Majestic Numerical Simulation of Simultaneous 3 Resonances --- OLR,

OILR and IILR

Chien-Chang Yen

嚴健彰July 28, 2003

Lunch Talk

Outline

•Object --- Galaxy (Stars, gas, dust)

•Performer --- Bars and spirals

•Mechanism --- Resonance excitation/Lindblad Resonance

•Theory --- Spiral density theory

•Approach --- Numerical simulation / parallel computation

•Result --- 3 resonances simulation

•Conclusion

GalaxyGalaxies --- They are large systems of stars and interstellar matter, typically containing about trillion stars. Their masses are also several trillion times that of our Sun.

Types --- Edwin Hublle (in 1936) classifies galaxies using ``tuning-fork diagram.” Originally, he thought it was an evolution sequence. We now know that it is NOT. Elliptical, Spirals (regular and barred), & Irregulars. Most galaxies are small and faint-at far distances only see luminous galaxies.

The Milky Way

Hubble Tuning-fork Diagram

Bars & Spirals

Barred Spiral Galaxy NGC 1300

Bars & Spirals

Barred Spiral Galaxy NGC 7479

Type SBb Galaxy in Pegasus with several distant companions

A long bar-shaped structure and spiral arms that are not symmetrical are visible in its central region.

Bars & Spirals

Barred Spiral Galaxy NGC 151

BarsBars are density waves which develop spontaneously in a galactic disc subject to its own gravitation. All the matter takes part in this gravitational instability, interstellar gas and stars. A disc is the more unstable as it is "cold", i.e. its velocity dispersion (or disordered motions) is small, and its rotational velocity (ordered motion) is high.

The bar is a perturbation which breaks the axial symmetry of the galactic disc, and consequently creates tangential gravity forces. These forces result in gravity torques on the spiral arms of the galaxy, and help to transfer the angular momentum of the interstellar gas towards the outer parts, which allows a large gas mass to fall towards the center. The action of the bar is thus to trigger bursts of star formation in the centers of galaxies, when those are fueled with gas.

Bars in Early and Late-Type GalaxiesElmegreen & Elmegreen (1985, 1989) have shown that the nature of bars changes along the Hubble sequence. First, spiral galaxies with early Hubble types tend to have bars that are longer relative to the size of the optical disk and the turnover radius of the rotation curve, compared to late-type barred galaxies; second, early-type bars are more uniform in intensity relative to the outer disk light profile, compared to bars in later Hubble types. The spirals surrounding these bars also differ, with a tendency toward grand design two-armed spirals occurring for the early types, while the later types have a more multiple-armed or flocculent structure, as in nonbarred galaxies. The bar and spiral-arm amplitudes differ along the Hubble sequence too, with stronger bars and radially decreasing spiral-arm amplitudes in early type SB or SAB galaxies, and weaker bars with radially increasing spiral-arm amplitudes in late types.

Spiral StructuresSpiral arms come in different "flavors":

•~10% grand-design (two well-defined spiral arms)

•~60% multiple-arm

•~30% flocculent spirals (no well-defined arms at all)

Spiral arms seem to be trailing arms, but this is very hard to determine. In some rare cases, spiral arms may lead.

Spiral Structures

grand-design

flocculent

Spiral Structures

The Winding Problem Spiral rotation curves are not "solid body" rotation curves. What would a solid body rotation curve look like? Spiral rotation curves are flat: V ~ constant. Therefore the angular rotation rate goes like V/r, so that the outer edge takes longer to complete an orbit than the inner regions.

So no solid arm could last very long -- it would quickly get "wound up":

The Winding Problem It looks like a spiral pattern, what's wrong with that? It happens too fast. It happens in a few rotation periods (say, half a billion years). The age of the Universe is something like 10-15 billion years -- if spirals made their arms this way, they should all be so tightly wrapped we wouldn't see them as spiral arms anymore!

So spiral arms can't be physical objects. What else could they be?

Spiral Density Wave Theory

The formation of the spiral patterns is still a mystery. A generally accepted mechanism for producing the spiral structure is called the density wave which is thought to be a disturbance that gently travels around the galaxy compressing gas in its wake. There are several good competing theories explaining the origin of the density wave:

•gravitational interactions between neighboring galaxies might be able to trigger the spiral density waves

•asymmetric mass distributions in the center of the rotating galaxy could generate the required density wave

Spiral Density Wave Theory

Wave is a propagating pattern of disturbance and it has propagating direction. A wave, due to a local increase in the gravitational field, that produces a series of alternate compressions and rarefactions as it propagates with fixed angular velocity in a rotating galaxy. The compression also acts on interstellar gas in the galaxy, which is triggered to form stars on the leading edges of the spiral arms. The large-scale structure of spiral galaxies can be understood in this way.

Spiral Density Wave Theory

Waves are also the angular momentum transport between the bar and disk. As waves attenuated by viscosity, the angular momentum will be deposited into the disk. This will cause the disk matter moving inward or outward, depending respectively on whether the angular momentum carried by the waves is negative or positive.

Rotation of the Gas Disk in M87

Rotation Curve of GalaxiesThe rotation curve of a galaxy is the variation in the orbital circular velocity of stars or gas clouds at different distances from the center. The manner in which the velocities vary with radius reflects the distribution of mass in the galaxy:

Rotation Curve of Galaxies

Milky Way

Resonance

Every object has a unique natural frequency of vibration.

A periodic force occurring at the same frequency as the natural frequency of vibration of an object may cause the object to vibrate. This is called mechanical resonance.

Epicyclic

1. In Ptolemaic cosmology. A small circle, the center of which moves on the circumference of a larger circle at whose center is the earth and the circumference of which describes the orbit of one of the planets around the earth.

2. Mathematics. A circle whose circumference rolls along the circumference of a fixed circle, thereby generating an epicycloid or a hypocycloid.

Lindblad Resonance

Numerical Simulations

The numerical simulation bases on gas dynamical system with periodic forcing on Cartesian coordinates. Furthermore, the isothermal gas is under assumption.

The total of grid number is the square of 1024. We need a lot of storage, for example, it needs 20 GB to save the data.

Motivation for Parallel Computing

The traditional scientific paradigm is first to do theory (say on paper), and then lab experiments to confirm or deny the theory. The traditional engineering paradigm is first to do a design (say on paper), and then build a laboratory prototype. Both paradigms are being replacing by numerical experiments and numerical prototyping. There are several reasons for this.

•Real phenomena are too complicated to model on paper (eg. climate prediction).

•Real experiments are too hard, too expensive, too slow, or too dangerous for a laboratory (eg oil reservoir simulation, large wind tunnels, overall aircraft design, galactic evolution, whole factory or product life cycle design and optimization, etc.)

Why parallelism is essential

The speed of light is an intrinsic limitation to the speed of computers. Suppose we wanted to build a completely sequential computer with 1 TB of memory running at 1 Tflop. If the data has to travel a distance r to get from the memory to the CPU, and it has to travel this distance 10^12 times per second at the speed of light c=3e8 m/s, then r <= c/10^12 = .3 mm. So the computer has to fit into a box .3 mm on a side. Now consider the 1TB memory. Memory is conventionally built as a planar grid of bits, in our case say a 10^6 by 10^6 grid of words. If this grid is .3mm by .3mm, then one word occupies about 3 Angstroms by 3 Angstroms, or the size of a small atom. It is hard to imagine where the wires would go!

Why writing fast programs is hard

Why does the speed depend so much on the problem size? The answer lies in understanding the memory hierarchy. All computers, even cheap ones, look something like this:

Parallel Computations•IBM RS/6000 SP in ASCC

•MPI (Send/Receive), Message Passing Interface

Message PassingEach processor is running a (possibly) different program (in C, Fortran, or another serial language), and whenever processor Pi wants to send data x to processor Pj, Pi calls a subroutine ``send(x,Pj)'', and Pj must also call a subroutine ``receive(x,Pi)''. It is low level and error prone; for example if Pj never does ``receive(x,Pi)'', Pi waits forever. But currently it is the most common and most portable parallel programming style. Message passing subroutine libraries (which implement ``send'' and ``receive'', among other functions) include CMMD or the Connection Machine Message Passing Library (just on the CM-5), PVM, or Parallel Virtual Machine (widely portable, but generally slow), and MPI, or Message Passing Interface (a recent standard adopted by the industry).

Parallel Computations•MPI Example---

If (MYID .eq. 0) THEN

CALL MPI_SEND(A,N,REAL8,1,99,0,IERR)

ELSEIF (MYID .eq. 1) THEN

CALL MPI_RECE(A,N,REAL8,0,99,ISTATE,IERR)

ELSE

.

.

ENDIF

Result

Result

Conclusion•A full view of numerical simulations on 3 resonances, OLR, OILR and IILR, have been demonstrated here.

•OLR produces tight spiral structure and IILR produces the open spiral structure. Furthermore, OILR and IILR will suggest a ring-spiral structure.

•Parallel computation is quite useful for large scale simulations.

Thanks•I am very appreciate Prof. Yuan’s advice and patience.

•ASIAA is a good place to do research.

•Thanks for your attention.