The Expanding Photosphere Method: Progress and Problems
arXiv:0704.0552v1  [astro-ph]  4 Apr 2007
The Expanding Photosphere Method: Progress
and Problems
József Vinkó and Katalin Takáts
Department of Optics & Quantum Electronics, University of Szeged, Hungary
Abstract. Distances to well-observed Type II-P SNe are determined from an updated version of
the Expanding Photosphere Method (EPM), based on recent theoretical models. The new EPM
distances show good agreement with other independent distances to the host galaxies without any
significant systematic bias, contrary to earlier results in the literature. The accuracy of the method
is comparable with that of the distance measurements for Type Ia SNe.
Keywords: supernovae; core-collapse; distances
PACS: 97.10.Vm, 97.60.Bw
INTRODUCTION
Distance is one of the most fundamental quantities in astrophysics, and it is especially
true for supernovae. Type Ia SNe are thought to be the most reliable distance indicators,
even up to z ∼ 1.5 redshift, and they play major role in determining the expansion of the
Universe as well as the cosmic equation of state. On the other hand, accurate distances
to SNe are crucial in understanding not only their physical properties, but also revealing
their progenitor objects and the possible explosion mechanisms.
The Expanding Photosphere Method (EPM) is a tool for measuring distances to SNe
that have large amount of ejected material [1]. The concept of EPM is based on a few
assumptions about the general physics of the expanding ejecta. These are the followings:
1. The expansion of the ejected material is spherically symmetric.
2. The ejecta is expanding homologously, i.e. R(t) = v(R) · (t − te), where R(t) is the
time-dependent radius of a particular layer in the ejecta, v(R) is the (constant)
expansion velocity of this layer and t − te is the time elapsed since the moment
of explosion (te).
3. The ejecta is optically thick, i.e. there exists a layer where the optical depth τλ ∼ 1.
This layer is the „photosphere” (Rphot ). Because of the expansion, the location of
the photosphere moves inward the ejecta, so its velocity (vphot ) is decreasing with
time.
4. The photosphere radiates as a blackbody, so the shape of the emergent flux spec-
trum is Planckian with a well-defined effective temperature Te f f . However, the ab-
solute flux value differs from that of the blackbody due to the dominance of scatter-
ing opacity over true absorption in the ejecta. The deviation from the blackbody can
be described with a simple scaling, i.e. Fλ = ζ 2πBλ (T ). where Fλ is the surface
flux, Bλ (T ) is the Planck function and ζ is the correction (or “dilution”) factor.
http://arxiv.org/abs/0704.0552v1
These assumptions are most likely to be valid in Type II-P SNe. These eject a massive,
hydrogen-rich envelope that remains optically thick for ∼ 100 days after explosion, and
the emergent spectrum is indeed close to be Planckian. Thus, EPM is expected to work
best for such SNe.
Based on the assumptions, the instantaneous radius of the photosphere can be ex-
pressed as Rphot = vphot(t) · (t − te) (the radius of the progenitor is usually neglected).
Meantime, the observed flux is fλ = θ 2 · ζ 2πBλ (T ), where θ = Rphot/D is the angular
radius of the photosphere from distance D. Combining these two equations, one gets the
basic equation of EPM [2, 3]:
t = te+D ·
vphot
. (1)
Since θ and vphot can be determined from observations, te and D are the only unknowns
in Eq.1. These can be derived via least-squares fitting to the observed quantities.
If the SNe under study are at high redshifts, the equations should be slightly modified
[4]. The definition of the angular radius is connected with the angular distance DA,
while in the expression of the observed flux the luminosity distance DL enters. At high z
DL = (1+ z)
2DA, so the angular radius of the photosphere can be inferred from
fλ (1+ z)
πBλ ′(T )
, (2)
where λ ′ = λ/(1+ z).
One particular advantage of EPM is that it does not require initial calibration, i.e. a
sample of objects with a priori known distances. However, the computation of the ζ
correction factors needs detailed model atmospheres, which makes the method essen-
tially model-dependent. Currently, there are two independent sets of model atmospheres
of Type II-P SNe in the literature, which were used to compute correction factors as a
function of Te f f [5, 6]. The former one was used in detailed studies of SN 1999em (the
most extensively studied SN II-P so far) that resulted in DEPM ≈ 8±1 Mpc [2, 3, 7] . This
is in significant disagreement with the Cepheid distance to the host galaxy NGC 1637
being DCep = 11.7±1 Mpc [8]. This problem has been solved in [9] by using a new set
of correction factors based on the NLTE radiative transfer code CMFGEN which gave
DEPM = 11.5±1.0 Mpc for SN 1999em.
NEW EPM DISTANCES TO SNE II-P
The method outlined above has been implemented in a new code that needs observed
BVRI light curves, radial velocities (determined from the absorption minima of certain
spectral features, see below) and reddening information (typically E(B−V )) as input.
As in any method based on photometry, the magnitudes must be dereddened, but fortu-
nately the results of EPM are quite insensitive to reddening errors, compared with other
methods [5].
 3000  4000  5000  6000  7000  8000  9000
Wavelength (Å)
SN 1999em (+8 d)
Tbb = 16862 K
 4  6  8  10  12  14  16  18  20
T (103 K)
FIGURE 1. Left panel: Result of fitting a blackbody (solid line) to broadband BVI fluxes (filled
symbols) of SN 1999em [3]. The R−band flux is also in good agreement with the fitted blackbody. The
flux-calibrated spectrum obtained simultaneously (dotted line) is shown for comparison. Right panel: The
correction factor as a function of Te f f from [6] (filled circles) and [5] (open circles).
At each epoch, the angular radius has been computed by a simultaneous fitting to the
dereddened B, V and I fluxes, as described in [2]. The corresponding effective temper-
ature has been derived by fitting a blackbody curve to the broadband fluxes converted
from the dereddened magnitudes. Our experience shows that the best results can be
achieved by considering all optical+NIR (i.e. BVRI) fluxes simultaneously. Earlier stud-
ies were sometimes limited to the usage of B and V bands only, which may result in
increased systematic errors due to the large deviation of the B-band fluxes from the
blackbody curve at later phases. The left panel of Fig.1 illustrates the optimum fitting of
a blackbody to either photometric, or precisely calibrated spectroscopic fluxes.
From the resulting Te f f , the correction factor ζ has been computed from the ζBV I(T )
function of Dessart & Hillier [6] for data obtained less than 40 days after explosion. For
data measured between 40 - 60 days after explosion, the function given by Eastman et al.
[5] was applied. As noted above, the usage of the function of Dessart & Hillier produces
better distances, but their models are valid only during the first month after explosion,
before the hydrogen starts to recombine. The ζBV I(T ) functions are plotted in the right
panel of Fig.1.
Beside the correction factors, the other important quantity is the photospheric velocity
vphot , because the resulting distance is very sensitive to the velocities that appear in the
denominator in Eq.1. Thus, the problem of finding an optimum method to infer vphot
from Type II-P SNe spectra has been addressed in several studies (see [2, 3, 6]).
We have studied this problem by computing model spectra with the parametrized
spectral synthesis code SYNOW [10]. SYNOW computes the emergent spectrum in a
homologously expanding atmosphere assuming LTE and pure scattering line formation.
The input parameters are the velocity and the blackbody temperature at the photosphere
(vphot and Te f f ), the exponent of the atmospheric structure, the list of ions contributing
to the spectral features, and the optical depth of one strong line for each ion.
Four sets of spectra have been defined corresponding to phases +10, +15, +50 and
+95 days after explosion, respectively. The list of ions contained H, He I, Na I, Fe II, Sc
 3000  4000  5000  6000  7000  8000
Wavelength (Å)
+10 d
+15 d
+50 d
+95 d
 0.96
 0.98
 1.02
 1.04
 1.06
 1.08
 2  4  6  8  10  12  14  16
vobs (10
3 km/s)
HeI 5876
FeII 4924
FeII 5018
FeII 5169
ScII 5526
FIGURE 2. Left panel: SYNOW model spectra of Type II-P SNe. The phase of each spectrum (ex-
pressed in days after explosion) is indicated. Right panel: The ratio of the true photospheric velocity (an
input parameter of a SYNOW model) to the „observed” velocity (derived from the absorption minimum
of P Cygni lines) as a function of the „observed” velocity. Different symbols mean different photospheric
lines indicated on the righ-hand side.
II, Ti II and Ba II, because these ions are thought to be responsible for the strongest lines
in the optical [3]. The input parameters except vphot were tuned to match real Type II-P
SNe spectra. Then, several model spectra were synthesized with different input vphot for
each phase. The left panel of Fig.2 shows representative spectra of all phases.
The synthesized spectra were used to compute “observed” radial velocities by mea-
suring the Doppler-shift of the absorption minima of selected lines. For P Cygni line
profiles, this should give exactly vphot if the line is isolated and optically thin. However,
in reality the features in a SN spectrum are all blends and may not be optically thin.
Therefore, vobs will differ from vphot .
In the right panel of Fig.2 the ratio of vphot/vobs is plotted as a function of vobs for
the features shown. It is seen that in almost all cases vphot is slightly underestimated.
The explanation of such a phenomenon is discussed in [6] for the Hα line. However, the
relative difference is below 5 %, thus, these lines are expected to represent vphot with 2 -
4 % accuracy. Motivated by these results, we have selected the He I λ5876 and the Fe II
λ5169 features to infer vphot from early-phase (< +20 days) and late-phase spectra of
real SNe, respectively.
In order to apply the method to real SNe, we have collected the available data of Type
II-P SNe from the literature (details and references will be published in a forthcoming
paper). Eq.1 was fitted to the observed data via least squares using either t or θ/vphot
as the independent variable. The two results for D and te were averaged to obtain their
final value. Whenever possible, the fit was restricted to data obtained between +5 – +40
days after explosion, and the angular radii were calculated using the Dessart & Hillier
correction factors (see above). In a few cases only late-phase (t ∼ 40− 60 days) data
were available. The Eastman et al. correction factors were applied for those SNe.
The EPM distances are plotted against the “reference” distances to their host galaxies
(mostly Tully-Fisher or SBF-distances for the nearby ones and Hubble-flow distances for
the more distant ones) in the left panel of Fig.3. As a comparison, the distances coming
 1000
 1  10  100  1000
Dref (Mpc)
 26  28  30  32  34  36  38  40
FIGURE 3. Left panel: the comparison of EPM (filled circles) and SCM (open triangles) distances with
the reference distances of the host galaxies. Right panel: residuals of the distance moduli of Type II-P SNe
from EPM (filled symbols) and Type Ia SNe (see text).
from the „Standard Candle Method” (SCM) [11] for nearly the same observational
sample are also shown. The scattering is very similar for both EPM and SCM. It is
concluded that these two methods provide distances to Type II-P SNe with ∼ 15−20 %
accuracy.
The accuracy of the new EPM distances is also similar to that of individual SNe Ia
distances. This is illustrated in the right panel of Fig.3, where the difference between
the distance moduli of Type II-P SNe (from this paper) and the low-redshift subsample
of Type Ia SNe (from [12]) are plotted against the reference distance moduli (adopting
Dre f = cz/H0 for Type Ia SNe). Again, the scattering of the data is similar for the two
samples. Thus, the concept of EPM combined with the present knowledge of Type II-P
SNe atmospheres may provide consistent and reliable distances, which may be extended
toward higher redshifts in the future. This could be a very important, independent test of
the Type Ia SNe distance scale.
This work was supported by Hungarian OTKA Grants No. T 042509 and TS 049872.
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