February 23, 2005

Avi, Andreas, and Heino,

   These calculations are not as straight forward
as we might have thought, but I've made quite a bit
of progress.  I thought I'd send you some things to
look at before we talk.

   Here is where things stand at the moment.  
Attached are some plots (.ps) and print outs (.prt) 
in a gzipped tar file.

=======
The Nbody program now does the following:

1) integrates the 3 major bodies (M31, M33 & MW) backwards
   in time for 10 Gyr to get their starting locations
   and velocities, then it

2) puts in test stars in M33 between 2.5 and 7.5 kpc radius,
   and run things forward in time to the present.

3) I've added "mass tracing" objects to M33 to keep track
   of tidal stripping of (dark) matter from M33.  I use
   these to update a table that gives the gravitational
   potential (assumed spherical) of M33 as a function
   of radius.

=======
Results:

The .ps files include 3 types of plots:

i) m31m33mw_0000_0000.ps:  Shows positions of the centers of
   the three galaxies over time.  The file name, in this 
   case "...0000_0000", gives M31's Vx_Vy (zero here).
   The bottom panel gives the separation between M33 and M31
   (red) and between MW and M31 (green).

ii) tracers_0000_0000.ps:  Shows the positions of stars in
    M33 on a large scale.  The actual stars used to study 
    tidal stripping are a tight clump that is barely resolved
    on this scale.  The others are the "mass tracing" objects 
    used to determine the gravitational potential of M33 on
    large scales.  They are very sensitive to tidal stripping.

iii) stars_0000_0000.ps: Shows the positions of the stars in
     M33 on a small scale.  These are the stars used to trace 
     tidal stripping. They started between 2.5 and 7.5 kpc
     of the center of M33, all in the XY plane.  The tilting
     of the planes, the thickness of the other planes, and
     stars >8 kpc from the center are all a result of tidal
     effects.

iv) There are some files for zero velocities of M31 that are
    for lower dark matter masses: eg, stars_0000_0000_0.33mass.ps
    The labeling should be clear.

------------
The .prt files are summaries of many Nbody runs with trial
values of M31's (Vx,Vy):

a) Look at "grid_results_mass1.00.prt":

   The dark matter halos of the three galaxies were defined
   following the email that Avi sent a while ago:

   M31: Vrot(kms), M(Msun), R(kpc) = 250.0 3.43E+12 235.4
   M33: Vrot(kms), M(Msun), R(kpc) = 110.0 2.92E+11 103.6
   M-W: Vrot(kms), M(Msun), R(kpc) = 220.0 2.33E+12 207.1

   This file is a grid of values which give the percentage
   of stars stripped from M33.  The horizontal axis 
   is for values of the unknown proper motion of M31
   in the East-West direction.  The vertical axis is
   for the North-South direction.  The proper motions
   are in km/s.

   For M31's (Vx,Vy) = (-66,+33) km/s, the grid value
   is 95, meaning 95% of M33's stars were stripped.
   My definition of stripped is either
      i) they moved beyond 9 kpc radius, or
     ii) they moved out of the XY starting plane
         by more than 20% of their XY-radius.

   For  M31's (Vx,Vy) = (+66,-100) km/s, the grid value
   is 0, meaning 0% of M33's stars were stripped.

   One can see that for large regions of (Vx,Vy) space
   that significant tidal stripping has occurred.

   Also, with so much mass in M31 and the MW, they tend
   to collide over the past 10 Gyr.

   One "problem" in the calculation is that the distance
   between M31 and the MW does not always "come back" to
   its current observed value.  I believe this occurs 
   in cases where M33 is strongly tidally stripped by
   close encounters with M31.  Currently, when I integrate 
   the positions of the three galaxies backwards in time,
   to get the starting conditions, I assumed that M33
   retains its full dark matter halo.  However, going
   forward in time, that halo can be stripped away, which
   (I believe) changes the motion of M31 with respect to
   M33.

b) "grid_results_mass0.67.prt"

   This grid was run the same way as above, except that
   the total size and mass of the dark matter halos of
   the galaxies was reduced by a factor of 0.67.
   This brings the total masses of M31 and the MW down
   to 2.3 and 1.6 x 10^12 Msun, respectively.  These
   values are closer to values favored from recent 
   studies of the Local Group kinematics and globular
   cluster motions.

   The tidal stripping in the Nbody simulations with these 
   lower total masses is decreased.  However the overall
   pattern of tidal stripping as a function of M31's (Vx,Vy)
   is similar.

c) "grid_results_mass0.33.prt:

   With even lower dark matter masses for the galaxies,
   now 33% of the original values, tidal stripping hardly 
   occurs.


Mark