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Title: Population Study of Hot DB White Dwarfs in the Sloan Digital Sky Survey
Article Type: Article
Corresponding Author: Mr. David J. Corliss, M.S.
Corresponding Author's Institution: University of Toledo
First Author: David J. Corliss, M.S.
Order of Authors: David J. Corliss, M.S.
Abstract: Data from the Sloan Digital Sky Survey (SDSS) / Release 4 have indicated the presence of a
small number of hot DB White Dwarfs with temperatures in the range of 30,000 - 45,000 K, a region
known as the DB Gap. Statistical analysis on these rare objects indicates variation in the population
distribution with temperature. While no DB white dwarfs are seen in the Upper DB Gap, from 40,000 45,000 K, they are found to be significantly more common in the lower third than in the middle of this
temperature range. It is proposed that the origin of this small population is due to DB white dwarfs
with an insufficient concentration of helium to support masking of the objects as DA stars throughout
the range of the DB Gap. Further statistical analysis on the population distribution with temperature is
performed, with autocorrelation analysis, an ARIMA model and the Kolmogorov-Smirnov Test all
indicating a measurable decrease in the population of DB white dwarfs extending as low as 25,000 K.
Manuscript
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Population Study of Hot DB White Dwarfs in the Sloan Digital Sky Survey
David J. Corliss
Department of Physics and Astronomy, University of Toledo, Toledo, OH 43606-3390, USA
[email protected]
ABSTRACT
Data from the Sloan Digital Sky Survey (SDSS) / Release 4 have indicated the
presence of a small number of hot DB White Dwarfs with temperatures in the range
of 30,000 - 45,000 K, a region known as the DB Gap. Statistical analysis on these rare
objects indicates variation in the population distribution with temperature. While no
DB white dwarfs are seen in the Upper DB Gap, from 40,000 - 45,000 K, they are found
to be significantly more common in the lower third than in the middle of this temperature range. It is proposed that the origin of this small population is due to DB white
dwarfs with an insufficient concentration of helium to support masking of the objects as
DA stars throughout the range of the DB Gap. Further statistical analysis on the population distribution with temperature is performed, with autocorrelation analysis, an
ARIMA model and the Kolmogorov-Smirnov Test all indicating a measurable decrease
in the population of DB white dwarfs extending as low as 25,000 K.
Subject headings: stars: White Dwarf — DB Gap — stars: Population Study — ARIMA
Model
1.
1.1.
Introduction
White Dwarfs and the DB Gap
In conventional theories, White Dwarf stars (WDs) are regarded to be electron-degenerate
masses of carbon and oxygen with a thin outer layer of hydrogen and helium. The gas comprising
the thin surface produces lines in the emergent spectrum. The spectra of DA White Dwarfs show
strong hydrogen lines while the DB stars show neutral helium lines. Both DA and DB WDs occur
with photometric temperatures up to c. 50,000 K. Above this temperature, white dwarfs are known
to exist to very high effective temperatures as DO stars.
DA WDs are seen at a continuum of temperatures below the DO WDs (Dreizler & Werner
1997). In contrast, DB WDs between 30,000 K and 45,000 K are extremely rare, even though
many have been observed at lower temperatures (Beauchamp et al. 1999). This anomalous nearabsence is known as the DB Gap. The physics underlying thisphenomenon is not well understood.
–2–
It is noted (Fontaine & Wesemael 1987; Shibahashi 2006) that the upper and lower ranges of this
critical temperature range correspond to convection zones. The high temperature limit of the DB
Gap at 45,000 K corresponds to the He+ /He++ convection zone. In conventional models for the
DB gap (Shibahashi 2006), the thin hydrogen and helium outer shell separates into layers in the
convectively-stable atmosphere, with a hydrogen surface layer effectively masking the helium-rich
nature of these objects. Upon cooling to the He/He+ convection zone at c. 30,000 K, conventional
models have the surface hydrogen layer dissipating as it is mixed with the underlying helium and
the star emerging from the Gap as a DB star once again; as many as 20% of DA white dwarfs are
believed to re-present helium-dominated atmospheres at the lower end of the DB Gap (Bergeron
& Liebert 2002).
The identification of white dwarfs with hybrid spectra, e.g., DAB white dwarfs (Burleigh et
al. 2001), has strengthened this view. These stars are of considerable interest, as they may be
examples of helium-dominated white dwarfs incompletely masked by a gravitationally-separated
hydrogen surface layer within the DB Gap (Bergeron & Liebert 2002) or stars in the process of representing as DB white dwarfs at the lower end of a DA phase in their evolution. Once a sufficient
number of these objects are identified, statistical techniques applied to the population distribution
of these stars may reveal more about their physical characteristics.
1.2.
SDSS Data
The Sloan Digital Sky Survey (SDSS), begun in 2000, is an optical survey comprising 25% of
the sky undertaken by a consortium led by the Alfred P. Sloan Foundation (York et al. 2000). The
primary SDSS resource is a 2.5m telescope on Apache Point, NM, equipped with a 120 megapixel
camera with a field of 1.5 square degrees. Data from the SDSS have been made publicly available
in a series of releases beginning in June, 2005; the most recent, Release 7, became available in June,
2009 (Abazajian et a. 2009).
The SDSS already has provided a great wealth of objects for consideration, in many cases
allowing for the first time the use of statistical methods of analysis. Indeed, until very recently,
no DB white dwarfs at all were known to exist in the DB Gap. The very large number of objects
available in SDSS data has yielded information for many rare objects including the first hot DB
white dwarfs in the DB Gap (Eisenstein et al. 2006a).
2.
2.1.
Population Analysis
SDSS Population in the DB Gap
Using data from the SDSS, Release 4, Eisenstein at al. (Eisenstein et al. 2006b) have identified
a small number of objects in this region. The temperature was measured by both ugriz photometry
–3–
and optical spectroscopy; sometimes temperature one is slightly lower for a given object and sometimes the other. Selected data pertaining to these objects, including the mean of the lower and
higher temperature values, are given in Table 1; Figure 1 graphs the population in bins of 5,000
K. The population appears to show a variation with temperature: of the 28 WDs in the present
sample, there are more in the lower end of the DB Gap and fewer as effective temperature increases.
In the range of 40,000 K to 45,000 K - a region which may be described as the Upper DB Gap there are no reported DB white dwarfs even among the 9,316 WDs of the SDSS Release 4 data.
It appears that the small population of the DB Gap may decrease toward the Upper DB Gap and
increase as the temperature approaches the lower end of the range at 30,000 K.
SDSS Designation
J084916.1+013721
J093759.5+091653
J090232.1+071929
J153852.3−012133
J093041.8+011508
J215514.4−075833
J141349.4+571716
J141258.1+045602
J222833.8+141036
J234709.3+001858
J143227.2+363215
J212403.1+114230
J095256.6+015407
J154201.4+502532
J123750.4+085526
J164703.4+245129
J084823.5+033216
J001529.7+010521
J090456.1+525030
J211149.5−053938
J040854.6−043354
J140159.1+022126
J092544.4+414803
J134524.9−023714
J074538.1+312205
J113609.5+484318
J081546.0+244603
J081115.0+270621
Table 1: Objects in Eisenstein et al. 2006b
Type
TLowerV alue (K) THigherV alue (K)
DB
28,500
29,500
DB
27,600
31,500
DB
29,500
30,500
DBA
28,000
32,000
DB
27,000
34,000
DB
29,000
32,000
DB
30,000
31,000
DB
30,000
31,500
DB
28,500
33,000
DB
28,700
33,000
DB
29,000
32,700
DB
30,000
32,000
DB
30,800
34,000
DB
32,000
33,000
DB
31,000
35,000
DB
32,000
34,000
DB
32,000
35,000
DB
35,000
36,000
DB
33,500
38,500
DBA
35,000
37,000
DB
35,000
40,000
DB
36,500
38,900
DB
38,000
39,000
DB (DBO?)
37,000
41,000
DO
37,800
41,800
DO+M
45,000
46,400
DO
43,500
48,500
DO
45,000
50,000
Mean (K)
29,000
29,550
30,000
30,000
30,350
30,500
30,500
30,750
30,750
30,850
30,850
31,000
32,400
32,500
33,000
33,000
33,500
35,500
36,000
36,000
37,500
37,700
38,500
39,000
39,800
45,700
46,000
47,500
–4–
Due to the very small size of this sample, there arises some question as to whether this variation
accurately reflects underlying physics or is simply noise. Statistically, a null hypothesis may be
presented that there is no real variation and the true population density in each range of the DB
Gap is the same. Were the sample a bit larger - more than 35 or so - a Gaussian distribution
would be used to test the null hypothesis. With only 23 observations, the sample is too small to
be analyzed using p-values from a Gaussian distribution: a t-test must be used. Given no WDs
in this sample in the Upper DB Gap (40,000 - 45,000 K) among 23 random trials, a two-tailed
t-test yields a p-value of 0.00134. Examining the Lower DB gap (30,000 - 35,000 K), 15 WDs are
observed in this sample among 23 random tris, yielding a p-value of 0.0397. In both cases, we
are led to strongly reject the null hypothesis. A stronger test may be performed: we do not test
to see if the population density is different than the overall average: we are specifically testing to
see if the density is higher from 30,000 - 35,000 K and lower from 40,000 - 45,000 K. Therefore, a
one-tailed t-test should be used. Testing the null hypothesis that the the population density from
30,000 - 35,000 K is not higher than the average density gives a p-value of 0.0145; testing the null
hypothesis that the the population density from 40,000 - 45,000 K is not lower than the average
density gives a p-value of 0.00074. Thus, we can accept the proposition that the Upper DB Gap has
a lower than average incidence of DB WDs and that the Lower DB gap has a higher than average
incidence with a very high degree of statistical confidence.
This variation within the sparse population of the DB Gap may provide some understanding
of the underlying physical process. The conventional explanation of the DB gap relies upon the
separation of a hydrogen layer at the outer surface of the star upon cooling to the onset of the
convection zone at 45,000 K. This layer then dissipates at the lower convection zone at 30,000 K.
This model is inconsistent with the observation of DB white dwarfs confirmed to exist in the DB
Gap. It may be expected that some DB WDs are too poor in hydrogen to support such a layer or
to sustain it throughout the range of 45,000 to 30,000 K. Should a very small number of WDs be
too hydrogen-poor to support a separated surface layer of pure hydrogen below a certain critical
temperature, the small but monotonically increasing population of the DB Gap with decreasing
temperature reflects the variation in hydrogen concentration in the most hydrogen-poor DB white
dwarfs.
2.2.
DB White Dwarfs in SDSS Release 4 at all Temperatures
Eisenstein et al. (2006b) began with more than 9,300 WDs in the SDSS Release 4 data. Further
criteria stipulated a spectral class of DB and an autofit temperature between 30,000 and 45,000
K. Addition of these two criteria allows for the identification of 23 objects within the DB Gap.
Applying the same spectral class criterion to the entire Eisenstein et al. 2006a catalog, one finds
530 DB white dwarfs at all temperatures. Of these, there are 424 DB white dwarfs below 20,000
K. This subset, comprising the bulk of the DB white dwarf population (∼ 80%) and well below the
DB Gap and its effects, can be used to establish statistical features of the DB population outside
–5–
of the DB Gap. A histogram of the population distribution below 20,000 K is given in Figure 2.
This subset has a mean autofit temperature of 16,700 K with a standard deviation of 1,700 K and
a median at 16,708 K, near the mean.
At temperatures above 20,000 K, a decrease in the population of DB white dwarfs is observed
as low as 25,000 K, below the 30,000 K limit usually associated with the DB Gap. There are 73 DB
white dwarfs listed in the Eisenstein et al. 2006a SDSS catalog with autofit temperatures between
20,000 and 25,000 K but only 17 between 25,000 and 30,000 K, equal to only 23% of the number
between 20,000 and 25,000 K.
An autocorrelation plot indicates the strength of trends either up or down with a value near
one when the data are smooth and dropping where it discontinuous or random. Figure 3 shows
such a plot, indicating a discontinuity in the population trend near a temperature of 25,000 K. The
increasing autocorrelation at 30,000 K indicates a new trend with a smaller population count has
already been established at a lower temperature.
2.3.
Comparison of DA and DB Population Distributions
As seen in Figure 3, the population counts show a strong autocorrelation except in DB White
Dwarfs at the edge of the DB Gap. The populations are also asymmetric in their distribution by
temperature. In these circumstances, ordinary regression is unsuitable for modeling the distribution
(Cerrito 2008). Regression methods implement the Central Limit Theorem and therefore contain
implicit assumptions of randomness and also symmetry in the population distribution, inconsistent
with the highly-skewed and autocorrelated distribution seen here. Under these circumstances,
ARIMA models may provide a better choice. This statistical technique examines trends in data by
using several successive terms to make a prediction of the subsequent value. An error term is then
calculated as the difference between the model prediction and the measured value for each succesive
data point. While ARIMA models are almost invariably used in time series analysis, the only
structural requirement is that of evenly-spaced intervals in the independent variable. The binning
of the population counts into equal-sized bins meets this structural requirement; such a model might
be termed a Non-Temporal ARIMA model. Here, the DA and DB white dwarf populations in the
Eisenstein catalog of SDSS data are binned by temperature into equal intervals of 2,500 K. Unlike
Regression techniques, ARIMA models are not rendered invalid by significant autocorrelation or
asymmetry in the distribution of the data. ARIMA models using a single autoregressive term
without a moving average component are designated to be of type (1,1,0); these models actually
leverage autocrrelations in data, performing well for analyzing small variations in fairly smooth
trends where strong autocorrelation is seen. This makes a (1,1,0) ARIMA model very well suited
to study the population distribution of hot DB white dwarfs. The ARIMA models in this study
were developed using SAS statistical software, the ARIMA procedure being modified for use with
non-temporal data.
–6–
Simple (1,1,0) ARIMA models use the autocorrelation of several successive terms to predict
a subsequent value without using an additive constant to reduce the magnitude of error terms.
Such a model is able to fit the population data for DA White Dwarfs but not the DB population
(Figure 4). The error terms in the model for the DA population distribution are small and random,
reflecting a good fit by the model. When the same process is applied to DB White Dwarfs, the error
terms are consistently negative. As shown in Figure 5, a constant must added to the model -only
for the DB populaion - to produce error terms in the model that are small and random, indicating
a good fit with the data.
The size of the constant term in the ARIMA Model shown in Figure 5 is 14.5 ± 2.0 stars per
2,500 K range, indicating a decrease in the DB population distribution in this amount. It may be
that some DB white dwarfs are being masked by the separation of a hydrogen rich surface layer
due to convection, with the few DBs that are observed in this very large sample from the SDSS
being too hydrogen poor to support such a surface layer. However, when the same methodology
is applied to DA, DAB and DBA white dwarfs in the Eisenstein et al. 2006a catalog, where such
masked DBs may be expected to be found, no corresponding increase is observed. The same is seen
when counting the much smaller population of just DAB and DBA white dwarfs. While there may
appear to be a remaining slight downward trend to the error terms with increasing temperature,
the magnitude of the slope of a χ2 best fit is only ∼ 0.25 stars per 2,500 K range, much less than
the variability in the error terms. Therefore, no second order correction may be asserted with
statistical confidence.
While the overall population of DA white dwarfs is larger than the DB population, normalization allows the two distributions to be compared directly. In Figure 6, the population in each
temperature range for DB white dwarfs is given as a percent of the entire population between
15,000 K and 52,500 K; this is shown against the percent of total population for each interval in
the DA population.
In this normalized population distribution, the DB Gap is more clearly visible. While visual
inspection of this graph may lead one to suspect an increase in the DA population corresponding to
the decrease in the DB population, the magnitude of the variation in the DA population distribution
is larger than the decrease in the DB population. No inference with any statistical confidence is
possible on this question with the present data.
Normalization of the DA and DB population distributions enables the use of the KolmogorovSmirnov (K-S) statistic to test whether a real difference exists between the two distributions (Table
2). This K-S test was developed and implemented in SAS. The probability of the DA and DB
distributions not matching is 0.645, about one standard deviation for a Gaussian distribution. This
K-S test therefore indicates that the distributions are, on the whole, neither very different nor
very similar. However, over the interval from 22,500 - 25,000 K, the distributions are very similar
and contain 7.1% of the DA population and 6.4% of the DB population. In the next interval,
from 25,000 - 27,500 K the two distributions strongly divereg. Themaximum deviation between
–7–
the two population distributions reaches a maximum at this point. While the fraction of total DA
White Dwarfs remains much the same, declining from 7.1% to 6.8% of the total population, the
DB population distribution drops more than two-thirds, from 6.4% to 2.1%.
3.
Conclusions
Within the DB Gap, the population of DB white dwarfs is found to be small and to decrease
with increasing temperature. Analysis using as t test on variations in the population of DB white
dwarfs supports this relationship between population and temperature with a very high degree of
statistical confidence.
Autocorrelation analysis, an independent ARIMA model and the Kolmogorov-Smirnov test
concur in identifying a drop in the DB population distribution with increasing temperature near
25,000 K. While the size of this data sample is insufficient to measure a corresponding increase in
the DA population, the temperature range of this population anomaly is consistent with DB White
Dwarfs having an insufficient concentration of hydrogen to support masking as DA stars through
some part of the DB Gap.
While the population distribution shows a clear discontinuity in the vicinity of 25,000 K, there
is less evidence for such a difference at the lower convection zone near 30,000 K. The population
observed between 25,000 - 30,000 K is not much more than the 15 seen in the range from 30,000 35,000 K.
Using these statistical methods, a measurable decrease in the population of DB white dwarfs is
observed as low as 25,000 K, below the 30,000 K limit usually associated with the DB Gap. While
this might be taken as evidence that some DB white dwarfs are being masked by the separation
of a hydrogen rich surface layer due to convection, any corresponding increase is too small to be
discerned in the DA / DAB / DBA population in this study. This could be due to a large overall
population of DAs absorbing the relatively small numbers of DB white dwarfs. The lack of any real
change in the small DAB / DBA population could then be accounted for by DAB / DBA white
dwarfs masking as DAs in about the same numbers as DB white dwarfs masking as type DAB /
DBA stars.
4.
Further Investigation
While the generally accepted end-points of the DB Gap correspond to convection zones at
30,000 and 45,000 K (Shibahashi 2006), this analysis of population trends finds a discontinuous
change at 25,000 K and none at 30,000 K. Thus, the effects causing the DB Gap remain poorly
understood and may not be accounted for by convection alone.
DB white dwarfs with effective temperatures above 25,000 K need to be carefully examined
–8–
and modeled to identify observable qualities reflecting the true underlying nature of these stars
without the effects of masking that may be affecting some stars presenting as DA white dwarfs
in the DB Gap. At the upper end of the DB gap, the hottest DB white dwarf in the present
sample (J134524.9-023714) has an effective temperature of ∼39,000 K. The possibility of extremely
hydrogen-poor white dwarfs re-presenting as spectral type DB at even higher temperatures may be
investigated. Consequently, any DB white dwarfs that may be found in the Upper DB Gap from
40,000 - 45,000 K should be very closely examined.
It has not been possible to quantify any secord-order correction in the ARIMA model for
the reduction in the number of DB stars with any statistical confidence using the present sample.
As more data becomes available in an analytic format, further details of the DB Gap population
distribution with temperature may be investigated.
Acknowledgements This work was submitted at Wayne State University as a Master’s Essay
under Dr. David Cinabro, to whom thanks are due for many very helpful discussions and suggestions. The author also wishes to thank Nancy Morrison and Rene Bellwied for their helpful
questions and comments.
5.
References
Abazajian, Kevork, N., et al., 2009, ApJS, 182, 543A
Beauchamp, A., Wesemael, F., Bergeron, P., Liebert, J., & Saer, R. A. 1996, in ASP. Conf. Ser.
96, Hydrogen-Deficient Stars, ed. S. Jeery & U. Heber (San Francisco: ASP), 295
Bergeron, P., & Liebert, J., 2002, ApJ, 566, 1091
Burleigh, M., Bannister, N., Barstow, M., Maxted, P. 2001, in ASP. Conf. Ser. 226, 12th European Workshop on White Dwarf Stars, ed. J. L. Provencal, H. L. Shipman, J. MacDonald, & S.
Goodchild (San Francisco: ASP), 135
Cerrito, P 2008, ”The Difference Between Predictive Modeling and Regression”, Proc. MWSUG,
ed. C. Lee & D. J. Penix
Dreizler, S., & Werner, K. 1997, in White Dwarfs, Proc. 10th European Workshop on White
Dwarfs, ed. J. Isern, M. Hernanz, & E. Garcia-Berro (Dordrecht: Kluwer), 213
Eisenstein, D.J., et al., 2006, ApJS, 167, 40 (Eisenstein et al. 2006a)
Eisenstein, D.J., et al., 2006, ApJ, 132, 676 (Eisenstein et al. 2006b)
Fontaine, G., & Wesemael, Fr., 1987, Second Conference on Faint Blue Stars, eds. A.G.D. Philip,
D.S. Hayes, & J. Liebert, Schenectady NY: L., Davis Press, p.319
Shibahashi, H., 2006, Proceedings of the International Astronomical Union, 2: 274–279
York, D.G., 2000, AJ, 120, 1579
–9–
6.
Figures
– 10 –
Fig. 1.— Population distribution of hot DB white dwarfs in the DB gap. The data for individual
stars are compiled in Eisenstein et al. 2006b and tabulated here in Table 1. The population
distribution with temperature shows a statistically significant variation, with a lower population
density at the upper end of the DB Gap - 40,000 to 45,000 K - and inceasing as temperature
decreases.
– 11 –
Fig. 2.— Population distribution of DB white dwarfs at temperatures well below the DB gap.
Data for the 424 individual stars in this sample are compiled in the Eisenstein et al. 2006a catalog;
the maximum temperature in this subset is 20,000 K. The data to the right of the peak in this
truncated dataset may be used to establish the trend in the population density with increasing
temperature in the absence of any effects associated with the DB gap.
– 12 –
Fig. 3.— Autocorrelation plot of DB white dwarf Population. The autocorrelation is calculated on
the basis of three successive intervals. Discontinuous change in the population density is indicated
near 25,000 K, lower than the lower limit of the DB gap in conventional model at 30,000 K.
– 13 –
Fig. 4.— ARIMA Model error terms for DA and DB populations. The error terms for the DA
population model (diamonds) are generally small and randomly distributed about zero, indicating
no need for further correction. By contrast, the error terms for the DB population model (crosses)
are consistently negative, illustrating the DB Gap. When a linear correction term is added to the
DB model (asterisks), the small and random error terms reflect a good fit by the corrected model.
– 14 –
Fig. 5.— Normalized population distribution of DA and DB white dwarfs. Populations are expressed as a percentage of the total number of stars (by type) above 15,000 K. The relative decrease
in DB versus DA white dwarfs is seen to extend below the 30,000 K convection zone usually associated with this phenomenon.