Lab 4

Radial Velocity Determination of

Membership in Open Clusters

Sean Lockwood1, Dipesh Bhattarai2, Neil

Lender3

December 2, 2007

Abstract

We used the Doppler velocity of 29 stars

in the open clusters NGC 752, Kronberger 1,

& NGC 7063 in order to confirm their status as

such. We also determined that 4 of these target

stars are likely to be non-members in the fore-

ground or background of the clusters.

1. Introduction

Groups of stars that appear in close prox-

imity to one-another on the sky may or may not

be physically associated. One way to determine

if they do indeed form an open cluster is to mea-

sure the radial velocity component of each mem-

ber and study the distribution (Rutledge, 1997).

If a cluster exists, this process can also be used

to eliminate individual stars from the field that

are non-members. If the distribution shows no

correlation between the velocities, the apparent

group of stars are not physically associated, and

are known as an asterism.

Doppler velocities can be measured by ob-

serving a region of the stellar spectrum with

1email: lockwood@bu.edu

2email: dipesh@bu.edu

3email: nlender@bu.edu

enough emissions lines and comparing its wave-

length offset to other stars.

2. Observations

On 2007 October 21-23, we observed open

clusters and a possible extrasolar planetary tran-

sit. We used the PRISM instrument on the

Perkins 72” telescope in Flagstaff, AZ. The

weather was reasonably clear on all nights, with

the exception of a few clouds. The weather was

not consistent enough for absolute photometry,

but was acceptable for spectroscopy and differ-

ential photometry. Gusts of wind occurred on

the second night, but remained within accept-

able parameters.

2.1. Subframing

Subframing is a process where unnecessary

pixels are not read out of the CCD in order to

save read time and disk space. We imaged our

clusters in full frame in order to find targets for

spectroscopic analysis, and then we proceeded

with sets of images in appropriate subframes.

We defined three types of subframes to aid

in our data collection. Subframe–1 was a small

thumbnail always used with the slit in and the

grism out in order to check stellar centering

before the spectrum was taken. They were

also used in analysis in an attempt to deter-

mine velocity corrections due to centering er-

rors. Subframe–2 was used with both the slit

and grism to obtain a spectrum of the centered

star (see fig. 3). This subframe was fairly large,

so we were able to obtain spectra from the night

sky for calibration purposes. Finally, subframe–

4 was used without the slit or grism in order to

– 2 –

image our transit target and calibration stars.

2.2. Image Calibration

Each night we took 9 bias images in each

subframe. We subtracted the median of each

row’s overscan region to remove local variations

in bias due to voltage fluctuations during read-

out. When subtracting the bias from other im-

ages, the images’ overscan regions were also sub-

tracted in this way. In order to reduce read-noise

and cosmic-ray interference in individual bias im-

ages, we stacked each set and median smoothed

over all pixels.

Sets of flat-field images were obtained for

each night in all applicable filters and with the

grism. Each flat-field image was overscan and

bias corrected. Then, the flat-field images were

median smoothed similarly to the bias image. In

the case of the grism flat-field, we needed to re-

move the blackbody curve induced by the tem-

perature of our dome lamp (see fig. 1). In order

to do this, each row was divided by its median

value. Non-uniformities in CCD sensitivity, as

well as the remaining interference fringe pattern

(see fig. 2) were successfully divided out of spec-

tral images using these grism flats. Finally, all

flats were normalized so that their median value

is 1 in order to keep flux values in calibrated im-

ages near their original scale.

In order to reduce an image, we apply:

Image =Image

o− Bias

(Flat − Bias)/median(Flat − Bias)

Fig. 1.— A bias-subtracted, median-stacked

dome flat-field image with the grism in place.

The overall shape is characteristic of the dome

lamp’s blackbody temperature. In order to re-

move this unwanted effect, this image needs to

have each spectral row divided by its median

value. Finally, the entire array needs to be renor-

malized.

2.3. Spectral Extraction

One-dimensional spectra were then ex-

tracted from the two-dimensional spectral im-

ages, removing spatial information about the

star’s location in the slit. Data points located

too far away from an image’s target spectrum

contribute unwanted noise to the signal from the

sky and other stellar sources in the slit. So, we

fit a polynomial to the slit’s position over wave-

length space (see fig. 3). For each spectral row,

31 points centered on this fit were summed to-

gether to find stellar flux versus pixel. How-

ever, a problem was noted with this method—

quantization errors occurred when the fit shifted

between discrete pixels (see fig. 4). In order to

fix this, each row was shifted via cubic interpola-

tion the appropriate fractional number of pixels

from the fit (with the cubic parameters set to op-

– 3 –

Fig. 2.— A close-up view of a median-stacked

dome flat-field image with the grism in place.

Note the interference fringes and other non-

uniformities that are divided out of spectrome-

try images. While the interference fringe pat-

tern is never quite the same between images,

flat-fielding tends to decrease its presence in our

data.

timize flux conservation). To get a sense of the

flux distribution about the slit, see fig. 6. Along

with the reduced one-dimensional flux (see fig.

7), we are also able to extract background sky

lines from most exposures.

This method of spectra extraction assumes

that a horizontal row of our array always corre-

sponds to a constant in wavelength space. While

this approximation has proved to be fairly ac-

curate, a more prudent approach would involve

fitting background sky lines at different places

along the spectral dimension in order to remove

their curvature. If this is of concern, we rec-

ommend using IDL’s poly 2d and polywarp

routines to remove any such curvature before ex-

tracting spectra.

Fig. 3.— A bias-subtracted, flat-fielded spectral

image in subframe–2 with an overlaid (red) fit

used to extract the spectrum.

2.4. Baseline Removal

As the goal of our reduction process is to

prepare for cross-correlation between multiple

images, it was useful to remove the broad black-

body shape from each spectrum so that it would

not interfere with the comparison of stellar ab-

sorption lines. To do so, we fit a fourth-order

polynomial to a median-smoothed version of

each stellar spectrum and subtracted this base-

line (see fig. 8). While not perfect, this process

proved fast and effective for our purposes. Note

that absorption lines will now appear as negative

flux, as one would expect.

2.5. Wavelength Solution

An approximate wavelength solution for all

of our spectra was found by manually identify-

ing the pixel location of known background sky

emission lines (Hanuschik, 2003) and fitting a

second-order polynomial. The process of identi-

fying emission lines was aided by iterating solu-

– 4 –

Fig. 4.— The central stellar spectrum with

each spectral row shifted by a discrete amount,

as determined by a fit to the stellar spectrum.

Note the quantization errors that occur where

the shading becomes discontinuous when the fit

crosses pixel boundaries in the spatial dimension.

tions with more and more lines. Thus, we were

able to generate a rest–wavelength solution as a

function of pixel number:

λ = 4877.9209 + 1.4803y − 8.0880 × 10−5y2

We resampled the spectra via interpola-

tion over a standard wavelength range and step

in order to remove the nonlinear components

that would decrease the effectiveness of cross-

correlation algorithms.

Note that the y-intercept of this wavelength

solution is not entirely accurate. So for this pa-

per, we adopt the definition that NGC 7063 A (as

observed in 071022.020.fits) has zero Doppler

shift. It happens that this star has one of the

highest Doppler shifts of the observation run,

leading to negative values for almost all other

shifts.

Fig. 5.— By shifting each row by a fractional

number of pixels (via cubic interpolation), as

determined by a fit to the stellar spectrum, we

are able to avoid the quantization errors that oc-

curred in fig. 4.

2.6. Velocity Conversion

In order to convert wavelengths, λ, to ve-

locities, v, we used the non-relativistic Doppler

shift equation:∆λ

λ=

v

c

where we chose λ = 6809 A, the central wave-

length of our spectra.

– 5 –

Fig. 6.— A three-dimensional view of the

flux contained in the spectral line, shifted to a

straight line via interpolation.

Fig. 7.— The stellar spectrum (above) and cor-

responding background sky emission spectrum

(below), as summed up along the spatial dimen-

sion. Note the presence of sky emission lines su-

perimposed on the stellar spectrum.

Fig. 8.— A fourth-order polynomial was fit to

the stellar baseline and subtracted off in order to

remove the blackbody temperature dependence

from each stellar spectrum.

– 6 –

2.7. Cross-correlation

Using IDL’s c correlate function, we

compared the relative shift of each spectrum to

a reference spectrum, NGC 7063 A. This refer-

ence spectrum was selected based on its high

maximum cross-correlation coefficient with most

other spectra. We shifted the target spectrum

relative to the reference spectrum by a few pixels

in discrete steps. In order to find the fractional-

pixel shift, we fit a quadratic to the top few

points from this shift and found the peak (see

fig. 9).

Fig. 9.— The degree to which two arrays cor-

relate with one–another is found as a function

of shift between them. This best shift is then

quadratically interpolated to a fractional pixel.

The red curve is the quadratic fit to the top 5

points, and the red line marks the peak of this

fit.

While the wavelength solution of any par-

ticular spectrum had errors on the order of

Angstroms, the differential error between the po-

sition of any two spectra usually turned out to

be small. In a few cases, we took multiple obser-

vations of the same object without moving the

telescope. By comparing the pixel shifts of each

of these, we find that errors in cross-correlation

range from 0.06 pixels to 0.5 pixels (in one case),

corresponding to velocity errors of 3900 m/s to

32,000 m/s. A reasonable error could be 0.15

pixels, corresponding to 9,800 m/s. While in-

dividual shift errors undoubtedly depend on a

number of independent factors, each set of ob-

servations that can be used to find these errors

consisted of only two or three data points, lead-

ing to a great amount of uncertainty in the esti-

mate of this error.

2.8. Velocity Corrections with

“Thumbnail” Images

One problem with high-precision spec-

troscopy that we anticipated was the occurrence

of random shifts in wavelength due to misalign-

ment of the target star with the slit. While this

error might not be very big, it is important to

note that the shifts we are looking for are very

small—on the order of pixels.

Fig. 10.— A sample thumbnail image. The

centering of the central star relative to the slit

turned out to range over only a few pixels.

It was our hope to develop a method to char-

– 7 –

acterize this spectroscopic shift due to star mis-

alignment by imaging each target without the

grism before taking the longer spectroscopic ex-

posure. These exposures were small subframes,

nicknamed “thumbnail” exposures (see fig. 10).

Since we did not move the telescope between

these two types of exposures, pointing errors

caused by moving the telescope were not present.

We parameterized the shift (“thumbnail pa-

rameter”) by first determining the center of the

target star with a Gaussian fit. We then fit a

line to the slit’s position across the array, ignor-

ing points contaminated with starlight. Then,

we removed any vertical component of the slit

by interpolating each row of the array by the

proper amount. Next, we summed the regions

1.5-3*FWHM from the center and fit a Gaussian

in order to determine the star’s centering rela-

tive to the horizontal slit. This process yielded

values that correspond to the number of pixels

that the star is shifted above or below the slit.

We observed a calibration source, M67 MMJ

6480, at various slit positions by moving the tele-

scope various amounts and taking sets of thumb-

nails and spectroscopy exposures (see fig. 11).

By cross correlating the reduced spectra from

these observations, we were able to determine

how many (fractional) pixels the spectrum had

shifted and plot this against the thumbnail pa-

rameter. Unfortunately, the scatter in this cali-

bration curve is too great for the purpose of cor-

recting the rest of our data. To correct this prob-

lem in the future, we recommend taking more

calibration images of these types over a larger

range of slit misalignments.

Fig. 11.— An attempt to calibrate misalignment

when centering the slit on a target star. Due to

the large amount of uncertainty in our linear fit,

we chose to ignore this calibration in our spec-

tral reduction pipeline. It should be noted that

the calibration range is smaller than the range

of “thumbnail parameters” over which we had

hoped to apply this correction, which would have

led to questionable extrapolation in many cases.

3. Results

Table 1 shows the mean and standard devi-

ation for our three target clusters. Furthermore,

figures 12, 13, & 14 display these distributions

as histograms. By noting the Gaussian-shaped

groupings of velocities, it appears that all three

target clusters are indeed clusters. It is usu-

ally visually apparent that a few stars fall out-

side the expected Gaussian distribution of veloc-

ities, meaning that they could possibly be non-

members of the cluster in the foreground or back-

ground. See table 2 for information on individual

stars within these clusters.

– 8 –

Fig. 12.— Based on this histogram, we conclude

that the star on the right is the interloper. How-

ever, this result may be incorrect due to incom-

plete coverage of the cluster, leading to what ap-

pears to be a gap in velocities.

Fig. 13.— It appears that Kronberger-1 is indeed

an open cluster. The star on the right (labeled

“g”) appears to be a promising candidate as an

interloper.

Fig. 14.— Note that the bump on the left is

actually two stars due to the size of the bins.

– 9 –

Unfortunately, the quality of these results

must be questioned because we do not have

enough redundant measurements to accurately

determine the errors due to cross-correlation and

slit misalignment. We must be careful, as adapt-

ing our reduction pipeline to minimize errors a

priori in the few redundant measurements we

do have will not necessarily reduce overall er-

ror. There are just not enough of these points to

accurately describe the statistics.

It appears that our initial goal of opti-

mizing exposure times for Doppler velocity

determination cannot be achieved because all

of our images exceed the criteria needed to

determine this value to reasonable accuracy. We

recommend that future work on this question

focus on observing fewer targets, but with

parameters such as exposure time and slit

alignment position studied in further depth. To

circumvent part of this problem, we could add

random noise to our spectra to simulate the

effects of a lower signal-to-noise ratio on our

results.

Cluster v σv

NGC 752 -99,200 13,877

Kronberger 1 -1,617 9,434

NGC 7063 -26,413 30,981

Table 1: Cluster statistics.

Cluster Star |v − v|/σv

NGC 752 a 0.06

b 1.16

c 0.07

d 0.29

e 0.27

f 1.25

g* 1.71

Kronberger-1 a 0.69

b 0.83

c 0.07

d 0.27

e 0.40

f 0.19

g* 2.33

h 0.38

NGC 7063 a 0.86

b 0.68

c 0.28

d 0.68

e 0.15

f 0.60

g 0.90

h 0.34

i 0.09

j 0.22

k 0.47

l 0.39

n* 2.03

o* 2.32

Table 2: Distance from mean in standard units.

Probable interloping star denoted by *.

– 10 –

4. References

Rutledge, et al. Galactic Globular Cluster Metallicity Scale From the Ca II Triplet, PASP 109,

883R (1997).

Hanuschik R.W. A flux-calibrated, high-resolution atlas of optical sky emission from UVES.

<Astron. Astrophys. 407, 1157 (2003)>

5. Acknowledgments

We would like to thank Prof. K. Janes of Boston University for organizing, participating in,

and providing guidance for our observation run.