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Galaxies without dark matter halos?



 
 
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Old October 30th 03, 10:42 PM
Tom McGlynn
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Default 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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