The Sigma-D Relation for Planetary Nebulae: Preliminary Analysis
Serb. Astron. J. } 174 (2007), 73 - 76
Preliminary report
THE Σ − D RELATION FOR PLANETARY NEBULAE:
PRELIMINARY ANALYSIS
D. Urošević1, B. Vukotić2, B. Arbutina1,2 and D. Ilić1
1Department of Astronomy, Faculty of Mathematics, University of Belgrade
Studentski trg 16, 11000 Belgrade, Serbia
2Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia
(Received: February 22, 2007; Accepted: March 30, 2007)
SUMMARY: An analysis of the relation between radio surface brightness and
diameter, so-called Σ−D relation, for planetary nebulae (PNe) is presented: i) the
theoretical Σ − D relation for the evolution of bremsstrahlung surface brightness
is derived; ii) contrary to the results obtained earlier for the Galactic supernova
remnant (SNR) samples, our results show that the updated sample of Galactic
PNe does not severely suffer from volume selection effect - Malmquist bias (same
as for the extragalactic SNR samples) and; iii) we conclude that the empirical
Σ −D relation for PNe derived in this paper is not useful for valid determination
of distances for all observed PNe with unknown distances.
Key words. planetary nebulae: general – Radio continuum: ISM – Methods: ana-
lytical – Methods: statistical
1. INTRODUCTION
The relation between radio surface bright-
nesses and diameters of supernova remnants (SNRs),
the so-called Σ−D relation, has been subject of the
extensive discussions in the last more than fourty
years. Due to improvements of the observational
techniques (radio-interferometers), the several hun-
dreds planetary nebulae (PNe) were resolved in the
last two decades at radio frequencies, but the Σ−D
relation for PNe was not discussed until now. By
using radio data, some statistical methods were es-
tablished in order to determine distances to PNe.
The main method was related to the correlation be-
tween radius of PNe and brightness temperature –
R − Tb relation (Van de Steene and Zijlstra 1995,
Zhang 1995, Phillips 2002). The different samples
of Galactic PNe with known distances were defined
in these papers. All the obtained empirical R − Tb
relations were used for determination of distances to
PNe for which the independent distances (in order of
R− Tb dependence) were not obtained earlier.
The samples of Galactic PNe are better for
statistical analysis than the samples of Galactic
SNRs. The selection effects should be smaller in
the case of PN samples. However, the selection ef-
fects surely influence the Galactic PN samples and
the statistical determination of distances to Galactic
PNe has to be highly uncertain.
The main objectives of this paper are the fol-
lowing:
i) to derive a simple form of the theoretical
Σ−D relation for PNe by analyzing the evolution of
radio bremsstrahlung surface brightness,
ii) to discuss whether the updated sample of
radio PNe is affected by the selection effects, and,
iii) to check whether the Σ−D relation is valid
for determination of distances to PNe.
http://arxiv.org/abs/0704.0421v3
D. UROŠEVIĆ et al.
2. ANALYSIS AND RESULTS
2.1. Theoretical Σ−D relation for PNe
The thermal bremsstrahlung mechanism is re-
sponsible for radiation of HII regions at radio wave-
lengths. The bremsstrahlung volume emissivity εν of
a PN can be shown to be (Rohlfs and Wilson 1996):
εν [ergs s
−1 cm−3 Hz−1] ∝ n2T−1/2, (1)
where n is the volume density and T is the thermo-
dynamic temperature of interstellar medium (ISM).
The surface brightness can be expressed as:
Σν ∝ ενD, (2)
where D is the diameter of PN. Combining Eqs. (1)
and (2), we obtain:
Σν ∝ n
D. (3)
Our next step is to express dependance of n and T
on D. For a constant velocity mass flow the den-
sity distribution is ̺ = Ṁ
4πr2v
, i.e. n ∝ D−x, where
x = 2. Moreover, for the isothermal envelope with
a power-law electron density distribution there is re-
lationship between the shape of the density distri-
bution and the power-law index of the radio contin-
uum spectra (see Gruenwald and Aleman 2007, and
references therein). Supposing that n ∝ D−2 and
T=const. (HII regions are approximately isothermal
at T ∼ 104 K), we obtain the simplest form of the
theoretical Σ−D relation for PNe:
Σν ∝ D
−3. (4)
This is a standard power-law form of the Σ − D
relation which can be written in general form as
Σ = AD−β , that is the same as in the case of SNRs.
It is possible that x in density distribution is
slightly higher, x & 2, and that the temperature is
not strictly constant throughout the nebula. We can
expect to see temperature gradients in PNe arising
from radiation hardening. More energetic photons
will travel further and when they are absorbed by the
PN they will impart greater kinetic energy to the ions
thereby producing a higher temperature. Using the
numerical model results given by Evans and Dopita
(1985), we calculate the dependence between logTe
and logD and find the low slope (≈ 0.1). Therefore,
this only slightly changes the slope of the theoretical
Σ−D relation. The value β = 3 is then a theoretical
lower limit, and the Σ − D relation could only be
steeper, as one can see from Eq. (3).
2.2. The empirical Σ−D relation for PNe
The most important prerequisite for deriving
a proper empirical Σ − D relation is defining of a
representative sample of PNe. The distances to the
calibrators have to be determined by accurate meth-
ods, e.g. trigonometric or spectroscopic parallaxes of
central stars in PNe, or by a method that uses the
expansion of nebulae. On the other hand, all sam-
ples suffer from the severe selection effects that arise
from limitation in sensitivity and resolution, but the
most severe selection effect for the Galactic samples
of PNe is Malmquist bias; i.e. intrinsically bright
PNe are favored because they are sampled from a
larger spatial volume compared to any given flux lim-
ited survey. The result is a bias against low surface
brightness nebulae such as highly evolved old PNe.
In this paper we use the updated sample of PNe at
the distances less than 0.7 kpc collected by Phillips
(2002). The influence of Malmquist bias in this sam-
ple is limited because of the limitation in distances to
PNe. In addition, we assume that the distances are
accurately determined for this sample of relatively
close PNe. The empirical Σ − D relation at 5 GHz
for 44 calibrators with distances less than 0.7 kpc
(Phillips 2002) has the form:
Σ56Hz = 2.33
+0.88
−0.64 · 10
−2.07±0.19
. (5)
The parameters A and β are calculated by least-
squares fitting procedure with correlation coefficient
−0.86. The corresponding Σν −D diagram is shown
in Fig. 1.
 0.01  0.1  1  10
D [pc]
Fig. 1. The Σ−D diagram at 5 GHz for 44 Galac-
tic PNe with distances less than 0.7 kpc.
The form of Eq. (5) is very close to the so-
called trivial Σ−D form with β = 2 (for details see
Arbutina et al. 2004). The additional test in order to
estimate the validity of Eq. (5) pertains to the possi-
ble dependence between the luminosity and diameter
of PNe. The Lν−D diagram is shown in Fig. 2. The
scatter in Lν − D plane shows that the correlation
between Lν and D is poor (correlation coefficient = -
0.06) and therefore the physical dependence between
L and D could not be confirmed by this statistical
procedure.
THE Σ−D RELATION FOR PLANETARY NEBULAE
 0.01  0.1  1  10
D [pc]
Fig. 2. The L −D plot at 5 GHz for 44 Galactic
PNe with distances less than 0.7 kpc.
3. DISCUSSION
The theoretical Σν −D relation (Eq. (4)) for
PNe, derived in this paper, describes a trend of de-
creasing radio surface brightness with increasing di-
ameter of an object. The radiation mechanism used
in this simple derivation is thermal bremsstrahlung.
This is the basic process of production of the radio
radiation in HII regions. The theoretically derived
slope (β = 3) is steeper than the slope from the em-
pirical relation given by Eq. (5). This discrepancy
can be explained by the low quality of the sample of
Galactic PNe or by the assumptions used in deriva-
tion of theoretical relation. Due to small variation
in power-law density distribution with x & 2 (Gru-
enwald and Aleman 2007, and references therein)
and approximately constant temperature of expand-
ing envelope of PNe, theoretical slope can be slightly
steeper than in Eq. (4). Therefore, we conclude
that the theoretical relation has the correct form,
but our empirical relation is under influence of bi-
ases that could make the slope shallower. On the
other hand, there are some attempts to show that
evolution of PNe are not linear in log-log scales (e.g.
Phillips 2004). These different dependences cannot
be derived from the thermal bremsstrahlung radia-
tion formula (Eq. (1)).
A very interesting feature regarding the em-
pirical relation for Galactic PNe (Eq. (5)) is that
the slope is approximately equal to the slope of triv-
ial Σ − D relation. Therefore, we conclude that
Malmquist bias is not so severe as in cases of Galac-
tic SNR samples. This slope (β ≈ 2) was obtained
for the extragalactic samples of SNRs (except M82
sample) where Malmquist bias is small, because all
the SNRs are at the approximately same distance
(see Urošević 2002, Urošević et al. 2005).
The large scatter in Lν − D plane (Fig. 2)
suggests that the slope in Eq. (5) does not have
real and valid physical interpretation. It is a kind
of luminosity-diameter scattering artefact which pro-
duces the trivial Σ ∝ D−2 form. Therefore, the rela-
tion defined by Eq. (5) is not precise enough for de-
termination of valid distances to Galactic PNe. This
is due to the different biases: the limitations in sen-
sitivity and resolution of radio surveys, the source
confusion, Malmquist bias (in mild form), mixture
of different types of PNe in the same sample, and
insufficient precision in determining the distances to
the 44 calibrators.
4. SUMMARY
The main results of this paper may be sum-
marized as follows:
i) The theoretical Σν −D relation for the radio
evolution of thermal bremsstrahlung surface
brightness of PNe in form of Σν ∝ D
−3 is de-
rived.
ii) Our results show that the updated sample of
Galactic PNe does not severely suffer from vol-
ume selection effect - Malmquist bias (same
as in cases of the extragalactic SNR samples).
This is opposite to results obtained earlier for
the Galactic SNR samples.
iii) Due to analysis of the Lν −D dependence, we
conclude that the Σν −D relation for Galactic
PNe is not useful for reliable determination of
distances for all observed PNe with unknown
distances.
The above observation leads to the more gen-
eral comment that PNe may have very different ini-
tial conditions leading to independent evolutionary
paths. These paths could follow the same theoreti-
cal Σ−D curve but with varying intercepts, leading
to the scatter such as the one found in this paper.
Acknowledgements – The authors would like to thank
the referee Prof. Nebojsa Duric for valuable com-
ments which have improved this paper. This research
has been supported by the Ministry of Science and
Environmental Protection of the Republic of Serbia
(Projects: No 146002, No 146003, No 146012, No
146016).
D. UROŠEVIĆ et al.
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Σ − D RELACIJA ZA PLANETARNE
MAGLINE: PRELIMINARNA ANALIZA
D. Urošević1, B. Vukotić2, B. Arbutina1,2 and D. Ilić1
1Department of Astronomy, Faculty of Mathematics, University of Belgrade
Studentski trg 16, 11000 Belgrade, Serbia
2Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia
UDK 524.37–77–54
Prethodno saopxteǌe
Prikazana je analiza tzv. Σ − D re-
lacije izme�u povrxinskog sjaja na radio-
frekvencijama i dijametra planetarnih
maglina (PM): i) izvedena je teorijska Σ −D
relacija za evoluciju povrxinskog sjaja
stvorenog zakoqnim zraqeǌem; ii) suprotno
rezultatima dobijenim ranije za uzorke
saqiǌene od Galaktiqkih ostataka super-
novih, naxi rezultati pokazuju da najnovije
formirani uzorak Galaktiqkih PM ne trpi
veliki uticaj zbog zapreminskog selekcionog
efekta, tzv. Malmkvistovog selekcionog
efekta (isto vaжi za vangalaktiqke uzorake
ostataka supernovih); i iii) zakǉuqujemo da
empirijska Σ − D relacija za PM izvedena
u ovom radu nije upotrebǉiva za pouzdana
odre�ivaǌa daǉina do svih posmatranih PM
sa nepoznatim daǉinama.
