← The Anatomy of Spiral Arms ↑ →
In this simulation with 2 010 stars, each star follows a rosette. This is the form of orbits predicted under Newtonian gravity for mass distributed symmetrically in the galactic plane and in the halo. Rosettes are aligned because of mutual gravity. The gravity of the arm causes stars to follow the arm during the ingoing part of their orbit.
The simulation uses orbits with random eccentricities between 0.10 and 0.18, corresponding to observations of local stars in the Milky Way. The pattern created is a grand-design two-armed spiral. To see an orbit, follow the path of one of the giant stars. The spiral pattern seems to shrink, but really it is rotating slowly backwards compared to stellar orbits. If one were to scale this galaxy to the Milky Way, with the bar taking up the central section, the Solar orbit would be about midway in the spiral. The diameter of the galaxy would be about 130,000 light years. The Sun has been moving down a spiral arm for about 150 million years, and is now crossing outwards through this arm, prior to leaving the arm, crossing the other arm, and rejoining the original arm. The time for the animation is about 200 million years.
Longer, 85 MB simulation with more stars and a bar.
These movies may be freely distributed, provided that a credit is given to Charles Francis and Erik Anderson as authors of the model and the animation. Please cite arXiv:0901.3503. The relationships between speed of rotation of the bar (bar pattern speed) the speed of rotation of the spiral (spiral pattern speed) and orbital velocities depend on the mass distribution of the Galaxy and are not known.
As shown by Newton, orbits about a massive body are ellipses, aligned with the massive body at a focus. The eccentricity vector (or, equivalently, the Laplace–Runge–Lenz vector) is defined pointing in the direction from apocentre toward pericentre, and with magnitude equal to the eccentricity of the ellipse.
In a spiral galaxy, mass is distributed thoughout the disc and also in the halo. As a star moves towards pericentre, the gravitational mass drawing it towards the galactic centre is less than it would be if all the mass of the galaxy was concentrated at one central point. As a result the orbit of the star is less curved at pericentre than an ellipse, and the orbit opens out into a rosette.
In each part of the orbit, the motion is approximated by elliptical motion. It is still meaningful to define the eccentricity vector. The eccentricity vector is seen to regress; it rotates backwards with respect to orbital motion. It also changes in magnitude; eccentricity is less near to pericentre (change in magnitude will not concern us in this analysis). If we imagine looking at the motion from above from a platform rotating with the eccentricity vector, the orbit will once again appear closed — the orbit will be approximately elliptical with the galactic centre at the focus, as in the previous image.
Now let us assume that orbits precess at the same rate at any orbital radius. Use coordinates rotating with the eccentricity vectors. In these coordinates orbits are approximate to ellipses aligned at a focus. If the major axis is rotated for larger orbits, they can be aligned in such a way that more than half of each orbit lies on an equiangular spiral.
In practice, regions of higher density exert more gravity, and attract more stars. The gravity of the arm ensures that stars rejoin the arm near apocentre, where they move more slowly. Thus gravity of the arm maintains the required orbital alignment to preserve the spiral pattern, and ensures that orbits do precess at the same rate, the rate of spiral pattern speed.
The pitch angle of a given spiral galaxy is directly related to the orbital eccentricities of stars in that galaxy. Higher eccentricity orbits fit spirals with higher pitch angles. The pitch angle of this arm is 11°. Orbits have eccentricity 0.3. Eccentricities in the range ˜0.25 to ˜0.35 can be fitted to an arm with this pitch angle.
The gravitational potential of a spiral galaxy can be compared to a giant, spiral-grooved, funnel. The gravity of the spiral arms creates the grooves in the potential. A star near apocentre, the slowest part of its orbit, will tend to fall into a groove and then follow the groove toward the centre of the spiral, picking up momentum as it goes. Eventually, the star gains enough momentum to jump free of its groove. It crosses over the next-highest groove, then falls back to a higher point in its original groove. At the same time, the funnel rotates slowly backwards due to orbital precession.
As stars are drawn into an arm, the gravitational field of the arm grows stronger, making the groove deeper and drawing greater number of stars into the arm. Thus, mutual gravity between stars reinforces spiral structure. The potential field of the arms locks the rate of orbital precession to spiral pattern speed for a wide range of orbits.
As stars follow orbits of different sizes and at different rates, accidental alignments lead to regions of greater mass, which attract more stars. In this way, flocculent galaxies form. Focculence consists of spiral segments, formed out of alignments in orbital rosettes. This will tend to happen more on the outer part of stellar orbits, where orbital velocity is less and stars spend more time. In consequence, trailing spirals are found in galaxies.
NGC 4414 as observed by the Hubble Space Telescope. Credit: HST/NASA/ESA
As a galaxy evolves, the gravity of larger spiral segments in a flocculent galaxy attract increasing numbers of stars, and spiral segments join up to form arms.
The Pinwheel Galaxy, M101, as observed by the Hubble Space Telescope. Credit: HST/NASA/ESA
Under gravity, gas clouds follow similar motion to stars, going in towards pericentre along the arm, and coming away from pericentre in a diffused manner. Gas in the arm is in turbulent motion, as gas clouds seek to cross in the arm and gain velocity as they approach pericentre. As shown in the simulation, stars rarely collide because of their small size compared to space between them. Stars pass outwards towards apocentre through the arm, but when outgoing gas from one arm meets ingoing gas in another arm, collisions between gas clouds create regions of higher pressure, and greater turbulence. Pockets of extreme pressure due to turbulence generate the molecular clouds
in which new stars
The Anatomy of Spiral Arms ↑ The Velocity Distribution of Local Stars →
In a multi-arm spiral, outgoing gas meeting an arm would outweigh ingoing gas in the arm. This would tend to remove gas from the arm. In a two armed spiral, the gas in the arm has greater mass. Thus, a two-armed gaseous spiral can be stable, whereas multiarmed gaseous spirals cannot.
Outgoing gas applies pressure to the trailing edge of a spiral arm with an inverse proportionality to radius. If one gaseous arm advances compared to the bisymmetric position, the pressure due to gas from the other arm will be reduced. At the same time, pressure on the retarded arm due to outgoing gas from the advanced arm will be increased. Thus gas motions preserve the symmetry of two-armed spirals.
In a flocculent galaxy, gas will also be attracted into spiral segments. When gas clouds meet they combine to form larger clouds, adding more quickly to the mass of the segment. An underlying bisymmetric structure is observed, even in flocculent spirals, due to the formation of gaseous arms (containing gas as well as stars). Because of the pressure on the trailing edge of a gaseous arm, it will regress more slowly than a non-gaseous arm. Thus non gaseous arms are absorbed into a gaseous arm and multi-armed spirals evolve into bisymmetric spirals.