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Numerical Modeling in Astronomy By Astronomers who sleep at night Slide 2 Theory Can we explain the things we see with fundamental physics? Chemistry? Slide 3 Slide 4 The 2-body problem Start with a star and a planet Slide 5 The 2-body problem or 2 stars, orbiting each other Slide 6 The 2-body problem plug in Newtons laws, and solve! The differential equations above can be solved to get the positions and velocities of both particles This is our easiest problem! Slide 7 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 Slide 8 The N-body problem It gets worse! Each body has interactions with every other body in the system! Slide 9 Switch to N-body simulations If we cant 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 Slide 10 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 10 20 kg (Vesta), m2 = 1.7 x 10 18 kg (Lutetia), r = 10 km = 10,000 meters F = 3 x 10 20 Newtons Slide 11 Example of an N-body simulation Now lets use Newtons second law to find the acceleration of each asteroid as a result of the force of gravity: Slide 12 Example of an N-body simulation Then we can use the acceleration to update the velocity a time-step later (well use 0.5 second time-steps): Assume both start from zero velocity Slide 13 Example of an N-body simulation And finally, use the velocity to update the positions: Slide 14 Example of an N-body simulation Now, the asteroids are 9956.3 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 Slide 15 The N-body problem or the BILLIONS of stars in a galaxy Slide 16 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, thats over 10 13 calculations! Galaxies form and evolve on billion year timescales for over 10 16 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! Slide 17 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 doesnt take too long! We need to understand the physics of collisions and to check for collisions on EVERY time-step! Slide 18 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 Slide 19 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 Slide 20 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 (well 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 Slide 21 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 Slide 22 Planet formation Slide 23 Slide 24 Planet migration *Co-rotating frame of reference Slide 25 The Grand Tack hypothesis