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Galaxies without dark matter halos?
greywolf42 wrote:
John Chandler wrote in message .... discussion of what is required to determine the orbit of .... an object... .... John Chandler: "... the 3-D orbit can be determined directly [from] astrometry ... without the help of radial velocities." greywolf42: "One cannot determine inclination of a stellar orbit just from astrometry -- even in the rare cases where you can watch and measure the describing of a full ellipse by the orbiting body. The projection of an ellipse is still an ellipse." Perhaps this simpler discussion might help... Consider an orbit of unknown real inclination and ellipticity which happens to appear circular in our line of sight. We can easily determine the real inclination and ellipticity of the orbit with the help of Kepler's laws. We measure the position of the star carefully all along its apparent circular orbit and infer the projected angular velocity of the star in this projected path. If it is constant, then the orbit is indeed circular and our sightline is perpendicular to the system. If not then the ratio of the minimum to maximum angular velocities is just the ratio of the pericenter and apocenter distances of the star from the center of mass. Simple geometry then tells us the inclination. We're just using Kepler's law that cross-product of the orbital velocity and the radius vector is a constant, i.e., conservation of angular momentum. E.g., suppose the orbit really is very elliptical but is projected to a circle. The star will appear to be move very slowly around that circle when it's at the apocenter of the orbit, and very quickly when near pericenter. The situation is not quite as trivial with an orbit that projects to an ellipse, but the same basic principles apply. There will be only one rotation that will transform the measured angular velocities and distances to ones that match Kepler's laws. So we can get the inclination without radial velocity information. That doesn't get you the scale of the system, i.e., a conversion from arc-seconds to meters. The easiest way to get the scale of the system is certainly to get the distance. But it's not the only way, at least in principle. E.g., if the mass of the primary can be measured directly (e.g., by a gravitational redshift), then the duration of the orbit can be used to get the scale. [You can also infer the distance to the object without a parallax, but that isn't needed to get the actual scale of the orbit.] The ambiguity with regard to the inclination comes up more in cases where the orbit is not derived from astrometry, but from the measurements of the changing radial velocities of stars (i.e., most planetary detections). In this case, the ambiguity cannot be resolved without further measurements, afaik. Regards, Tom McGlynn |
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