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Fitting Spiral Structure to the Milky Way

It is straightforward to observe spiral structure in other galaxies, but extremely difficult to observe it within our own galaxy, as recently illustrated by observations by the Spitzer space telescope showing that stellar concentrations are not found at the positions where two arms were thought to be. There have been two principle methods for locating spiral arms, based on the distributions of neutal and ionised hydroden. The usual four-armed spiral is derived principally from the distribution of ionized hydrogen (Georgelin & Georgelin, 1976; Russeil, 2003), but in fact the distribution is so sparse and irregular that it is difficult to be certain that anything has really been fitted. The neutral hydrogen distribution was famously mapped by Oort, Kerr and Westerhout (1958), and more recently by Levine, Blitz and Heiles (2006). Levine, Blitz & Heiles fit (slightly irregular) four-armed spirals, but comment that other fits are possible.

The Local Standard Of Rest

The local standard of rest (LSR) is defined to mean the velocity of a circular orbit at the Solar radius from the Galactic centre. The definition idealizes an axisymmetric galaxy in equilibrium, ignoring features like the bar, spiral arms, and perturbations due to satellites. An accurate estimate of the LSR is required to determine parameters like the enclosed mass at the solar radius and the eccentricity distribution which is of importance in understanding galactic structure and evolution.

It is customary in kinematic analyses of the stellar population to denote velocity in the direction of the galactic centre by U., in the direction of rotation by V., and perpendicular to the galactic plane by W. The solar motion relative to the LSR is (U₀, V₀, W₀). The usual way to calculate the LSR is to calculate the mean velocity of a stellar population, and to correct V₀ for asymmetric drift. The method assumes a well-mixed distribution, but the observed kinematic distribution is highly structured, and divides into six populations each with distinct motion and stellar composition. Motions of thin disk stars in the W-direction may be treated as a low amplitude oscillation due to the gravity of the disc, and as independent of orbital motion in the U-V plane. It is thus not unreasonable to calculate W₀ as the mean motion of a population. However, in the absence of knowledge of the causes for streams, there is no way to relate the statistical properties of their motion to U₀ and V₀.

Spiral structure draws orbits into eccentricities dictated by the pitch angle of the arms. As a result, very few stars are found on circular orbits, leading to a minimum in the velocity distribution at the local standard of rest. In Calculation of the Local Standard of Rest from 20 574 Local Stars in the New Hipparcos Reduction with Known Radial Velocities, Erik Anderson and I used this minimum to find the LSR, obtaining (U0, V0, W0) =  (7.5 ± 1.0, 13.5 ± 0.3, 6.8 ± 0.1 ) km s−1.

The Eccentricity Distribution

For an elliptical orbit the eccentricity vector is defined as the vector pointing toward pericentre and with magnitude equal to the orbit’s scalar eccentricity. It is given by

where v is the velocity vector, r is the radial vector, and μ = GM is the standard gravitational parameter for an orbit about a mass M. For a Keplerian orbit the eccentricity vector is a constant of the motion. Stellar orbits are not strictly elliptical, but the orbit will approximate an ellipse at each part of its motion, and the eccentricity vector remains a useful measure (the Laplace-Runge-Lenz vector, which is the same up to a multiplicative factor, is also used to describe perturbations to elliptical orbits). We smoothed the eccentricity distribution by replacing each discrete point with a two dimensional Gaussian function and finding the sum. Standard deviation is used as a smoothing parameter. A standard deviation of 0.005 gave a clear contour plot. In a well-mixed population, eccentricity vectors will be spread smoothly in all directions, with an overdensity at apocentre and underdensity at pericentre, because of the increased orbital velocity at pericentre and because stars at apocentre come from a denser population nearer the galactic centre. This is not seen in the plot. The distribution is concentrated at particular values, corresponding to stream motions.

Eccentricity distribution for the entire population, for stars closer to apocentre (green) and stars closer to pericentre (red), as defined by position with respect to the semi-latus rectum. The bulk of local stars have eccentricities in the range 0.07 to 0.2

The Neutral Hydrogen Distribution

The modal value of eccentricity in the Milky Way is a little above 0.1. This indicates a much lower pitch angle than is used in four armed spirals. We fitted a symmetric double spiral with a pitch angle 4.92° and 8.2 kpc bar to the hydrogen maps of Oort, Kerr and Westerhout and those of Levine, Blitz and Heiles (2006). There is a subjective element in the quality of such a fit, but the two-armed spiral seems to us to better follow the line of the hydrogen clouds, while the more open four armed spirals appear to follow clouds bridging the true line of the arms. The top plot shows density, the second shows height.

We constructed a symmetric two-armed spiral with this pitch angle from ellipses using an angular increment 33° for each 105% enlargement. For the calculated value of the LSR, current solar eccentricity is 0.138. The Sun is 16.3° before pericentre (as determined by the current eccentricity vector), at which point it should lie near the inner edge of the arm, and be heading outwards through the arm. We were not able to make a meaningful map showing the positions of stars with velocities in the arm with respect to the Sun, because the data from Hipparcos is not sufficiently comprehensive over large enough distances, and because the data for which we have radial velocities is strongly weighted to the northern hemisphere, but, in agreement with typical estimates, we found indications in the data that we are in the arm, about 100-150pc from the inner rim, and too far to be able to detect the outer rim.


The solar eccentricity, 0.138, has been used for the ellipses in the diagram. Good fits are obtained with eccentricities from about 0.1 to about 0.18. A two-armed spiral necessitates a little care to avoid confusion in naming the arms, because traditionally named sectors with the same name lie on different arms. Orion is not a separate spur, but is a part of a major arm connecting Perseus in the direction of rotation to Sagittarius in the direction of anti-rotation. We have called the Orion Arm the major spiral arm containing the Sun as well as Norma, Perseus, Orion, Sagittarius, and Cygnus sectors. The Centaurus arm contains Sagittarius, Scutum-Crux, Cygnus, and Perseus sectors. The solar orbit is shown in approximation, together with the major axis and latus rectum.

Streams and Spiral Structure

It is now possible to understand that the major local stellar streams are not dissolved clusters, but actually show the spiral structure of the Milky Way. The Alpha Ceti stream is large and disperse. It consists of stars, like the Sun, in the Orion arm. As can be seen in the spiral structure animation, stars move on the inward part of their orbits along the arm with widely varying velocities. The Hyades stream consists of stars from the Centaurus arm, crossing the Orion arm at the same part of their orbit. The Pleiades stream consists of new born stars, with low eccentricities, created when outgoing gas clouds from the Centaurus arm meet with the Orion Arm. The Sirius stream consists of moderately young stars with relatively high eccentricities, whose orbits have not yet settled into alignment with the arms.

The Age of the Milky Way Spiral

It is found from isochrone aging (the position of stars on a Hertzsprung-Russell diagram that the Hercules stream, and the high eccentricity stars in the Alpha Ceti stream consist of old stars, with ages predominantly greater than about 9 Gyrs. Eccentricities are too high for normal spiral arm motion. Orbits with the right eccentricity (~0.29) can be aligned to spiral structure spanning both arms, as in the diagram. The stability of these of configurations shows that the Milky Way has been a grand-design two-armed spiral for about 9 Gyrs.

The Spiral Structure of the Milky Way ↑ ptolemyisdeadalreadysogetoveritokay.com →