to go - yet I am not aware of a very recent such analysis. The best I currently have at hand is this text by Davis et al. on the stability of systems.

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  • $\begingroup$ But don't we expect small planets closest to the star, as well as a small planets furthest out? How can it be a power law fit when it seems to me the planet mass at r approaches 0 as r approaches both 0 and infinity, no? $\endgroup$ Oct 5 '21 at 14:28
  • $\begingroup$ Where do you see a power law fit of planet mass vs. radial distance? The diagramm showing planetary mass over orbital period does not give raise to the assumption that planetary mass depends on oribital distance of the planet. The apparent slope, leaving the lower right area free, is the detection threshold and not physical. $\endgroup$ Oct 5 '21 at 15:59
  • $\begingroup$ Note also that the models on the solar nebular give a power law distribution of the initial radial mass density (that's not quite hydrostatic equilibrium, or the star couldn't form, but that's where such powerlaw comes from) - but they don't imply that the planetary mass scales the same, as the evolution processes from disk to planets mixes things. And as the observations of the exoplanet systems show, our own solar system is not necessarily representative (even though systems similar to ours are still very hard to detect, if at all - thus again: observation bias probable there) $\endgroup$ Oct 5 '21 at 16:03
  • $\begingroup$ Ok I see, but purely from a theoretical standpoint, ignoring planetary migration (so I am interested in initial mass distribution very early on in formation) and assuming most of the mass of the planetary disk near the center forms the star, pretending it to be a point mass for the moment, a planet very near the star only has a smaller and smaller ring of material to work from, to the theoretical limit where a planet at 0 would have no mass. Likewise for a planet at ∞. The formula you showed for planetary mass available however scales with r^(3/2), which is unbounded as r approaches infinity. $\endgroup$ Oct 5 '21 at 21:24

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