How do various astrophysical models deal with so many orders of magnitude in scale? Are there completely different models for different scales?

Question

If we just model our solar system's dynamics, one could use a straightforward numerical integrator with all n(n-1)/2 gravitational interactions along with some torques and non-gravitational forces. However it was found to be no less accurate (limited by the availability of measured data with with to fit) to treat certain tightly bound systems as somewhat isolated. For example, forces between the smallest moons of two widely separated planets don't need to be treated explicitly, and using the Jovian system's barycenter is sufficient to perturb Mercury's orbit.

Similarly, in the dynamics within a single galaxy numerical models don't treat all n(n-1) two-body forces when n=10¹¹. Instead models can use some hybrid techniques - treating each of a large number of small volumes of the galaxy as having an average density, while at the same time using different models that operate on what's put inside each of those volumes.

Those are just examples of my meagre understanding of the topic.

But I'd like to ask, and this is a hard question to formulate and may be hard to answer - how do various astrophysical models deal with so many orders of magnitude in scale? Have they sort-of settled on some standard ways of chopping things up?

Just for a hypothetical example "class I" for solar systems, class II for stellar clusters or groups (IIa and IIb might be with or without a black hole) class III for entire galaxies, class IV for galactic clusters and class V for bigger/biggest?

You might embed one class in another, a galactic model might also contain say 100,000 stars with randomly initiated solar systems as sort of "test particles" within a fluid representing the rest of the matter.

Or is it better to characterize the current state of the art as still in the process of trying out all kinds of ways of dealing with scale?

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For a nice illustration of how many orders of magnitude are involved, have a nice experience zooming out at https://htwins.net/scale2/

• distance from Earth to Moon at 10^8.8 meters
• distance from Voyager to Earth at 10^13.5 meters
• observable universe at 10^26.5 meters

Answer 1

An attempt at a partial answer, or perhaps "one possible answer":

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If I understand correctly, the problem you describe appears in numerical simulations where you have to choose between simulating a large enough volume, and fine enough details. If so, then the answer is subgrid physics.

As an example, let's say a cosmological hydro-simulation uses $10^{11}$ particles to simulate a cubic gigalightyear. If I've done my maths right, this would mean that each particle would represent $\sim10^7\,M_\odot$. With a minimum of 10 (poor) to 100 (preferred) particles to represent a galaxy, this is acceptable if you're interested in, say, number statistics or clustering statistics of galaxies above $10^{8\text{–}9}\,M_\odot$.

But $10^7\,M_\odot$ is only the upper limit of the mass range of the molecular clouds from which stars are born, so how can you ever simulate star formation, which is important not only if you're interested in stellar populations and galaxy luminosities, but actually also affects the galaxy masses through their feedback which may push gas out of the galaxies?

The answer is that you use results from observations, analytical models, and lower-volume, higher-resolution simulations that teach you how stars are formed and how they affect the gas around them. From these you know for instance that gas needs to be 1) dense and 2) cold, in order to form a star. In your huge simulation you can then say "If the density of gas particles exceeds a certain threshold, and their temperature is below some threshold, and if they have a negative velocity gradient, then (a fraction of) the gas particles should be converted to star particles."

Subsequently you can say that stars are treated as either 1) low-mass or 2) high-mass. For the purposes of the simulation, given the size of a time step low-mass stars live forever while high-mass stars die instantly, being converted back into gas particles. A gas particle has an associated metallicity (and possibly dust), which has now increased by some yield, and because massive stars die not with a whimper, but with a bang, a certain amount of energy is now dumped into the surrounding gas particles as either heat, or momentum, or a mixture thereof, giving rise to feedback.

The scenario described above uses particles as an example, but the same approach is used for cell-based simulations. In both cases, we're dealing with physics that is not resolved by the particles/cells, and hence is referred to as subgrid physics.

The highest-resolution hydro-simulations of molecular clouds contain individual stars, but the stars themselves are not resolved. So how do we know how much energy they produce when they explode, how much metals they produce, what kind of spectrum they emit, etc.? Yes, by observations, analytical models, and even higher-resolution simulations that tell you how stars evolve.