Numerical Modeling in Astronomy By Astronomers who sleep at night

Theory Can we explain the things we see with fundamental physics? Chemistry?

The 2-body problem Start with a star and a planet…

The 2-body problem …or 2 stars, orbiting each other…

The 2-body problem …plug in Newton’s laws, and solve! The differential equations above can be solved to get the positions and velocities of both particles This is our easiest problem!

The 3-body problem Unsolved for hundreds of years! An approximate solution exists in the “restricted” 3 body problem For planets, an approximate solution is the disturbing function

The N-body problem It gets worse! Each body has interactions with every other body in the system!

Switch to N-body simulations If we can’t solve the equations explicitly, we can solve them by iteration Given some initial conditions (e.g. velocity, position of asteroids), calculate the change in those conditions a short time later Use that change to update the interesting quantities Repeat

Example of an N-body simulation Take 2 asteroids (so N = 2…) Calculate the force of gravity between them from our old friend… m1 = 2.6 x kg (Vesta), m2 = 1.7 x kg (Lutetia), r = 10 km = 10,000 meters F = 3 x Newtons

Example of an N-body simulation Now let’s use Newton’s second law to find the acceleration of each asteroid as a result of the force of gravity:

Example of an N-body simulation Then we can use the acceleration to update the velocity a time-step later (we’ll use 0.5 second time-steps): Assume both start from zero velocity

Example of an N-body simulation And finally, use the velocity to update the positions:

Example of an N-body simulation Now, the asteroids are m apart. Use that to find the new force of gravity between them! Imagine doing this for the 152,942 asteriods mentioned in the book…

The N-body problem …or the BILLIONS of stars in a galaxy…

The advent of computation …for thousands or millions or billions of years The solar system formed on million year timescales. For our 0.5 second time-steps, that’s over calculations! Galaxies form and evolve on billion year timescales for over calculations! You can quickly see that computers are necessary to get anywhere In fact, we were not able to do this kind of thing ~30 years ago!

A few comments… How big should the time-steps be? What happens when stuff collides/accretes? Small enough to accurately describe all the physics… …but large enough that the simulation doesn’t take too long! We need to understand the physics of collisions… …and to check for collisions on EVERY time-step!

A few comments… What is the smallest particle you can have? The solar system formed from >10 50 atoms! WAY too many to model, even for a supercomputer. We need to be smart about size

What is this used for? Solar system/planet formation – going from gas and dust to planets, stars, asteroids, and comets Galaxy formation – going from HUGE clouds of gas and dark matter to disks of stars Evolution of the solar system – predicting where the planets will be in x years Evolution of the universe – motion of galaxies, dark matter halos, and dark energy

What other types of simulations are there? Interiors and evolution of stars – how nuclear fusion takes place, how they go from dwarfs to giants, etc Spectra of light from stars, galaxies, nebulae, and planets (we’ll talk more about light later this week) Radiative transfer – how light gets around, and carries energy through the universe Planets and moons – climates, geological activity, magnetic fields, tidal heating, etc. Novae, Supernovae, Hypernovae, gamma-ray bursts – how stuff ExPLoDeS!! and why

Planet formation Giant diffuse cloud of gas collapses into star and disk (track the gas particles in the simulation) Metals and rocks precipitate out of the gas to form dust grains (model the thermodynamics and track the dust grains) Dust grains collect and build up to form planetesimals/asteroids (still tracking all gas, dust and planetesimals) Planetesimals accrete more dust, smaller planetesimals, and finally light gases (at ~12x the mass of the Earth) Solar winds begin and clear all light particles (gas) out of the disk Migration of planets may occur at almost any time Small stuff (asteroids, comets) continue getting kicked around by (mainly) Jupiter for billions of years, but gradually tapers

Planet formation

Planet migration *Co-rotating frame of reference

The Grand Tack hypothesis