Kinematical membership criteria
• Members follow the motion of the cluster
center of gravity
• Internal velocity distribution
• From best to …
1. Radial velocity and proper motion
2. Radial velocity
3. Proper motion
Clemens, 1985, ApJ, 295, 422
Sofue, 2021, Galaxies, 8, 37
Van Bueren, 1952, BAN,
11, 385
Hyades
After the
correction of the
solar motion
Be careful about the
sign/direction of X
and U
Motion of the Sun
• Peculiar Apex motion:  = 18h28m,
 = +30o
; l = 56.24o
, b = +22.54o
• (U,V,W) = (-10, +5, +7) km s-1
• vorbit = 220 km s-1
• Local Standard of Rest (LSR)
Determination of the kinematical
membership
• Three possibilities:
1. Observation of the position at two
difference times (= epochs), with a very
large time basis. First photographic plates
around 1860, largest time scale about 160
years
2. Proper motions of stars in the direction of
the Declination  and Right Ascension 
3. Radial velocity measurements
Mathematical method
• Measurement of the position (X, Y) at two different
epochs t1 (´) and t2 (´´) for each star
• Calculate the absolute distance in X and Y for both
epochs and each star individually
• Determine the differences of the absolute distances
• Plot the histograms of the differences of the
absolute distances. The members have to group
around the minimum of the distributions (ideal case:
minimum = zero).
Example from Javakhishvili et al. (2006, A&A, 447, 915) for
Collinder 121
Now we need a mathematical formalism to describe
the membership probability from the distributions
• Calculate the absolute distance in X and Y for both
epochs and each star individually
• Plot the histograms of the differences of the
absolute distances
• The distributions are fitted with Gaussian functions
• The probability p, if a star is member of the star
cluster is defined as
Javakhishvili et al.,
2006, A&A, 447, 915 for
Collinder 121
From these diagrams, the
membership probability can
be exactly determined
• In the same way, the proper motions in  and  can
be used, the basic equations and the determination
of the membership probability is exactly the same
Now, with the new Gaia data we can even
investigate very distant star clusters using proper
motions
Klein Wassink, 1927, Publications of the Kapteyn Astronomical Laboratory Groningen, Vol. 41, 1
Praesepe
d = 190 pc
Field stars
Common proper motion
Credit: Tristan
Cantat-Gaudin
Dramatic improvement by Gaia even for
overlapping star clusters
Radial velocities
• Advantages:
1. Correlated with the galactic rotation only
2. Possible to measure for most distant
cluster members
• Disadvantages:
1. High-resolution high S/N spectrum
needed
2. Faintness of members for distant clusters
In total, 321 open clusters
Mean radial velocity [km s
-1
]
Determination of the radial velocity
• Doppler shift of spectral lines
• Determine the central wavelength of the shifted line
• Better accuracy if
1. Instrumental resolution (/) is higher
2. Signal-To-Noise ration (S/N) is higher
3. vsini of star is lower
4. The number of measured lines is higher
c
vR
 =
5 10 30 100
3500 0,058 0,117 0,350 1,167
4000 0,067 0,133 0,400 1,333
4500 0,075 0,150 0,450 1,500
5000 0,083 0,167 0,500 1,667
5500 0,092 0,183 0,550 1,833
6000 0,100 0,200 0,600 2,000
6500 0,108 0,217 0,650 2,167
7000 0,117 0,233 0,700 2,333
7500 0,125 0,250 0,750 2,500
8000 0,133 0,267 0,800 2,667
RV
[km s-1] [Å]
 [Å]
Instrumental profile defined by the resolution:
Rotational broadening:
with
with
for  = 4200Å
for  = 4200Å
Situation – Gaia DR2
Can we use this for studying the internal velocity
distribution (rms = 4.3 km s-1 )?
How good can be mean RVs determined?
Situation – Gaia DR2
Huge spread and only
a few members per
cluster
Situation – Gaia DR2
Not
useable for
internal
distribution