1. Orbital Period of the Pre-Cataclysmic Variable: NN Serpentis
Carlos Osorio∗
(Physics 134L)
(Dated: June 3, 2016)
Abstract. A pre-cataclysmic variable is a binary star system consisting of a white dwarf and a
less massive star, whose separation is not small enough to allow mass transfer between them. These
systems possess some interesting characteristics, for which they are commonly a subject of study.
On this lab a pre-cataclysmic variable by the name of NN Serpentis (NN Ser) is observed using a
CCD sensor located at Las Cumbres Observatory Global Telescope Network in Goleta, California.
After the extraction of the data from FIT files into catalogues, and the isolation of the NN Ser
data points from these catalogues, the light curve for this binary system in the V-band filter was
successfully plotted. By then plotting a waveform to fit the data, the period of the orbit of such
system was calculated to be 3.206±0.023 hrs. This value, along with some others acquired from
the scientific literature, were used to find a total of 7 parameters defining the binary, including the
mass and radius of the red dwarf, as well as the separation between the two stars. The calculated
parameters were within the scope of the previously published values, thus rendering the observation
as successful.
CONTENTS
I. INTRODUCTION 1
A. Roche Geometry 1
B. NN Ser 2
II. METHOD 2
A. Data Acquisition 2
B. Data Extraction and Calibration 2
C. Graph Generation and Curve-Fitting 3
III. RESULTS 4
A. Orbital Angular Frequency and Eclipse
Duration 6
B. The NN Ser Orbit and the Reflection Effect 8
IV. CALCULATIONS 9
V. CONCLUSION 10
VI. CURRENT RESEARCH 10
VII. REFERENCES 11
I. INTRODUCTION
A. Roche Geometry
A closed binary system is a system of two stellar
objects (such as stars, or dwarfs) that orbit around
∗ Also at Physics Department, University of California Santa Bar-
bara; cosorio@umail.ucsb.edu
each other due to a gravitational attraction. Closed
binary systems also possess the trait of having other
significant, non-gravitational interactions taking place
between the two stellar objects (Warner et al., 1995).
Among the most diverse and commonly studied types
of closed binary systems are the cataclysmic variables.
These are binary systems composed of a white dwarf
and some other, less massive, type of star. These two
stellar objects are usually referred to as the primary
and secondary stars, respectively. Due to the small
separation between the two stars, and the high density
of the white dwarf, the secondary star of a cataclysmic
variable becomes highly distorted and is no longer
spherical, but instead has a tear-like shape pointing
towards the primary star. The geometry that dominates
this type of system is called Roche geometry.
If the stars are close enough so that the gravita-
tional attraction of the primary star at the radius of the
secondary is greater than that of the secondary itself,
the mass of the secondary star begins flowing into the
primary, creating an accretion disk around it. This
mass flow is the defining characteristic of a cataclysmic
variable. The maximum region a secondary star could
occupy, with still complete gravitational binding over
its material, is called the Roche lobe, and it is bounded
by the critical gravitational potential at which mass
transfer can happen.

2. 2
B. NN Ser
Now, if the secondary star of this type of binary
system does not fill its Roche lobe completely, no mass
transfer can happen between the stars, and the system
is said to be detached. This binary is what is commonly
referred to as a pre-cataclysmic variable. An example of
such a binary system is the NN Serpentis, located about
500 parsecs away from earth in the Serpens constellation.
First referenced in the Palomar-Green Survey in 1980,
and originally catalogued as PG 1550+131, the NN Ser
system has a Right Ascension of 15h 53m 31.051s and
a declination of +120◦
57’ 40.13” in FK5 coordinates
(Simbad, 2016). It is a pre-cataclyismic binary system
composed of a red dwarf of low mass and a white dwarf
of about half the mass of the sun. With an inclination
of almost 90◦
, and a primary star much smaller than its
companion, the NN Ser undergoes through deep periodic
eclipses (Parsons et al., 2010).
The purpose of this lab is to observe the NN
Serpentis system through a CCD sensor to obtain the
period at which the stars orbit each other. This result
will then be used, along with some published parameters,
to calculate the radius and mass of the stars, as well
as the separation between the dwarfs and the orbital
velocity of the red dwarf.
II. METHOD
A. Data Acquisition
The NN Ser binary system was observed using a CCD
sensor located at Las Cumbres Observatory Global Tele-
scope Network in Goleta, California. The observation
took place on May 8th
, 2014 at 6:05am and lasted for
about 4 hours until 10:04am of the same day. The data
acquired consisted of 331 FIT images, 303 of which were
taken with a V band pass filter, while the rest were taken
with a B-filter. All 331 images were obtained with an
exposure time of 30 seconds, and had an average of 47
seconds between recordings. Due to the small amount of
data collected with the B-filter, this data was not ren-
dered useful for any of the calculations involved, and was
thus disregarded.
B. Data Extraction and Calibration
After the data was collected, the following step was
to extract from the FIT images the information that
was needed to obtain the orbital period of the binary
system. For this we used SExtractor to get a catalogue
of each FIT file. These newly created cat files contained
the time of observation, the aperture magnitude and
isophotal flux of each star in the image with their
associated errors, and each stars’ right ascension and
declination.
Now, in order to isolate the information of the
NN Ser system from the rest of the data points in
the cat files, a python code was written that would
recognize and extract the data on this binary system
from within each of the files. To do this an algorithm
was implemented in which the program would select
from each file the data point with the closest right
ascension and declination to that of the NN Ser system.
This was done by finding the data point within each file
that had the minimum square difference with the NN
Ser’s coordinates, as given by the following formula;
d = (ra − RA)2 + (δ − D)2 (1)
In here d represents the total distance of each data
point to the NN Ser’s coordinates, ra and δ are the right
ascension and declination of each data point respectively,
and RA and D the constant coordinate values for the
NN Ser, as given in section 1B.
After secluding the data for the NN Ser from
the cat files, the next step was to calibrate the aperture
magnitudes obtained from the CCD sensor to represent
the apparent magnitude of the stars. To do so we chose
two stars from the fit files and compared their aperture
magnitudes to the apparent V-Tycho magnitudes given
to them in the Sky-Map.net catalogue. Before this,
however, the Tycho magnitudes given by Sky-Map were
converted to the standard Johnson-Cousins passband
magnitudes by using the conversion formulas provided
by Mamajek et al. (2002). The function used to signal
out these two stars from each cat file and compare
them to their Sky-Map values was the same as that
used above to extract the NN Ser data points. The
average of the differences between the data and Sky-Map

3. 3
V-magnitude values, for each cat file, was then added to
all the aperture magnitudes of the NN Ser data points
in their corresponding cat file in order to convert them
into calibrated apparent magnitudes.
C. Graph Generation and Curve-Fitting
The last stage was to plot the graphs of the data col-
lected, and, using an existing curve-fitting function from
Python, draw a line of best fit for the data. Matplotlib’s
pyplot object was used to create four graphs in total.
The first (Fig.1) was a plot of the NN Ser’s uncalibrated
V-magnitudes against time, with the second plot (Fig.2)
being the same but with calibrated magnitudes instead.
The third graph (Fig.3) was a plot of the isophotal flux
as a function of time. Finally, the fourth plot (Fig.4) was
a scaled version of the second graph, with a sine wave
plotted to fit the data points. For all of these plots, the
time of each data point was specified to be the mid-point
of the exposure time, that is, 15 seconds after the expo-
sure started. To get a curve to fit the third plot, scipy’s
curve fit function was used. For the function to work, a
sine wave function was created and given as argument,
as well as an initial guess for the sine wave’s parameters.
This function would then return the fitted parameters of
the curve, as well as the errors in calculating such pa-
rameters.
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
Time of Observation [hr.]
8.5
8.0
7.5
7.0
6.5
6.0
UncalibratedApertureMagnitude
Uncalibrated V-Magnitude Light Curve
FIG. 1. Plot of the aperture magnitude of NN Serpentis, taken from Las Cumbres Observatory Global Telescope Network, as
a function of time. This plot shows some evidence for a fluctuation of NN Ser’s apparent magnitude with time, as well as a dip
in magnitude around 8.7 hours that is characteristic of a deep eclipse.

4. 4
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
Time of Observation [hr.]
16.0
16.5
17.0
17.5
18.0
18.5
CalibratedApparentMagnitude V-Magnitude Light Curve
FIG. 2. Plot of the calibrated apparent magnitude of NN Serpentis as a function of time. This plot shows a significant increase
in resolution from the uncalibrated plot in Fig.1. We can now see more clearly the periodic fluctuation of the binary system’s
apparent magnitude. The dip in magnitude caused by the red dwarf eclipsing the white dwarf around 8.7 hours is also more
evident. There exist a number of outliers between 8 and 8.5 hours that indicate the visibility of the sky around that time might
have been affected due to weather.
III. RESULTS
The graphs discussed in the previous section are
plotted below. The results given by these graphs turned
out to be very interesting, and it is valuable to run them
down one by one. As stated before, Fig.1 shows the
uncalibrated V-magnitude light curve. At first glance
this graph does not seem to have a concrete periodic
oscillation. Although this plot does show sign of a rela-
tion between the V-magnitude of NN Ser with time, the
seemingly rough ends and uneven spacings between the
minimums give little hope for finding a fitting model to
calculate the period of the binary’s orbit. Nevertheless,
by calibrating the aperture magnitudes into apparent
ones, in the process mentioned in section 2.B above, this
apparent chaotic result turns into a smooth waveform,
whose sinusoidal appearance is hard to overlook (Fig.2).
Looking at this graph we don’t only see with clarity a
periodic oscillation of the apparent magnitude, but there
also appears to be a hole in the plot once the data points
get close to the waveform’s minimum. This gap on the
outline of the graph is not due to lack of measurements,
but is instead the outcome of what is called a primary
eclipse. Since the white dwarf of this binary system is
almost half the size of its companion, and the inclination
of the system relative to us is close to 90◦
, when the
red dwarf’s orbit passes in front of its primary star,
the light emitted by the latter is completely blocked.

5. 5
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
Time of Observation [hr.]
10.0
9.5
9.0
8.5
8.0
7.5
7.0
6.5
6.0
IsophotalFlux
Time dependece of Isophotal Flux
FIG. 3. Graph of the Isophotal Flux of NN Serpentis, captured with an exposure time of 30 seconds by a CCD sensor, and
plotted as a function of time. This plot strongly correlates with the plot of apparent magnitude in Fig.2 as is expected since
the magnitude of a system is just the integral of the flux density over the filter’s banpass range. We see once again sinusoidal
fluctuations of the signal, as well as strong evidence of a deep eclipse around 8.7 hours, and signs for bad weather conditions
between 8 and 8.5 hours.
Therefore, for those brief minutes, we receive only the
light from the red dwarf, whose low temperature makes
its apparent magnitude so high (and its brightness so
low) that the system becomes no longer visible in the
V-bandpass region for the CCD sensor used. Such a
drastic change of magnitude during the eclipse is why
this phenomenon is also termed as a deep eclipse. This
effect can also be witnessed in Fig.6, in which the binary
star appears to completely fade away from the FITs
image, as compared to a few moments before, in Fig.5,
in which it ws completely visible. The error bars of both
these diagrams were taken to be the aperture magnitude
errors recorded for the binary in the cat files.
The next graph (Fig.3) serves as a verification of
the effects seen in the light curve of Fig.2. In it, the
sinusoidal aspect of the binary’s signal, as well as the
primary eclipse’s dip in signal around 8.7 hrs can be
recognized almost immediately. Notice also the set of
outliers between 8 and 8.5 hours present in both of these
graphs. The appearance of this outliers is thought to
be due to bad weather conditions during this time that
may have clouded the sky, thus making some of the
measurements produced in this range appear as if they
were dimmer. It is no surprise that the flux diagram
seems so closely related to the the light curve of Fig.2,

6. 6
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
Time of Observation [hr.]
16.0
16.2
16.4
16.6
16.8
17.0
17.2
CalibratedApparentMagnitude V-Magnitude Light Curve
Best Fit Curve
FIG. 4. A scaled up version of the light curve shown in Fig.2. This plot also has a sine wave of the form A sin(ω×t+φ)+b plotted
to fit the data. The sine wave seems to fit the data very closely, and thus serves as a good model for the periodic oscillation of
the binary star’s magnitude. With this fit, the angular frequency of the orbit was determined to be ω = 1.960 ± 0.014 rad−1
,
and thus the period to be P = 3.21 ± 0.11 hrs. The duration of the eclipse was also calculated to be ∆t = 0.176 ± 0.025 hrs.
after all, the magnitude of a system is just the integral
of its flux density over the specified band range.
A. Orbital Angular Frequency and Eclipse
Duration
The last graph presented (Fig.4) is a zoomed in
version of Fig.2 with a sine wave plotted to fit the data.
As explained in the methods section, this was possible
by using scipy’s curve fit function in python. The curve
of best fit generated through this method coincides very
accurately with the data points, as we see that the
curve passes through most of the data points’ error bars.
It is thus safe to say that this curve can serve as an
accurate model of the NN Ser’s light curve, at least in
the short-term range. The form of this sinusoidal curve
was A sin(ωt + φ) + b, and through curve fit the best-fit
parameters were found to be those shown below in Table
1. More specifically, the final wave plotted on this graph
was V = 0.293 sin(1.960t + 3.21) + 16.538.
There are three important aspects of these cal-
culated parameters. First of all, the angular frequency
of the magnitude, and thus the angular velocity of
the system’s orbit was calculated to be 1.960±0.014
rads/hr. This value will later be used to find the orbital
period of the NN Ser binary. The amplitude of the wave
described above also tells us the change in magnitude of

7. 7
-500-
500
FIG. 5. FITs image of the NN Serpentis system. In this image we can clearly see the appearance of the binary system, along
with some neighbor stars.
the system as it completes the orbit. The analysis of this
result will be shown in the next section (3.B). Finally,
we see that the waveform is centered at a magnitude
of 16.538±0.004 which relates very closely to the calcu-
lated mean V-magnitude of 16.8 by Haefner et al. (1989).
From the curve fit function it was also possible
to obtain the variance of the best fit parameters calcu-
lated. These values were then square rooted to give the
standard deviation of each parameter, and thus used
as their respective errors. From this graph it was also
possible to obtain the duration of the secondary eclipse.
This was calculated as being the time difference between
the midpoints of both the last data point before the
magnitude dip and the first recording after it, and the
last exposure of this event and the first data point back
near the fitted waveform. This way, the duration of the
eclipse (∆t) was found to be around 10.56 ± 1.5 min,
which is very close to the value found by Haefner et al.
(1989) to be around 12 min. The error in the calculation
of this value takes into account the amount of time
between the observations of the transition from the
waveform magnitude to the eclipsed value.

8. 8
-500-
500
FIG. 6. FITs image of the NN Serpentis system during a primary eclipse. This image is centered around the same location
in space as Fig.5, but NN Serpentis does not appear to be visible. This is a clear indication of the large change in magnitude
this binary undergoes through whenever the red dwarf passes in front of its primary, in what is generally called a deep eclipse.
More over, this total eclipse gives evidence for the inclination of the system being close to 90◦
.
B. The NN Ser Orbit and the Reflection Effect
One last thing to make sense of from the light curves
plotted in Fig.2 and Fig.4 is the respective orbital mo-
tion of the stellar objects in the system, along with the
reasoning behind this periodic motion of the apparent
magnitudes. As stated a couple of times before, the
minimum point in the light curves, where the dip in
magnitude occurs, corresponds to the eclipse of the
white dwarf by the red dwarf. It is known that the orbit
of these two stars are circular and not eccentric since the
tidal interactions of a pre-cataclysmic variable eliminate
any initial eccentricity in the system’s orbit (Warner
et al., 1989). This effect is also visible in the light curves
plotted, as any eccentricity in the orbit would cause
there to be different time displacements in between
extrema. With this being said, it is quite obvious that
the moment in which the red dwarf is situated behind
the white dwarf, in what is called a secondary eclipse, is
exactly one half-period away from the primary eclipse.

9. 9
This result is seemingly contradicting at first
due to the fact that the brightness of the binary system
seems to increase when the red dwarf is covered. The
reason behind this increase in brightness when the white
dwarf is in front of the red dwarf is what is usually
referred to as the reflection effect. This a phenomenon
caused by the red dwarf’s absorption and re-emission
of the white dwarf’s radiation (Hellier et al., 2001).
Now, since the red dwarf is tidally locked to the white
dwarf due to tidal interactions (J. Horner et al., 2012),
there is only one side of the red dwarf the absorbs all
the incoming radiation from the white dwarf, making
that side significantly hotter than the rest of its mass.
Since in stellar objects higher temperatures mean higher
brightnesses (and thus lower magnitudes), the side of
the red dwarf facing the white dwarf has a substantially
brighter surface than the rest of its body. That being
said, when the white dwarf is in front of the red dwarf,
the secondary’s brightest surface is also pointing directly
at Earth, and so we receive both the light of the white
dwarf, and it’s companion’s highest brightness, thus
reaching a maximum in the light curve.
With this known, we return to the amplitude
calculated for the light curve’s fitted waveform in Fig.4.
This amplitude tells us the change in magnitude of
the system as it completes the orbit. The only thing
changing throughout these stars’ orbit, in terms of the
brightness emitted towards the earth, is the angle at
which we face the hottest and brightest region of the
red dwarf. Therefore, the amplitude of the fitted sine
wave (0.293 ± 0.006) gives us a relation between the
brightnesses of the red dwarfs’ faces, and thus a relation
between each sides temperature. Specifically, this result
tells us that the brightest and dimmest faces of the red
dwarf have a magnitude difference of twice the derived
amplitude (0.586 ± 0.012). Although this result will not
be fully explored in this paper, it should pose as an
interesting calculation for any future observations on the
system’s temperature distribution.
IV. CALCULATIONS
Now that the sinusoidal aspect of the NN Ser system
was modeled and used to find the angular frequency of
the orbit, it is time to use these results to find some
defining parameters of the system. Unfortunately there
are very few things that can be done with only the data
that was used for this lab. One of the few parameters
that can be calculated on its own is the period of the
orbit. For this, the following equation was used, with P
being the period in hrs:
P =
2π
ω
(2)
The result of this calculation, along with all the ones that
will follow, are displayed in Table 1.1 below. With a value
of approximately 3.21 hrs, the period calculated closely
approaches its published value of 3.12 hrs (Brinkworth
et al., 2006). Notice also that all of the derived values
below take into account the errors of their calculations.
The propagation of this error was done by using the un-
certainties’ ufloat function in python, which was checked
throughout to show accurate error values.
Parameters Other sources Derived % Error
ω(rads/hr) - 1.960±0.014 -
A(magnitude) - 0.293±0.006 -
b(magnitude) ∗
16.8 16.538±0.004 1.6
φ(rads) - 3.21±0.011 -
∆t(min) 12 10.56±1.5 12
M1(M ) 0.535±0.012 - -
M2(M ) 0.111±0.004 - -
R1(R ) 0.0211±0.0002 - -
R2(R ) 0.149±0.002 0.184±0.023 23.5
P(hrs) 3.12 3.206±0.023 2.9
i(◦
) 0.149±0.002 -
q(M2/M1) 0.207±0.006 - -
a(R ) 0.934±0.009 0.947±0.016 1.4
Vt(km/s) - 359±6 -
TABLE I. List of all the parameter values acquired from other
sources, as well as the ones derived in this lab. The percent-
age error of the derived values with the accepted ones is also
shown. All the published values were taken from Parson et al.
(2010) and ∗
Haefner et al. (1989)
For the following calculations, three parameter values
were extracted from the literature, to compute a total of
six parameters. The first two of the parameters taken
from Parsons et al. (2010) are the mass ratio of the stars
and the mass of the primary star. The former of these two
has a value of about 0.207, as shown in the table above
as q. This ratio, along with the white dwarfs mass, also
given above, was later on used to calculate the separation
of the two stars, a, by using Keppler’s third law in the

10. 10
following form:
a3
=
GP2
orbitM1(1 + q)
4π2
(3)
This equation gave a separation between stars of
0.947 ± 0.016R , whose accepted value of 0.934R falls
within its errors.
The next step taken was to use the newly ac-
quired values of the period and star separation in an
equation to obtain the tangential velocity of the red
dwarf, Vt, modeled as if it travelled around the whit
dwarf in a perfect circle. As explained before, due
to tidal interactions, the orbit of this pre-cataclysmic
variable is circular and this model renders as an accurate
description of the system’s orbit.
Vt =
2πa
P
(4)
The tangential velocity of the red dwarf taken this way
was found to be 359 ± 6km/s.
Now, assuming that the white dwarf is small enough,
as compared to the red dwarf and the separation between
them, that it can be taken to be as a dot, the equation
to find the radius of the red dwarf becomes:
R2 =
Vt × ∆t
2
(5)
From which we see that the secondary star has a radius
a little less than a fifth of that of the sun. Notice how
this is the calculation with the biggest percentage error
with 23.5%. The big errors involved in this calculation
are thought to be a result of all the approximations that
were used to acquire this value.
By checking once again at the parameter values
in Table 1, we can see how close the calculated values
were to the already published ones, with each of them
having a percentage error of less than 3%, with the
exception of the red dwarf’s radius.
V. CONCLUSION
In this lab we were able to use FIT data files of the
NN Serpentis pre-cataclysmic variable, taken by Las
Cumbres Observatory Global Telescope Network, to
plot the V-band light curve of this binary system. After
calibration of the magnitudes, and curve fitting using
Python, the sinusoidal aspect of this binary’s light curve
was exposed. From here, the orbital period of the two
stars, P, was calculated to be 3.206±0.023 hrs, differing
only by 2.9% from the published value. Through this
process, the mean V-magnitude of the system, b, and
the amplitude of the magnitude change for each periodic
oscillation, A, were able to be computed. The light
curves plotted were also used to determine the length
of the periodic deep eclipse caused by the red dwarf
on the white dwarf primary, found to be 10.56 ± 1.5
min. Finally, these results, along with some parameter
values taken from the scientific literature, were used to
calculate the radius (R2) and tangential velocity (Vt) of
the white dwarf, as well as the separation (a) between
the two stars in the system to be 0.184 ± 0.023 R ,
359 ± 6 km/s, and 0.947 ± 0.016 R , respectively. All of
the calculated parameters were within the scope of the
accepted values, and, with the exception of R2, they all
had percentage errors of less than 3%.
Due to the general low percentage error of the
derived values in this observation, this experiment can
be termed successful. Nonetheless, the lack of data in
other bandpass filters (apart from V), and the short
amount of time of the recordings, exerted a constraint on
the potential of this observation’s findings. This should
be taken into account when taking future observations.
Also, from the amplitude change (A) of the system,
a more detailed analysis can be done to calculate the
temeprature distribution of the red dwarf.
VI. CURRENT RESEARCH
Recent studies surrounding the NN Serpentis binary
system concern about the existence of a planetary sys-
tem in its orbit. This theory was proposed due to some
small variations in the binary’s orbital period, which
could be explained by the gravitational interaction with
planets orbiting it. The latest studies suggest that there
are two gas planets orbiting the pre-cataclysmic variable,
each with an orbit of 15.5 and 7.7 years, and a mass of
6.9MJup and 2.2MJup, respectively, where MJup repre-
sents Jupiter masses (Beuerman et al., 2010). Studying
such a system offers the opportunity to understand bet-

11. 11
ter the evolution and formation of planets and stars in a
pre-cataclysmic variable.
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Why They Vary. London: Springer, 2001. Print.
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G. Tinney. ”A Detailed Investigation of the Proposed
NN Serpentis Planetary System.” Monthly Notices of
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V. S. Dhillon, S. P. Littlefair, B. T. Gnsicke, and R.
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White Dwarf and Low Mass M Dwarf in the
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(2010): 2591-608. Web.
- Warner, Brian. Cataclysmic Variable Stars.
Cambridge: Cambridge UP, 1995. Print.
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Littlefair, C. M. Copperwheat, and J. J. Hermes.
”Two Planets Orbiting the Recently Formed
Post-common Envelope Binary NN Serpentis.”
Astronomy and Astrophysics AA 521 (2010): n. pag.
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Liebert. ”PostT Tauri Stars in the Nearest OB
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