Simulation of Galactic Bridges and Tails with CUPS (and other miscellaneous things..): Simulation of Galactic Bridges and Tails with CUPS (and other miscellaneous things..) 20000139 류병기 20000333 이상훈


Contents: Contents Brief Introduction to the CUPS program Restricted Problem of Three Bodies 1. Galactic Bridges and Tails 2. Rings of a Planet 3. Asteroids and Resonances Motion of N-Bodies 1. Virial Theorem and N Attracting Bodies 2. Construction of Our Own Solar System


CUPS: CUPS Consortium for Upper-level Physics Software (by Pascal) We will use “Programs on the Motion of n-Bodies” Program of CUPS – Astrophysics Simulations cf) http://entropy.kaist.ac.kr/~leago/CUPS.zip Units : G = 1 → If M=1011 solar masses, R = 10 kpc T = 2π sqrt(R3/MG) = 103 years Computation 1) Restricted Problem of Three Bodies (f, g series) 2) Motion of N-Bodies (?) 3) DE solving : Runge-Kutta method


Programs on the Motion of n-Bodies: Programs on the Motion of n-Bodies Interaction between two galaxies The sun, Jupiter and asteroids Many-body motion Make your own solar system Play-back Orbital element demonstration


Restricted Problem of Three Bodies: Restricted Problem of Three Bodies Two bodies revolve around each other in Keplerian orbits, while a third, having mass too small to influence the first two, moves in their combined gravitational field ▽U = ▽( (1-μ)/ρ1 + μ/ρ2) where mass μ at ((1-μ),0,0) and mass (1-μ) at (-μ,0,0) (origin at CM) ρ1 = sqrt((x + μ)2 + y2 + z2) ρ2 = sqrt((x – 1 + μ)2 + y2 + z2)


Equations of Motion: Equations of Motion x’’ – 2y’ – x = ∂U/∂x y’’ + 2 x’ – y = ∂U/∂y z’’ = ∂U/∂z (a rotating reference system – Coriolis and centrifugal terms on the left) Lagrangian Points : Where the time derivatives in above equations are zero (resonance)


Lagrangian Points: Lagrangian Points Stability : L1, L2, L3 – unstable L4, L5 – stable cf)Trojan asteroids


Galactic Bridges and Tails: Galactic Bridges and Tails Models of Toomre, A., Toomre, J. Galactic bridges and tails. Astrophysical Journal 178:623-666, 1972 The bridges and tails seen in some multiple galaxies are just tidal relics of close encounters Key factors : Masses of two galaxies, Inclination angle i, Eccentricity e, … Simulations for different factors and compare with real galaxies (e.g., Arp 295, M51, …)


Tidal Force: Tidal Force Due to the gradient of the gravitational force See a classical mechanics textbook for details e.g.) “5.5 Ocean Tides” of Marion (4th ed.) FTx = 2GmMmx / D3 FTy = - 2GmMmy / D3


A Retrograde Passage: A Retrograde Passage For the time being, e = 1 (parabolic) & i=0 (flat) Equal masses, retrograde (equivalent to i=180°) * CUPS – tgret.pos


A Direct Passage (Equal Masses): A Direct Passage (Equal Masses) * CUPS – tgeq.pos


A Direct Passage (Quarter Mass): A Direct Passage (Quarter Mass) * CUPS – tgqu.pos


Passage of a Heavy Companion: Passage of a Heavy Companion Four times as massive as the “victim” * CUPS – tgfour.pos


Inclined Passages: Inclined Passages Geometry


Various Inclined Passages: Various Inclined Passages Equal Masses, i = 15° * CUPS – tgeq15.pos Equal Masses, i = 45° * CUPS – tgeq45.pos Quarter Mass, i = 30° * CUPS – tgq30.pos etc, …


Real Galaxies – Arp 295: Real Galaxies – Arp 295 A composite image of the optical light (green and yellow), star forming regions (pinkish white), and cold atomic hydrogen gas (blue) in the classic bridge-tail system Arp 295


Simulation of Arp 295: Simulation of Arp 295 i = 15°, ω = 45°, e=1, quarter-mass * CUPS – arp295.pos


Real Galaxies – M51, NGC 5195: Real Galaxies – M51, NGC 5195 The optical image (depicted by green and yellow colors in the above image) highlights the younger stars, as well as the dust; the latter can be seen as dust lanes in the spiral arms of M51 itself and in obscuring the eastern (left hand) part of its companion, NGC5195.


Simulation of M51, NGC 5195: Simulation of M51, NGC 5195 i = -70°, ω = -15°, e=0.8, one-third mass * CUPS – m51.pos


Real Galaxies – The “Mice,” NGC 4676: Real Galaxies – The “Mice,” NGC 4676 A composite image of the optical light (green), warm gas (pinkish white), and cold atomic hydrogen gas (blue) in the well known interacting pair of galaxies known as "The Mice", or NGC 4676. The nickname of this object derives from its optical appearance, specifically the narrow tails emering from oval shaped bodies, reminiscent of two mice .


Simulation of NGC 4676: Simulation of NGC 4676 i = 55°, ω = 0°, e=0.6, equal disks * CUPS – mice.pos


Real Galaxies – The “Antennae,” NGC 4638/9: Real Galaxies – The “Antennae,” NGC 4638/9 The image shows the optical starlight depicted in green and white, with the neutral atomic gas depicted in blue.


Simulation of NGC 4038/9: Simulation of NGC 4038/9 i = 0°, ω = 0°, e=0.5, equal disks * CUPS – antenna.pos


Rings of a Planet: Rings of a Planet


Stability of Ring Structure : Stability of Ring Structure Using the interacting galaxies model to test The mass of heaviest satellite Titan ≒ 10-3 times the mass of Saturn Titan is far from the ring Other satellites are much lighter * CUPS – saturn1.pos, saturn2.pos, saturn3.pos


Asteroids: Asteroids Rocky and metallic objects that orbit the Sun but are too small to be considered planets Most are contained within a main belt that exists between the orbits of Mars and Jupiter


Orbital Resonance: Orbital Resonance Ratio of revolutions p:q → ratio of radius p-2/3:q-2/3 by Kepler’s law 2:1, 3:2 (Hilda group), 1:1 (Trojans) Chaotic motion nearby the fractions e.g.) 13:6 * CUPS – aster21.pos, aster32.pos, aster136.pos


Trojan Asteroids: Trojan Asteroids Asteroids at the L4 and L5 Lagrangian points of the Sun-Jupiter system (Greece(L4) and Troy(L5)) * CUPS – trojan.pos


Motion of N Attracting Bodies: Motion of N Attracting Bodies Virial Theorem If U = krn+1, = (n+1)/2 See a classical mechanics textbook for details e.g.) “7.13 Virial Theorem” of Marion (4th ed.) For gravitational potential, n = -2 → = -2 * CUPS – Simulation now!


Construction of Our Own Solar System: Construction of Our Own Solar System Real Solar System * CUPS – realss.pos The mass of Jupiter 10 times * CUPS – ssj10.pos 100 times * CUPS – ssj100.pos 1000 times (≒ solar mass) * CUPS – ssj1000.pos The masses of other planets are negligible


Summary: Summary By using restricted problem of three bodies method and motion of N bodies, we can simulate simple astronomical situations and obtain qualitative results. The more sophisticated method we use, the more realistic result we will get. Astrophysics will progress with increasing computing power!


Reference: Reference Danby, Kouzes, and Whitney. CUPS – Astrophysics Simulations, 1995 Toomre, A., Toomre, J. Galactic bridges and tails. Astrophysical Journal 178:623-666, 1972 Marion, J., Thornton, S. Classical Dynamics of Particles and Systems (4th ed.), 1995 The Encyclopedia of Astrobiology, Astronomy, and Spaceflight – http://www.angelfire.com/on2/daviddarling/Lagpoint.htm National Radio Astronomy Observatory – http://www.nrao.edu/imagegallery/php/level2.php?class=Galaxy Students for the Exploration and Development of Space http://seds.lpl.arizona.edu/nineplanets/nineplanets/nineplanets.html Asteroid Introduction - http://www.solarviews.com/eng/asteroid.htm Wikipedia : The Free Encyclopedia – http://en2.wikipedia.org/wiki/Trojan_asteroid


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