Subject: As Per Olsen Request

From: Dave Monet

Submitted: Mon, 3 Mar 2003 15:42:03 -0700

Message number: 90 (previous: 89, next: 91 up: Index)

Knut Olsen asked (meaning tasked) me to scribble something about wiggles
and astrometry.  I don't have any profound insights, but there are a couple
of relationships that can be presented.

1) Planetary perturbations:  These are the limiting case of where the
   flux of the planet is negligible with respect to that of the star,
   and normalizing to things like the Sun and Jupiter make sense.

   FullAmp (mas) =  (a/5) * (10/D) * Mp / Ms

   where   a is the semi-major axis in AU
           D is distance in pc
          Mp is the mass of the planet in Jupiter masses
          Ms is the mass of the star in solar masses
   There are a semi-infinite number of other ways to think about, plot,
   and rummage around parameter space, but this one is pretty simple.
   Clearly, we win big if the astrometric error can be driven down
   into the mas region, but when you examine 1e10 (+/- a few) systems
   you are bound to find something.

2) Astrometric binaries:  For stellar systems, you don't measure the
   semi-major axis.  You measure the distance between the barycenter
   and the photocenter (see van de Kamp's Principles of Astrometry
   for details).  You still have

         m1a1 = m2a2

   but you measure

         FullAmp (mas) = 2000 * (alpha/D)

   where  alpha is measured in AU
            D   is measured in PC

   and

          alpha = a1*(m2/(m1+m2) - (l2/(l1+l2)))

   which is just the distance of the photocenter from the barycenter.
   If you prefer magnitudes, the second term can be written as

          l2/(l1+l2)  =   (1/(1 + 10**(0.4*DeltaMag)))

   The interesting part of LSST is that we will measure the astrometric
   wiggle in several colors, and this gives insight into the flux ratio
   of the stars in the different passbands.  Hence, we extend the serious
   study of a few systems with crude searches of billions of systems.
   My guess is that somebody can find the appropriate integrals over
   the various orbital elements to derive mean properties of binary stars.

3) In a series of widely ignored (and justifiably so) papers
   in the last 70s (some of which were not even published in those
   dark days before astro-ph), Monet attempted to remind the reader
   that Kepler's equation is solved by a Fourier series whose coefficients
   are Bessel functions (see Brower and Clemence's Methods of Celestial
   Mechanics for details).  Hence, one can design tuned filters for
   Keplerian motion (and even apsidal motion, unseen companions, etc.)
   that work over a surprisingly wide set of cases needing only Fourier
   analysis of unequally spaced observations.  Boring, but it might come
   in handy for large-scale searches for wiggles.  There are other
   boundary conditions (like getting the same period in both axes,
   using higher harmonics if the SNR allows, etc.) that might assist.

   An equally amusing, but apparently not to the referee in 1980, study
   was the degradation of SNR as a function of partial coverage of the
   period.  Obviously, very short coverage means that linear motion is
   just as good a fit so one cannot determine Keplerian motion.  Similarly,
   once you have sampled both peaks of the basically sinusoidal period
   (yes, large eccentricities are a nuisance but there is no divergence,
   and yes, large eccentricity looks a lot like noise and really needs
   a different algorithm), the Fourier coefficients are pretty well
   measured and you do pretty well.  In the region of where you have
   half a hump (Heffalump?), I found that the uncertainty in the semi-major
   axis is effectively increased by a factor of the fractional period
   coverage to the -3.5 power or so.  That is to say, if you have about
   half an orbit, then the derived parameters are about a factor of 10
   worse than if you had the same number of equal quality observations
   spaced over a whole orbit.  I believe that other investigators have
   found similar conclusions, but I haven't paid close attention.

   The reason for this history lesson is that I think that it
   will be pretty silly to attempt to estimate Keplerian elements
   for systems whose periods are more than 4X the duration of the
   observations.  For LSST, one can hope that we will be able to probe
   systems with periods as long as 40-50 years, and we might find
   interesting nearby systems with separations of many arcseconds.
   Again, it is the thoroughness of the LSST survey that is the big
   step forward.

-Dave

LSST LSST LSST LSST LSST    Mailing List Server   LSST LSST LSST LSST LSST LSST
LSST
LSST  This is message 90 in the lsst-general archive, URL
LSST         http://www.astro.princeton.edu/~dss/LSST/lsst-general/msg.90.html
LSST         http://www.astro.princeton.edu/cgi-bin/LSSTmailinglists.pl/show_subscription?list=lsst-general
LSST  The index is at http://www.astro.princeton.edu/~dss/LSST/lsst-general/INDEX.html
LSST  To join/leave the list, send mail to lsst-request@astro.princeton.edu
LSST  To post a message, mail it to lsst-general@astro.princeton.edu
LSST
LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST LSST