← ptolemyisdeadalreadysogetoveritokay.com ↑ →
Astonishing as it may seem, the study of Galactic orbits in the 20th century has been dominated by a theory of epicycles adapted from that of Ptolemy, which had been abandoned after Newton explained Kepler’s discovery that planetary orbits are elliptical. More seriously, the
density wave hypothesis is often treated as though it were established science, although it conflicts with Newtonian gravity, and can reasonably be described as a catalogue of elementary mathematical and physical mistakes.
Epicycles
Epicycles were revived by the Swedish astronomer,
Bertil Lindblad. Lindblad’s epicycle theory perturbs a circular orbit by superimposing an elliptical motion. As is seen in the diagram, for low eccentricities it is possible to approximate a precessing orbit. However, this is not useful. After the effort, and with the inaccuracy, of introducing the epicyclic approximation, no more has been said than that the orbit is a rosette. We already knew that.
To take the analysis any further, and calculate, for example, the rate of precession, we would need further corrections because the epicyclic approximation is only good for low eccentricities, and we would need to know the gravitational potential throughout the orbit. If fluctuations in potential due to spiral arms, the bar, and satellites, are to be considered, then analysis becomes very difficult, and is only possible by numerical methods. However, if a computer will be used to provide a numeric solution, there is no point in starting with an approximation. It would be better, and simpler, to find a numerical solution directly from the equations of Newtonian gravity.
When mathematicians have addressed the problem of orbits in a gravitational field differing from that of a central mass distribution, they have perturbed the
eccentricity vector, or
Laplace-Runge-Lenz vector, used in the analysis of
spiral structure. Lindblad’s epicycles offer no advantages, but more seriously they are commonly used in analyses fraught with simple mistakes.
Closed Orbits
A closed orbit is one which returns to apocentre at the same point after a number of cycles, and then repeats the same path. We can close the orbit in the above figure by using coordinates rotating at just such a rate that the point
B moves so as to coincide the previous position of
A after a number of orbits. We can calculate an equation for the rate of rotation of the coordinate system as follows:
Angular speed, or angular frequency, of circular motion: Ω.
Angular speed of radial oscillation (i.e. elliptical motion): κ.
Period of radial oscillation: 2π ⁄ κ.
Angular distance, AB, after m > 0 radial oscillations: AB = 2πn ± Ω 2πm ⁄ κ, for integer n.
Mistake 1: Standard treatments overlook the possibility of a minus sign. The orbit may be closed by rotating in either direction, through either the major or the minor arc AB.
Angular speed of rotating coordinates: f.
To establish a closed orbit in the rotating coordinates, after
m periods of the radial oscillation, we require,
f 2πm ⁄ κ = AB = 2πn ± Ω 2πm ⁄ κ,
f = nκ ⁄ m ± Ω.
The usual solution uses plus and
m = −2n.
f = Ω − κ ⁄ 2.
Mistake 2: This ignores imaging issues with m ≠ |n|, such that the rotating coordinate system shows more than one orbit.
Mistake 3: The use of plus means the rotation of coordinates is in the direction of orbital motion, but actually orbits regress.
The correct rate for spiral pattern speed requires minus and
m = n = 1, so that
f = κ − Ω.
Density Wave Theory
Spiral structure is usually explained using the density wave theory of
Lin &
Shu, according to which stars move through the arms, which consist of dense regions analogous to regions of heavy traffic on a motorway.
Mistake 4: A simple analogy with patches of heavy traffic fails because a wave effect would require that stars slow down when they approach a dense region, but the gravity of the dense region would cause them to speed up. Even if the theory were right, the analogy would be wrong.
Mistake 5: The claim that stars move through the arms on near-circular orbits is empirically incorrect. It is observed that stars move along the arm for a large part of their orbits.
|
Following Kalnajs, density wave theory is usually explained by means of a diagram constructed by enlarging and rotating ovals. The orbits are an epicyclic approximation in coordinates rotating at a rate Ω − κ ⁄ 2, and appear as ovals centred at the galactic centre. |
Mistake 6 (following from mistake 2): Even if the rotation of the ovals were correct (it is not), the diagram would show a single spiral twice, not a bisymmetric spiral.
Mistake 7 (to be replaced by mistake 8): The increase in density shown as spirals in the figure represents only a small proportion of the orbit. If stars were placed randomly, one on each oval, this would lead to a very small increase in stellar density on the “arms”, much less than is observed.
Mistake 8 ("correcting" mistake 7): The orbits in the figure are imagined as gas in laminar flow. Then the increase in gas density in the spiral pattern is assumed to instigate star creation in the arms. But this increase in density would not be sufficient to initiate star formation, and spiral arms contain a substantial increase in density of old as well as new stars.
Mistake 9 (following from mistakes 6 & 8): If the ovals were to represent gas motion, it could not be treated as laminar flow, because while paths do not cross in rotating coordinates, they do cross in physical space. Taken as the motions of gases, the Kalnajs diagram is impossible.
Summary
Understanding the cause of spiral structure has appeared a difficult problem for about eighty years, ever since it was clearly recognised that spiral nebulae are other galaxies. It is a many-body problem with unknown initial conditions, and its solution involves the turbulent motion of interstellar gas in addition to the motions of the stars. Nonetheless, once known, the solution is remarkably straightforward, and is confirmed by the empirical evidence of the local velocity distribution and the neutral hydrogen distribution.
The introduction of a working model for spiral galaxies will lead to important changes in the study of galactic dynamics. Many of the treatments found in textbooks are seen not to apply. Perhaps of greater concern to astrophysics as a science is that some of those treatments are not even internally consistent. When mathematicians have addressed the problem of orbits in a gravitational field differing from that of a central mass distribution, they have perturbed the eccentricity vector or the Laplace-Runge-Lenz vector. Lindblad’s epicycles have distracted astrophysicists from this type of analysis and substituted a model which is conceptually more complex, and which, because numerical solution is still required, adds nothing to what is already known of galactic orbits. The analysis has been based empirical assumptions about orbits which were not justified from data at the time, and which have since proven false, and it contains elementary mathematical mistakes which have been compounded by further mistakes in the development of density wave theory.
The implication to astrophysics is severe. The motions of stars are governed by known mathematical laws. Astrophysics is, or at least it should be, a mathematical science. One should therefore expect that theories in astrophysics are subjected to rigorous mathematical scrutiny. Regrettably, the degree of scrutiny applied to Lindblad’s epicycles and to density wave theory has been seriously lacking. Students should be made aware that these ideas can no longer be considered as science, and authors of textbooks should consider whether they merit anything more than a historical note.
ptolemyisdeadalreadysogetoveritokay.com ↑ Links to Topics →
There are no comments on this page. [Add comment]