Dust in the inner regions of debris disks around a stars

Abstract

We present infrared interferometric observations of the inner regions of two A-star debris disks, β Leo and ζ Lep, using the FLUOR instrument at the CHARA interferometer on both short (30 m) and long (> 200 m) baselines. For the target stars, the short-baseline visibilities are lower than expected for the stellar photosphere alone, while those of a check star, δ Leo, are not. We interpret this visibility offset of a few percent as a near-infrared (NIR) excess arising from dust grains which, due to the instrumental field of view, must be located within several AU of the central star. For β Leo, the NIR excess-producing grains are spatially distinct from the dust which produces the previously known mid-infrared (MIR) excess. For ζ Lep, the NIR excess may be spatially associated with the MIR excess-producing material. We present simple geometric models which are consistent with the NIR and MIR excesses and show that for both objects, the NIR-producing material is most consistent with a thin ring of dust near the sublimation radius, with typical grain sizes smaller than the nominal radiation pressure blowout radius. Finally, we discuss possible origins of the NIR-emitting dust in the context of debris disk evolution models.

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arXiv:0810.3701v1 [astro-ph] 20 Oct 2008
Dust in the inner regions of debris disks around A stars
R.L. Akeson1, D.R. Ciardi1, R. Millan-Gabet1, A. Merand2,3, E. Di Folco4, J.D. Monnier5,
C.A. Beichman1, O. Absil6, J. Aufdenberg7, H. McAlister2, T. ten Brummelaar2, J.
Sturmann2, L. Sturmann2, N. Turner2
ABSTRACT
We present infrared interferometric observations of the inner regions of two A-star debris
disks, β Leo and ζ Lep, using the FLUOR instrument at the CHARA interferometer on both
short (30 m) and long (>200 m) baselines. For the target stars, the short baseline visibilities
are lower than expected for the stellar photosphere alone, while those of a check star, δ Leo,
are not. We interpret this visibility offset of a few percent as a near-infrared excess arising from
dust grains which, due to the instrumental field of view, must be located within several AU of
the central star. For β Leo, the near-infrared excess producing grains are spatially distinct from
the dust which produces the previously known mid-infrared excess. For ζ Lep, the near-infrared
excess may be spatially associated with the mid-infrared excess producing material. We present
simple geometric models which are consistent with the near and mid-infrared excess and show
that for both objects, the near-infrared producing material is most consistent with a thin ring
of dust near the sublimation radius with typical grain sizes smaller than the nominal radiation
pressure blowout radius. Finally, we discuss possible origins of the near-infrared emitting dust in
the context of debris disk evolution models.
Subject headings: circumstellar matter — stars: individual(beta Leo, zeta Lep)
1.Introduction
The list of main sequence stars known to have
circumstellar material in the form of debris disks
has been greatly expanded over the last few years
by surveys at longer wavelengths and most re-
cently from Spitzer observations (see e.g., the re-
view by Meyer et al. 2007). Given the size and
1Michelson Science Center, Caltech, Pasadena, CA
91125
2Center for High Angular Resolution Astronomy, Geor-
gia State University, Atlanta, GA 30302
3current address:European Southern Observatory,
Alonso de Cordova 3107, Casilla 19001, Vitacura, Santi-
ago 19, Chile
4Observatoire de Geneve, Universite de Geneve, Chemin
des Maillettes 51, 1290 Sauverny, Switzerland
5Department of Astronomy, University of Michigan,
Ann Arbor, MI48109
6LAOG, CNRS and Universite Joseph Fourier, BP 53,
F-38041, Grenoble, France
7Physical Sciences Department, Embry-Riddle Aero-
nautical University, Daytona Beach, FL 32114
distribution of dust in these disks, the grains are
expected to have short lifetimes. Therefore, it is
generally believed that the dust is not remnant
from the star formation process, but is generated
through collisions of larger bodies. The majority
of known debris disks have cold (<100 K) mate-
rial located tens of AU from the central star in
an analog of our own Kuiper belt. In some cases,
this material extends to 1000 AU. A small frac-
tion (Rieke et al. 2005; Beichman et al. 2006) have
warmer dust located within 10 AU of the central
star.
The distribution of material in a debris disk is a
balance of collisions, radiation pressure, Poynting-
Robertson (PR) drag and the dynamical influence
of any large bodies in the system. In order to con-
strain models of these systems, the dust spatial
extent and grain size distribution must be mea-
sured. Observations of optical and near-infrared
scattered light have provided the most detailed
overall picture of the dust distribution. However,
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these scattered light observationsdo not have suffi-
cient resolution to characterizethe material closest
to the star, and this is where infrared interferom-
etry can provide a unique constraint.
Although many of these sources do not show a
clear near-infrared excess in their spectral energy
distribution (SED), limits set by spatially unre-
solved broadband photometry are generally not
better than a few to several percent.
warm dust component could be present if dust
generated by collisions migrated close to the star
or was produced by bodies in close orbits. If lo-
cated within a few AU of the central star, this dust
would be at temperatures which would produce
near-infrared emission and small grains would pro-
duce scattered light. Detection of (or stringent
limits on) warm dust will characterize the inner
portions of these debris disks. The spatial resolu-
tion of infrared interferometry can be exploited to
probe for warm dust in these systems. On long
baselines (> 100 meters) the central star is re-
solved and the visibility is primarily a measure
of the stellar photospheric size. On shorter base-
lines (< 50 meters) the photosphere is mostly un-
resolved and if the measured visibilities have high
accuracy, other emission components can be de-
tected by looking for deviations from the visibility
expected for the stellar photosphere. Any resolved
or incoherent emission will decrease the measured
visibility from the stellar value.
Teams using the Palomar Testbed Interfer-
ometer (Ciardi et al. 2001) and the Center for
High Angular Resolution Array (Absil et al. 2006;
di Folco et al. 2007; Absil et al. 2008) have de-
tected near-infrared extended emission around
known debris disk systems, including Vega, the
prototype debris disk. While a near-infrared ex-
cess was not known through broadband spectral
modeling, the interferometrically detected near-
infrared excess was consistent with the photomet-
ric uncertainties.Observations of other debris
disk sources revealed a small near-infrared excess
flux around τ Ceti and ζ Aql (di Folco et al. 2007;
Absil et al. 2008). In all these systems, the near-
infrared excess is consistent with emission from an
inner, hot dust component, although for ζ Aql, a
binary companion is also a likely origin.
In this paper we present infrared interferometry
observations of two known debris disk systems, the
A-type stars, β Leo and ζ Lep. The interferom-
A small,
etry observations, including determination of the
stellar angular diameter, and mid-infrared imag-
ing for β Leo are presented in §2. Possible origins
for the observed visibility deficit are discussed in
§3. In §4, we discuss the distribution of the ex-
cess producing grains and in §5 the origin of these
grains. Our conclusions are given in §6.
2.Observations and data analysis
2.1.Targets
The targets were chosen from the sample of
known debris disk systems with the V and K
brightness as the primary selection criteria. Table
1 lists the target and check star stellar properties.
β Leo was identified as having an infrared ex-
cess from IRAS observations (Aumann & Probst
1991).Mid-infrared imaging has not resolved
the disk (Jayawardhana et al. 2001, §2.4) although
differences between the IRAS and ISO fluxes led
Laureijs et al. (2002) to suggest that the disk
emission may be somewhat extended in the ISO
beam (52′′aperture). Chen et al. (2006) obtained
Spitzer IRS spectra of β Leo and found a fea-
tureless continuum spectra consistent with dust
at ∼120 K located 19 AU from the central star.
ζ Lep was also identified as a debris disk by
Aumann & Probst (1991) and has an unusually
high dust temperature (>300 K) (Aumann & Probst
1991; Chen & Jura 2001).
Moerchen et al. (2007) resolved the excess at
18 µm and their model comprises two dust bands
extending from 2 to 8 AU. As with β Leo, the
Spitzer IRS spectrum for ζ Lep is featureless
(Chen et al. 2006).
Recent work by
2.2. CHARA Observations
Observations were conducted with the FLUOR
fiber-optics beam combiner at the Center for
High Angular Resolution Array (CHARA) op-
erated by Georgia State University. CHARA is a
long-baseline, six-element interferometer with di-
rect detection instruments that work at optical to
near-infrared wavelengths (ten Brummelaar et al.
2005).These FLUOR observations were taken
in the K’ band and have an effective central
wavelength of 2.14 microns. The FLUOR beam
combiner produces high precision visibilities by
interfering the inputs from two telescopes af-
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Parameter
HD number
Spectral type1
Distance2(pc)
Radius3(R⊙)
Teff
Luminosity5(L⊙)
v sin i (km/sec)
β Leo
102647
A3Va
δ Leo
97603
A5IVn
ζ Lep
38678
A2Vann
21.5 ±0.32
1.60 ± 0.11
11.1 ±0.11
1.58 ± 0.018
17.7± 0.26
2.17 ± 0.073
90204
82961
99101
11.5 ± 1.115.5 ± 1.817.0 ± 2.3
1104
1736
2454
Table 1: Stellar properties of the sources
1NASA Stars and Exoplanet database: http://nsted.ipac.caltech.edu
2Distances taken from Hipparcos (Perryman & ESA 1997); we note a more recent reduction of Hipparocs data (van Leeuwen
2008) has yielded new distances which are within 1 σ of those listed here. We use the older values for consistency with previous
work.
3this work
4Chen et al. (2006)
5calculated from the radius and effective temperature
6Rieke et al. (2005)
ter spatial filtering through single-mode fibers
(Coude du Foresto et al. 2003).
wavefront aberrations are converted to photomet-
ric fluctuations which are corrected by simultane-
ous measurement of the fringe and photometric
signals from each telescope.
β Leo and a check star δ Leo were observed on 3
nights in 2006 May and ζ Lep on 2 nights in 2006
October and November. A check star for ζ Lep
was observed but due to its lower K band flux,
these data were not useful and are not included
here. Observations of the targets and check star
were interleaved with calibration observations to
determine the instrument response function, also
called the system visibility. The check star is an
additional target with roughly the same properties
as the main target, but no known excess emission
at any wavelength. Observations of the check star
are processed in the same way and with the same
calibrators as the main target and serve as a mea-
sure of systematic effects in the data. The calibra-
tors used, along with their adopted diameters are
given in Table 2.
The FLUOR data consist of temporally modu-
lated fringes over an optical path difference (OPD)
of 170 microns, centered around the zero OPD.
The coherence length (fringe packet size) in the
K’ band is of order 11 fringes, or approximately 25
microns. In addition to the fringe signal, FLUOR
In this design,
records simultaneous photometric channels, in or-
der to allow the correction of scintillation noise
and coupling variations in the input single mode
fibers. The photometric correction and flux nor-
malization were done using the numerical meth-
ods described in Coude Du Foresto et al. (1997).
Once the fringe signal was recovered, we estimated
the squared visibilities of individual frames as the
integrated power in the frequency domain.
To estimate the fringe power, we used a
time/frequency transform,
transform, instead of the classical Fourier ap-
proach (Coude Du Foresto et al. 1997). The clas-
sical Fourier method extrapolates the power under
the fringe peak using data collected at frequencies
outside the fringe peak (M´ erand et al. 2006). This
approach works well if the readout noise is white.
The wavelets approach isolates the fringe signal
in the OPD and in the frequency domains (as de-
scribed in Kervella et al. 2004a), allowing a mea-
sure of the off-fringe power at all frequencies and
therefore a direct measurement of the background
noise for each scan. The isolation of the fringe
signal in OPD is possible because the modulation
length used (170 microns) is much larger than the
coherence length (approximately 25 microns) and
the background noise is measured using the por-
tion of the scan situated more than 50 microns on
each side of the fringe packet (i.e. four times the
coherence length).
a Morlet wavelets
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Calibrator
70 Leo
ζ Vir
IRC 10069
η Lep
HR 1965
HR 1232
Diameter (mas)
0.770 ± 0.015
0.760 ± 0.015
1.342 ± 0.07
0.940 ± 0.020
1.272 ± 0.017
0.920 ± 0.020
Target
β Leo, δ Leo
β Leo, δ Leo
ζ Lep
ζ Lep
ζ Lep
ζ Lep
Diameter reference
SB relation, Kervella et al. (2004b)
SB relation, Kervella et al. (2004b)
M´ erand et al. (2005)
SB relation, Kervella et al. (2004b)
M´ erand et al. (2005)
SB relation, Kervella et al. (2004b)
Table 2: The calibrators used for the CHARA observations. The calibrator sizes are derived using optical
and infrared photometry and the surface brightness (SB) relation from Kervella et al. (2004b) or taken from
M´ erand et al. (2005).
The background noise arises from 3 compo-
nents: the photometric variation residuals (after
photometric correction), the photon shot noise
and the detector readout noise. The first com-
ponent is only present at very low frequencies,
since fringes are acquired at a frequency (100Hz)
higher than the scintillation and coupling varia-
tions (typically 25Hz at CHARA) and because the
photometric correction is very efficient. The sec-
ond component (photon shot noise) is white noise.
The third component, readout noise, is less pre-
dictable and can have transients or peaks at dis-
crete frequencies (electronic noise). As the wavelet
approach directly measures the background com-
ponent from the data, there are fewer residuals
than in the Fourier method where the noise esti-
mate is approximate. For the FLUOR data, the
wavelet method improved the consistency of the
results, although the basic results are the same
between the two methods.
Finally, the final squared visibility estimate and
the one sigma uncertainty for a given batch of
frames are obtained by the average and stan-
dard deviation of the bootstrapped average, as de-
scribed in Kervella et al. (2004a). The calibrated
target data obtained using this reduction method
are given in Table 3.
2.3.Stellar size and visibility deficit
If the measured visibilities were due entirely to
a resolved stellar disk, both the short and long
baseline data would be well-fit with a single uni-
form disk. However, as shown in Figure 1, the visi-
bility measured on the short baseline for β Leo and
ζ Lep is lower than expected from the stellar size
fit on the long baseline. Fitting a single stellar size
to both baselines yields a very poor fit as measured
by χ2
for the target stars, while the single-component fit
to both baselines for the check star, δ Leo, is good
(Table 4). Any additional flux component within
the field of view will decrease the measured vis-
ibility and will therefore make the model more
consistent with the short-baseline data. A par-
tially resolved emission component will increase
the discrepancy between the long and short base-
line visibilities as it would be more resolved, and
therefore have lower visibility, on the long base-
lines. An over resolved, i.e. incoherent, source of
emission will produce the same fractional decrease
in visibility for all baselines. For the simple case of
a star and an incoherent component, the measured
visibility, Vmeas, is
rin comparison to the single-baseline only fits
V2
meas=
?Vstar∗ fstar
fstar+ fincoh
?2
(1)
where Vstar is the visibility of the stellar photo-
sphere and fstarand fincohare the fractional stel-
lar and incoherent component fluxes. The visibil-
ity used here is a normalized quantity such that
an unresolved source has V = 1 while an incoher-
ent (i.e. completely resolved) source has V = 0.
We fit a single uniform diameter plus an incoher-
ent emission contribution to both baselines, which
gives a lower χ2
rfor β Leo and ζ Lep than the
uniform disk by itself. The visibility deficit for
ζ Lep is a tentative detection as the V2predicted
from the stellar size, 0.996±0.001, is only 2.3 σ
from the averagemeasured visibility, 0.966 ± 0.013
and the stellar size uncertainty is much larger as
the star is smaller and fainter than the other tar-
gets. We note that the stellar size for ζ Lep from
data reduced using the classical Fourier approach
is the same as for the wavelets approach, despite
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Object
β Leo
MJDBaseline(m)
32.531
33.234
33.801
313.083
312.858
293.126
33.823
34.069
33.940
286.055
295.815
218.336
223.077
232.879
24.739
26.062
27.252
Pos Angle (deg)V2
σ
53856.226
53856.270
53856.309
53864.185
53865.185
53865.236
53856.248
53856.290
53856.328
53864.233
53865.215
54040.479
54040.487
54040.506
54045.475
54045.498
54045.518
-12.980
-21.034
-26.735
74.485
74.288
68.316
-20.956
-27.244
-31.660
62.950
65.870
-57.363
-57.415
-57.237
-22.258
-26.679
-29.651
0.9487
0.9001
0.9285
0.0679
0.0503
0.0897
0.9726
1.0025
1.0356
0.2206
0.1790
0.7543
0.9403
0.5783
0.9742
0.9524
0.9696
0.0219
0.0269
0.0204
0.0079
0.0040
0.0032
0.0393
0.0173
0.0482
0.0253
0.0088
0.0713
0.1002
0.0772
0.0209
0.0235
0.0272
δ Leo
ζ Lep
Table 3: The calibrated visibility observations from CHARA.
the scatter in the long baseline data. Further ob-
servations are needed for confirmation of the vis-
ibility deficit of ζ Lep.
component corresponds to an excess flux of 2.7 ±
1.4 Jy for β Leo and 0.47 ± 0.41 Jy for ζ Lep. For
δ Leo the uniform disk fit is adequate, suggest-
ing no visibility deficit on the check star and no
substantial systematics in the observing or data
reduction process.
From the uniform disk fit, we can calcu-
late a limb-darkened angular and physical di-
ameter for these stars using the formula from
Hanbury Brown et al. (1974)
The best-fit incoherent
θLD
θUD
=
?
1 − µλ/3
1 − 7µλ/15
?1/2
,(2)
where the coefficient µ depends on the effec-
tive temperature and is taken from Claret et al.
(1995). The difference between the limb darkened
and uniform disk diameters is less than 2% for
our stars, with ratio values ranging from 1.011
to 1.014.The uniform disk and limb-darkened
diameters and the derived stellar radii are given
in Table 5, where the uniform diameter is taken
from the stellar + incoherent component model.
These limb-darkened diameters agree with values
calculated from the surface-brightness relation of
Kervella et al. (2004a) of 1.35, 1.17 and 0.73 mas
for β Leo, δ Leo, and ζ Lep respectively.
We note that these limb-darkening parameters
are appropriate for slowly rotating stars, which is
violated by the values for v sini given in Table
1. Aufdenberg et al. (2006) find limb-darkening
corrections 2.5 times higher for Vega, an A0 star
rotating at 275 km/sec. As our data are insuf-
ficient to separately derive the limb-darkening or
rotational velocity, we use the low rotation rate co-
efficients to allow for comparison to other works,
but note that even at 2.5 times higher, the limb
darkening corrections would be 4%, still much too
small to explain the difference between the short
and long baseline sizes given in Table 4.
The diameter of β Leo has been previously mea-
sured with interferometry observations. Hanbury Brown et al.
(1974) obtained a limb-darkened diameter of
1.33 ± 0.1 mas at a wavelength of 4430 ˚ A
with the Narrabri intensity interferometer, while
di Folco et al. (2004) measured 1.449 ± 0.027 mas
at 2.17 µm with the VLTI, which is inconsistent
with our diameter at the 3.7σ level. However, the
di Folco et al. (2004) fit did not include an inco-
herent component. If we include the di Folco et al.
(2004) data in our two component fit, both the
stellar diameter and incoherent flux level change
by less than 0.2 σ, thus the VLTI and CHARA
data are consistent.
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ObjectUniform disk
all data
Diam.(mas)
1.332 ± 0.014
1.148±0.025
0.70±0.15
Uniform disk
long baselines
Diam.(mas)
1.332 ± 0.009
1.149 ±0.012
0.69 ±0.09
Uniform disk
short baselines
Diam.(mas)
2.289 ± 0.31
0.0 ± 1.17
2.0 ± 0.65
Uniform disk + incoherent flux
all data
Diam.(mas)
1.323 ± 0.013
1.149±0.022
0.66 ±0.14
χ2
3.8
0.8
2.5
r
χ2
2.1
0.5
3.7
r
χ2
0.9
0.5
0.2
r
Inc. flux
0.024±0.013
0.0±0.006
0.015±0.013
χ2
1.6
1.0
2.0
r
β Leo
δ Leo
ζ Lep
Table 4: Uniform diameter and incoherent flux fit to data
Fig.
ζ Lep(bottom, left). For each object, the visibility curves for a uniform disk fit only to the long base-
line data (solid line with dotted line errors) and for a uniform disk plus an incoherent flux component fit to
all the data (dashed line with dot-dash line errors) are shown.
1.— The measured visibilities and errors (points) for β Leo (top, left), δ Leo (top, right) and
ObjectUniform disk diam.
(mas)
1.323 ± 0.013
1.149 ± 0.022
0.66 ± 0.14
LD coeffLimb darkened diam.
(mas)
1.339 ± 0.013
1.165 ± 0.022
0.67 ± 0.14
Stellar radius
(R⊙)
1.54 ± 0.021
2.14 ± 0.040
1.50 ± 0.31
β Leo
δ Leo
ζ Lep
1.012
1.014
1.011
Table 5: Measured uniform disk and limb-darkened diameters
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2.4.Mid-infrared imaging
Mid-infrared imaging observations of β Leo
were made on 2006 March 8 (UT) using the
Mid-Infrared Echelle Spectrometer (MICHELLE;
Glasse & Atad-Ettedgui 1993) on the Gemini
North 8-meter telescope. MICHELLE utilizes a
320 × 240 pixel Si:As blocked impurity band de-
tector, with a spatial scale of 0.′′1 pixel−1. Imag-
ing was obtained in the Qa filter (λc = 18.1µm,
∆λ = 1.9µm) with a standard off-chip 15′′ABBA
chop-nod sequence and a chop position angle of
30deg E of N. Two image sequences of β Leo were
taken with 30 ms frametimes and a total on-source
integration time of 325 s per image. Prior to and
following the β Leo observations, HD109511 (K0,
F18µm≈ 1.4 Jy) was observed with the same ob-
serving sequence to serve as a point spread func-
tion and flux density calibrator. The data were
reduced with custom-written IDL routines for the
MICHELLE data format.
β Leo appears unresolved in comparison to the
calibrator. At 18.5 µm, the excess for β Leo is
∼ 0.3 Jy (Chen et al. 2006) and at 19 AU from the
star, the radius inferred by Chen et al. (2006) for
the mid-infrared emitting material, we measured
an rms dispersion of the background in the β Leo
images of ∼ 0.55 mJy pixel−1. In §4.4, we will
use this limit to constrain the radial extent of the
mid-infrared emitting material.
3. Origin of the visibility deficit
In this section we discuss the possible origins of
the visibility deficit.
3.1.Companion
A companion anywhere within the 0.′′8 (FWHM)
field-of-view (FOV) will lower the measured vis-
ibility. A companion within the fringe envelope
(roughly 25 milliarcsec for these observations) will
produce a visibility modulation which is a func-
tion of the binary flux ratio and separation and
the projected baseline length and position angle.
A companion outside this separation range but
within the field-of-view will contribute incoher-
ent flux and the visibility decrease will be the
same fraction on all baselines. The flux ratio of
a companion which would produce the measured
visibility is the incoherent fraction listed in Table
4, which corresponds to ∆K = 4.0 ± 0.9 for β Leo
and ∆K = 4.5 ± 1.4 for ζ Lep. These flux differ-
ences would be produced by a main sequence star
of spectral type M0 for β Leo and M2 for ζ Lep.
Neither star has a known companion within a
few arcsec of the primary star. The Washington
Double Star (WDS) catalog lists 3 companions
for β Leo, located from 40′′to 240′′from the
primary (far outside the FOV) with V magni-
tude differences of 6.3 to 13 (Worley & Douglass
1997).None of these stars could affect the in-
terferometry observations due to the large an-
gular separation.ζ Lep has no listed compan-
ions in the WDS. Both objects have been imaged
in the mid-infrared (Jayawardhana et al. 2001;
Moerchen et al. 2007, §2.4) with no companion
detected. In our MICHELLE/GEMINI data, the
Q-band magnitude difference for a point source
which can be ruled out is 2.5 mag within 0.′′5 and
4 mag from 0.′′5 to 0.′′8. These data are sufficient
to detect a possible companion between 0.′′5 to 0.′′8
around β Leo for the derived companion spectral
type of M0.
The strongest constraints on close (<1′′) com-
panions come from the Hipparcos measurements.
β Leo was observed 64 times over 3.0 yrs with fi-
nal positional uncertainties of 0.99 mas (RA) and
0.52 mas (dec), and ζ Lep was observed 117 times
over 3.1 yrs with uncertainties of 0.51 mas (RA)
and 0.41 mas (dec) (Perryman & ESA 1997). As
neither source was detected to have any astromet-
ric motion by Hipparcos, these uncertainties can
be used to place limits on any stellar companions.
Using the secondary stellar types inferred from the
flux ratios, the companion stellar masses would be
approximately 0.5 M⊙for β Leo and 0.4 M⊙ for
ζ Lep. As the astrometric signature increases with
orbital distance, the astrometric uncertainty from
the Hipparcos data sets a lower limit to the ex-
cluded periods, while the sampling duration sets
the limit for longer-period companions.
timate the shortest period companion which the
Hipparcos data could detect, we assumed a mass
for each primary of 2.0 M⊙and quadratically com-
bined the positional uncertainties to obtain astro-
metric uncertainties of 1.12 mas for β Leo and 0.65
mas for ζ Lep. Setting a threshold of 5σ to ac-
count for the uneven time sampling, the minimum
detectable separations are 0.25 AU (β Leo) and
0.34 AU (ζ Lep), which correspond to periods of
32 days and 51 days respectively.
To es-
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The detection of longer-period companions is
limited by the overall time span of the Hipparcos
data. The orbital period and astrometric signa-
ture of a companion located at the edge of the
FOV would be 5.5 yrs and 250 mas for β Leo and
14.5 yrs and 200 mas for ζ Lep. For β Leo, the
Hipparcos data samples half a period and would be
sufficient to detect such a companion. For ζ Lep,
the Hipparcos data would sample 20% of the or-
bital period. For a circular orbit, the deviation of
this arc from a best-fit straight line would be 12
mas, detectable with the 0.65 mas uncertainty, but
detecting some phases of an elliptical orbit would
be more difficult. A very long period companion
with the relevant magnitude difference could have
escaped detection if the orbit is inclined on the
sky such that companion is currently too close to
the primary (within 0.′′5) for detection by imag-
ing.One probe of such a very long period or-
bit is the proper motion as a function of time.
Gontcharov et al. (2001) combined proper motion
data from ground-based catalogs starting in the
1930’s with the Hipparcos data. For β Leo and
ζ Lep, the combined proper motions were within
the uncertainties of the Hipparcos proper motions,
and both stars were classified as having no com-
panions within 10′′.
Any companion closer than the short-period
limit derived above would produce a substantial
radial velocity signature. Using the inclination an-
gles derived in §3.2, a companion at the short pe-
riod limits above would produce a radial velocity
of 8 km/sec for β Leo and 12 km/sec for ζ Lep.
Galland et al. (2005) made measurements of β Leo
with an uncertainty of 137 m/s, more than suffi-
cient to detection such a large signature, however,
the time sampling covered only a few hours and
is not sufficient to rule out companion periods of
tens of days. Observations of ζ Lep (Grenier et al.
1999, e.g.) have also been made with sufficient
precision, but not sufficient time sampling to find
a companion with a period of a many days.
In summary, neither target star has a known
companion within the CHARA FOV and Hippar-
cos measurements rule out companions with pe-
riods from tens of days to several years. A very
close companion (periods less than tens of days,
separations less than 0.35 AU) can not be ruled
out in either case, but would produce an easily
detectable (> 5 km/sec) radial velocity signature.
Although we can not definitively rule out a com-
panion as the source of the flux decrement, it is
unlikely given the above constraints on period and
magnitude difference. A less massive companion
would produce a smaller flux decrement, which
would require another flux component in the sys-
tem. Given the small phase space remaining for an
undetected companion and the fact that the two
mid-infrared excess sources (β Leo and ζ Lep) have
a near-infrared visibility decrement, while δ Leo
with no mid-infrared excess does not, we proceed
with the hypothesis that the flux decrement does
not arise from a companion.
3.2.Stellar rotational oblateness
Our analysis of the visibility deficit on the short
baseline relies on knowledge of the stellar size from
the longer baselines. If the star is oblate due to
rotation, the predicted size on the short baseline
may be incorrect as the short and long baselines
are nearly orthogonal (Table 3). We can calcu-
late the maximum possible effect by assuming the
short stellar axis is aligned with the longer base-
line, which would place the longer axis along the
short baseline, producing lower visibilities.
calculate the ratio of stellar radii, XRfrom
We
XR=Rpol
Req
=
?
1 +v2
eqReq
2GM
?−1
(3)
where Rpol and Req are the polar and equato-
rial radii, veq is the equatorial velocity, G is
the gravitational constant and M is the stel-
lar mass (Domiciano de Souza et al. 2002).
the most conservative calculation, we take veqto
be the maximum equatorial velocity inferred by
Royer et al. (2007) of a survey of A stars, which
are grouped by sub-class. These velocities are 300
km sec−1for β Leo and ζ Lep and 280 km sec−1
for δ Leo. The resulting oblateness is corrected
for viewing angle by deriving i from the measured
v sini and the assumed veqand approximating the
stellar shape as an ellipsoid (Table 6). The ob-
served stellar radii ratio Xobsis then given by
For
Xobs=
XR
R)cos2i)1/2
(1 − (1 − X2
(4)
Starting with the derived stellar size on the long
baseline (θlong, see Table 4), we calculated the
8

Page 9

V2that would be measured on the short baseline
(V2(θlong)). We then applied the observed oblate-
ness factor, Xobs to find the maximum possible
angular diameter, θlong/Xobs, and recalculated the
V2for the short baseline (V2(θlong/Xobs)). Be-
cause these angular sizes are at best marginally
resolved on the short baseline, the change in visi-
bility is less than 1% in all cases, even if the appar-
ent angular size changes by 20%, as predicted for
ζ Lep. For comparison, we also list the short base-
line size, θshortfrom Table 4. The measured visibil-
ity on the short baseline, V2
lower than either V2(θlong) or V2(θlong/Xobs) for
both β Leo and ζ Lep but not for the check star
δ Leo and thus stellar oblateness can not account
for the measured visibility deficit. We note that if
rotational axis of the star is aligned such that the
short stellar axis is along the short baseline, then
the true visibility decrement is actually slightly
larger than measured.
As these stars are rotating rapidly, they are
also subject to gravity darkening, which produces
a decrease in the effective temperature from the
pole to the equator. Since the limb-darkening de-
pends on the effective temperature, this effect is
also linked to the apparent oblateness. However,
this effect is very small compared to the oblateness
derived above.Using the effective temperature
difference found by Aufdenberg et al. (2006) for
Vega, an A0 star, of 2250 K, the limb-darkenening
correction for the pole is 0.3% larger than correc-
tion at the equator. This factor goes against the
rotational oblateness which makes the equatorial
radius larger and even with the factor of 2.5 for
a fast rotating star, is insufficient to explain the
ratios between diameters fit to the long and short
baselines of 1.72 ± 0.23 for β Leo and 2.9 ± 1.0
for ζ Lep.
measuredis significantly
3.3.Emission and scattering from dust
Dust grains within the field of view will pro-
duce a near-infrared excess through thermal emis-
sion and scattering. We assume that there is no
gas in these debris disks and therefore the inner
radial limit for the debris disk is the dust sub-
limation radius. For a sublimation temperature
of 1600 K and assuming large grains in thermal
equilibrium emitting as blackbodies, the sublima-
tion radius is 0.12 AU for β Leo and 0.14 AU for
ζ Lep. The 2 µm emission will be maximized for
dust at the sublimation temperature, so a lower
limit to the excess luminosity can be estimated
following Bryden et al. (2006)
Ldust
L∗
=Fdust
F∗
kT4
dust(ehν/kT− 1)
hνT3
∗
.(5)
where h and k are the Planck and Boltzmann con-
stants. For a temperature of 1600 K, the frac-
tional dust luminosity is 2.0±1.1×10−3for β Leo
and 9.8 ± 8.5 × 10−4for ζ Lep.
ison, Chen et al. (2006) calculated mid-infrared
dust luminosities of 2.7 × 10−5and 6.7 × 10−5
for β Leo and ζ Lep respectively. However, the
much larger near-infrared luminosity does not re-
quire substantially more mass than implied by the
mid-infrared excess since, as the fractional dust
luminosity represents the fraction of the star as
seen by the dust, the calculated fractional lumi-
nosities are highly sensitive to the dust location.
An estimate of the minimum mass of near-infrared
emitting grains can be calculated using the frac-
tional luminosity and assuming efficiently emitting
grains (Jura et al. 1995),
For compar-
Mdust≥16π
3
Ldust
L∗
ρar2
(6)
where ρ is the density, a is the grain radius and r is
the distance from the star. A minimum mass can
be calculated by using the Ldustvalues calculated
above for small dust grains located near the subli-
mation radius. For a grain radius of a = 1µm, r at
the dust sublimation radius and ρ ∼ 2 gm cm−3,
the minimum mass of the near-infrared emitting
material is 5 × 10−9M⊕for β Leo and 2 × 10−9
M⊕for ζ Lep. Chen et al. (2006) derived a mass
for the small grains in the mid-infrared producing
material of 4.2 × 10−6for β Leo and 5.6 × 10−6
for ζ Lep. So although the near-infrared excess
represents a higher fractional dust luminosity, this
can be produced by a much smaller mass than the
mid-infrared ring.
A ring of hot dust near the sublimation ra-
dius is not incompatible with the incoherent flux
model fit in §2.3, as the sublimation radius is large
enough to be resolved on even the short baseline.
Given the relative uncertainty in the incoherent
flux component fit, a component with V2< 0.2
would fit within the uncertainty. For β Leo, the
sublimation radius corresponds to 11 mas and a
ring of any width at this radius has a V2< 0.2 on
9

Page 10

β Leo
110
300
21.5
1.54
0.74
0.95
1.332±0.009
1.401±0.009
2.289 ± 0.31
δ Leo
173
280
38.1
2.14
0.70
0.84
1.149±0.012
1.368±0.014
0.0 ±1.17
ζ Lep
245
300
54.7
1.5
0.74
0.80
0.69±0.09
0.826±0.11
2.0 ± 0.65
v sini (km sec−1)
assumed veq(km sec−1)
i (deg)
Req(R⊙)
XR
Xobs
θlong(mas) (Table 4)
θlong/Xobs(mas)
θshort
V2on short baseline:
V2(θlong)
V2(θlong/Xobs)
V2
measured
0.976±0.0003
0.973±0.0003
0.938±0.015
0.981±0.0004
0.973±0.0006
1.001 ± 0.015
0.996±0.001
0.994±0.002
0.966 ± 0.013
Table 6: The calculated maximum visibility change due to rotational oblateness.
θlong/Xobs, V2(θlong), and V2(θlong/Xobs) include the uncertainty in the measured value of θlong but not
the uncertainty in Xobs, which is unknown.
The uncertainties in
all baselines in our observations. For ζ Lep, the
sublimation radius is at 7 mas and any ring wider
than 1 mas (0.02 AU) produces V2< 0.2 on all
baselines. If the inclination angles are close to the
values inferred in Table 3.2, these approximations
are sufficient. Thus, thermal emission from hot
dust near the sublimation radius could produce
the measured visibility deficit.
At larger angular scales than have been in-
vestigated with the interferometer (? 1′′− 10′′),
debris disks are often detected in scattered light
at optical and near-infrared wavelengths (e.g.,
AU Mic and Fomalhaut; Kalas, Liu, & Matthews
2004; Kalas, Graham, & Clampin 2005) and scat-
tering from within the field of view of the interfer-
ometer (? 0.′′8) could also produce the observed
visibility deficit. Scattering in the near-infrared
will dominate emission for grains at several hun-
dred degrees, depending on the grain size and
composition. To investigate the scattering from
warm dust, we used the debris disk simulator1
described by Wolf (2006) which calculates the
thermal emission and scattering given the dust
size, composition and distribution. For example,
small grains uniformly distributed from 1.0 to 4.6
AU (the β Leo FOV radius) will produce the ob-
served near-infrared excess given a total mass of
small grains of 1 − 7 × 10−5M⊕, depending on
1http://aida28.mpia-hd.mpg.de/∼swolf/dds/
the exact size and composition. This is more than
1000 times larger than the minimum mass needed
to produce the excess from hot grain emission.
As there is no known evidence for a companion,
we contend that thermal emission and scattering
from dust grains is the most likely origin of the
near-infrared excess. This is also consistent with
our finding that the two sources with a measured
visibility deficit have mid-infrared excess emission
while the control star, which has no known excess
does not have a visibility deficit. In the next sec-
tion, we explore the constraints on these grains
and discuss possible mechanisms for their origin.
4. Dust distribution and small grain origin
4.1.Dust grain sizes
Both β Leo and ζ Lep have a substantial mid-
infrared excess which has a characteristic temper-
ature much lower than dust which would produce
a near-infrared thermal excess and is therefore fur-
ther from the central star. Many authors (see e.g.
Dominik & Decin 2003; Wyatt 2005, and refer-
ences therein) have studied the dynamics of debris
disks similar to our targets and have found that
collisions are dominant over PR drag, i.e grains
collide and become smaller before PR drag sig-
nificantly decreases the size of their orbits. Ra-
diation pressure also plays a role as small grains
are subject to removal from the system.How-
10

Page 11

ever, clearing of small grains may not be abso-
lute. Krivov et al. (2000) modeled the β Pic disk,
which has a similar spectral type (A6V) and opti-
cal depth to the systems discussed here and found
that although grains at and below the canonical
blowout radius are depleted compared to a purely
collisional system, a population of small grains
persists in their model. For our target stars, the
radiation pressure size limit is ∼2 µm, the PR
drag timescale at 1 AU is 1000 yrs for 10 µm ra-
dius grains and the collisional timescale for these
same grains is 80 years (following the formula of
Backman & Paresce 1993).
A second constraint on the dust size is the lack
of a significant silicate feature in the IRS spectrum
for either source (Chen et al. 2006, see Figure 2),
although the excess for β Leo is not strong enough
at 10µm to provide as strong a constraint as for
ζ Lep, which has excess emission at shorter wave-
lengths.The lack of a silicate emission feature
requires the grain population to have radii larger
than a few microns if composed of silicates or to
be primarily non-silicate.
4.2.Modeling approach
As the interferometer data provide only an up-
per limit to the visibility and therefore a lower
limit to the size of the near-infrared flux region,
the strongest spatial constraint from the interfer-
ometry data is that the dust must be within the
FOV. However, there is another strong constraint
from the measured mid-infrared excess of these
sources. The dust producing the near-infrared ex-
cess will also produce mid-infrared excess, with the
exact flux depending of course on the dust temper-
ature and opacity.
We now begin to explore various specific mod-
els for the distribution of dust in these systems,
and examine whether these models fit within the
constraints provided by the near and mid-infrared
data.In all models, we assume optically thin
emission for the near and mid-infrared emission.
In this section, we consider the relative contribu-
tions of scattering and emission to the near and
mid-infrared excess flux. For the scattering, we
have used the debris disk models of Wolf (2006)
to calculate the emission and scattered light flux
for various grain radius and radial distributions
and two example grain compositions.
chosen grain compositions which will produce the
We have
featureless mid-infrared spectrum seen in the IRS
data and have substantially different emissivity ra-
tios between the near and mid-infrared. These two
populations are silicate grains with radii between
3 and 10 µm and graphite grains with radii from
0.1 to 100 µm. In both cases, we use a distri-
bution of grain radii, n(a) ∝ a−3.5appropriate
for collisionally dominated disks. For these toy
models, we concentrated on illustrative cases of
dust radial distributions and did not modify the
grain radius distribution for the effects of radiation
pressure. The possible presence of small grains is
discussed in more detail in §4.4. For each case,
the disk mass was determined by scaling to match
the observed near-infrared excess. These masses
are significantly higher than the minimum mass
derived in §3.3 as that estimate assumes the flux
comes only from small, hot grains which produce
much more near-infrared emission for the same
mass than a distribution of grain sizes and tem-
peratures can. In Table 7 we present the results
for the two grain populations over several radial
distributions, listing the ratio of emission to scat-
tering at 2 µm, the excess flux at 10 and 24 µm
and the mass in small grains. All models have
radial density profiles of n(r) ∝ r−1.5.
As expected, emission dominates for grains
close to the central star (<1 AU), while scat-
tering dominates for grains farther away.
mass in small dust grains necessary to produce the
near-infrared excess flux is higher for scattering-
dominated disks than for emission-dominated
disks.The scattering-dominated cases produce
too much mid-infrared flux, in some cases by more
than an order of magnitude. The 24 µm excesses
measured by Su et al. (2006) are 0.46±0.01 Jy for
β Leo and 0.53±0.02 Jy for ζ Lep and the 10 µm
excess from the IRS spectra are 0.002±0.004Jy for
β Leo and 0.18±0.01 Jy for ζ Lep from Chen et al.
(2006). The emission-dominated disks also pro-
duce too much mid-infrared flux, but not by as
large a factor. As the models which have sub-
stantial near-infrared emission have mid-infrared
fluxes close to the observed values, we assume that
emission is the primary mechanism for producing
near-infrared flux. In the following sections, we
will explore other density distributions and mod-
els to fit both the near-infrared and mid-infrared
excesses in detail.
The
11

Page 12

β Leoζ Lep
Radial
distribution
Rsub-FOV
Grain
type
silicate
graphite
silicate
graphite
silicate
graphite
Fem/Fsc
F10µm
(Jy)
1.7
5.4
231
2.0
10
4.7
F24µm
(Jy)
1.2
2.8
254
1.8
3.9
2.0
Msmallgr
M⊕
1 × 10−5
2 × 10−6
4 × 10−4
2 × 10−5
1 × 10−6
1 × 10−6
Fem/Fsc
F10µm
(Jy)
18
6.7
261
32
10.5
4.7
F24µm
(Jy)
16
4.6
400
35
4.0
2.1
Msmallgr
M⊕
1 × 10−4
2 × 10−5
4 × 10−3
2 × 10−4
5 × 10−6
5 × 10−6
6.1
8.6
0.019
4.9
8.6
9.2
8.5
10
0.02
4.1
8.8
12
1.0 AU-FOV
Rsub-1.0 AU
Table 7: The ratio of emission to scattering flux at 2 µm, the mass in small grains necessary to reproduce
the observed near-infrared excess and the 10 and 24 µm flux for several disk models. The radius of the FOV
corresponds to 4.6 AU for β Leo and 8.6 AU for ζ Lep.
4.3.Dust grain distributions
For both stars, we first considered the hypoth-
esis that the grains producing the near-infrared
excess were generated by collisions between larger
bodies in the belt which produces the mid-infrared
excess.These grains can then be dragged to-
wards the central star via PR drag and become
sufficiently heated to emit at near-infrared wave-
lengths. For a specific theoretical description of a
disk in which grains created in collisions in the
planetesimal belt migrate inward, we used the
model of Wyatt (2005), who calculated the steady-
state optical depth as a function of radius. In this
model, the disks are collisionally dominated, but
a small fraction of the dust created by collisions
in the planetesimal belt migrates inwards due to
PR drag and is subject to collisions as it migrates.
Assuming a single grain size, Wyatt (2005) found
the optical depth as a function of radius to be
τeff(r) =
τeff(r0)
1 + 4η0(1 −?r/r0)
(7)
where τeff(r0) is the optical depth of the planetes-
imal belt at r0, the radius of the planetesimal belt
and η0is a parameter balancing collisions and PR
drag, which had a value of 2.4 for β Leo and 6.7 for
ζ Lep. For η0 = 1 the collisional lifetime equals
the time it takes a grain to migrate to the star.
We assume optically thin, blackbody grains dis-
tributed with the optical depth given by eq. 7 and
starting at the sublimation radius. The value of
τeff(ro) is iterated until the optical depth within
the CHARA FOV produces the observed near-
infrared excess. We then calculate how much mid-
infrared flux would be produced and compare to
the measured mid-infrared excess.
For β Leo, the mid-infrared excess spectra
is well fit by a grain temperature T ≈120 K,
which implies a distance from the star of 19 AU
(Chen et al. 2006), well outside the FOV of our
observations (4.6 AU). Applying the model in eq.
7 with r0 = 19 AU and assuming an emissivity
wavelength dependence of λ−2(§4.4) produces
a 2/10 µm flux ratio of 1.6, while the observed
ratio, using our detection and the IRS data of
Chen et al. (2006) is > 140. A shallower grain
emissivity function with wavelength will produce
an even larger discrepancy between this model
and the data. Thus, the grains which produce the
near-infrared excess can not come from a smoothly
distributed population generated by collisions in
the mid-infrared belt.
Moerchen et al. (2007) resolved the 18 µm
emission from ζ Lep and modeled the distribu-
tion as arising from two rings with stellar distances
from 2-4 and 4-8 AU, which is contained within the
CHARA FOV of 8.6 AU. Using a radius of 4 AU
in eq. 7 produces a 2/6 µm excess of 0.8, while
the observed excess from our data and Chen et al.
(2006) is 18. Thus this model is not a good fit
for ζ Lep either. We note that Moerchen et al.
(2007) did not resolve the excess emission from
ζ Lep at 10 µm and concluded that the dust pro-
ducing this excess is interior to the resolved 18 µm
rings. They surmise that the 10 µm emitting dust
is migrating inward by PR drag from the belts
resolved at 18 µm.
12

Page 13

4.4.Geometric models
To further examine the constraints which the
near-infrared and mid-infrared excess place on the
dust distribution, we use a geometric model of the
dust distribution. The data are compared to the
models in a Bayesian approach designed to con-
strain the range of valid model parameters, rather
than finding a single best-fit model. The input
data are: 1) the near-infrared excess within the in-
terferometer FOV and the visibility limits for this
excess, 2) the IRS data from Chen et al. (2006)
and 3) the spectral energy distribution from 2 to
100 µm from the literature, including a 70 µm
Spitzer-MIPS measurement (K. Stapelfeldt, pri-
vate communication).We have constructed an
SED for each star using photometry from SIM-
BAD in order to determine the excess flux. The
stellar template was determined by fitting the op-
tical and near-infrared photometry to a grid of
Kurucz-Lejeune models (Lejeune et al 1997) cov-
ering the range of effective temperature and sur-
face gravity values appropriate for the main se-
quence stellar types of the target stars. Both stars
are nearby and have photospheric colors consistent
with AV = 0 (Chen et al. 2006).
The basic disk model is an optically thin ring of
dust. To simplify the calculations, we consider the
dust to be geometrically thin; however, we note
that to intercept ∼1% of the starlight, the dust
will need to have a finite vertical height. At a
radial distance of 0.1 AU, this corresponds to a
height h with h/r = 0.02. Such a vertical height
is smaller than a flared primordial disk at this ra-
dius (Chiang & Goldreich 1997, h/r = 0.09) and
smaller than the value h/r ∼ 0.05 derived for the
β Pic dust disk at larger (r > 15 AU) radii and
thus is plausible.
The excess flux ratio between 2.2 µm and the
shortest IRS wavelength can be used to set a limit
on the wavelength dependence of the grain emis-
sivity. For β Leo, F(2.2 µm)/F(10 µm) > 140,
implying an emissivity decreasing at least as fast
as λ−1.3if the emission is from a hot black-
body. For ζ Lep, F(2.2 µm)/F(6 µm) ∼ 18, im-
plying emissivity proportional to at least λ−0.9
for hot dust. As we wish to examine the range
of disk physical parameters, including the grain
size and composition, which can reproduce the
near and mid-infrared excess emission, we adopt
a analytic approximation for the grain proper-
ties to keep the calculation manageable yet self-
consistent. We have therefore chosen a power-law
representation of the radiative efficiency ǫ, follow-
ing work on β Pic by Backman et al. (1992) and
Backman & Paresce (1993). For a given grain ra-
dius a, the absorption and emission efficiency is
roughly constant, ǫ ≃ 1 − albedo, for radiation
at wavelengths shorter than a critical wavelength
λo and decreases for wavelengths longer than λo
(Backman & Paresce 1993). The relation between
the grain radius a and the critical wavelength de-
pends on the grain composition and shape and
varies from λo/a ∼ 2π for strongly absorbing
grains to λo/a ∼ 1/2π for weakly absorbing grains
(Backman & Paresce 1993).
We assume that the disk is optically thin to its
own radiation, therefore the stellar radiation is the
only input. Radiation from the early A spectral
types observed here (Teff≥ 9000 K) is dominated
by wavelengths < 1µm. Our input data are at 2.2
µm and longer, therefore we can not constrain the
value of the critical wavelength below 2 µm. The
grain emission efficiency decreases at wavelengths
much larger than the grain radius and given our
wavelength constraints, we assume that the ex-
cess is dominated by grains with a critical wave-
length of ≥ 1µm and therefore that the absorb-
ing efficiency is essentially constant. The emission
efficiency is assumed to follow a power-law such
that ǫe= ǫo(λo/λ)q. This formulation of the effi-
ciency does not account for spectral features, but
as neither object has such features, the approxi-
mation is appropriate. We investigate two values
of q: q = 1 which is appropriate for absorbing
dielectrics and amphorous materials such as sili-
cate and roughly matches the silicate population
considered in §4.2 and q = 2 which is appropriate
for conductive substances such as pure graphite
or crystallines and represents the graphite popu-
lation in §4.2. We then derived the temperature
of the grains as a function of radius from the star,
following Backman & Paresce (1993),
T(r) = 468 L1/5
T(r) = 685 L1/6
∗ λ−1/5
o
r−2/5K q = 1(8)
∗ λ−1/3
o
r−1/3Kq = 2(9)
(10)
where L∗ is in L⊙, λo is in microns and r, the
distance to the star is in AU. Assuming a power
13

Page 14

law for the radial distribution, the flux in an ring
is then (Koerner et al. 1998)
F(r,λ) = τro
?r
ro
?α
ǫo
?λo
λ
?q
B(T)2πrdr
D2
(11)
where τrois the optical depth and D is the distance
to the star from earth. The input parameters to
our models are the inner disk radius rin, the disk
radial extent ∆r, the optical depth, τro, the optical
depth radial exponent, α and the grain character-
istic wavelength λo. Unless the grain composition
varies with disk radius, the values of τroand ǫoare
degenerate. We set ǫo = 1, thus τrohere repre-
sents the emission optical depth and is only equal
to the geometric optical depth if the grains have
an albedo of 0.
For β Leo and ζ Lep we were unable to fit both
the near-infrared and mid-infrared excess with a
single ring of dust. This is not surprising as the
2 µm excess is higher than the excess flux at the
shortest IRS wavelengths, requiring a decrease in
emissivity at some intermediate radii. The next
level of sophistication is to add a second ring of
dust, with each ring following the physical descrip-
tion given above.
For each object, a grid of millions of models
was calculated and compared to the input data.
The results for each object are a range of param-
eters consistent with the data, given all possible
values for the other parameters. We found that
some parameters were not well constrained by the
data and others were degenerate, such as the opti-
cal depth and the disk radial extent. We initially
assumed that the inner and outer rings had the
same radial power-law α and the same character-
istic grain size, λo. We were unable to fit both
the near-infrared and mid-infrared data with a two
ring model if the characteristic grain size was the
same in each ring. Fitting for two values of α and
λowithin a single grid is computationally very ex-
pensive, so we fit just the IRS data to a single
ring to constrain the values of α and λo for the
outer ring. For β Leo, the value of αouter is not
well constrained, and we assume a value of -3/2
as predicted for collisionally dominated disks (e.g.
Kenyon & Bromley 2005).
For each star, we tried to match all the input
data with the emissivity power-laws of δ = 1 or
2. For β Leo, the data can not be matched with
δ = 1, not surprising given the flux ratio between
2 and 10 µm discussed above. For ζ Lep, models
with δ = 1 can match both the near-infrared and
mid-infrared data, but these models require that
the outer ring extend over 20 AU, which is much
larger than the extent derived by Moerchen et al.
(2007) in their imaging. We therefore place a prior
constraint of ∆router < 15 AU. With this con-
straint, only δ = 2 models provide adequate fits.
The range of parameter values which falls
within a 67% probability range (corresponding
to ±1σ for a normal distribution) is given for each
target in Table 8.An example model for each
object is shown with the SED and IRS data in
Figure 2. For β Leo, the outer ring rinvalues of
7.5-15 AU are smaller than the 19 AU found by
Chen et al. (2006) due to the different tempera-
ture law we used, but the inner and outer ring
are clearly separated by a gap of several AU. For
ζ Lep, the inner and outer ring are at similar radii
(< few AU) and although the inner ring must have
significantly higher opacity to produce the near-
infrared flux, it is possible to fit the data with
models in which the inner and outer ring overlap.
Interestingly, Moerchen et al. (2007) also required
a higher flux ratio in their inner ring (2-4 AU) as
compared to the outer ring (4-8 AU).
The optical depth and radial extent of the inner
dust ring in these models are degenerate param-
eters as the constraining data are the 2 µm flux
and the lack of strong mid-infrared flux. We have
deliberately limited these models to be optically
thin, but we note that the near-infrared flux could
also arise from a ring with a very small radial
extent which was vertically optically thick. The
strongest test of the radial extent for the inner ring
would be to resolve it interferometrically, which re-
quires observations on shorter baselines than the
data presented here. For example, a ring around
β Leo at 0.12 AU with a radial extent r/4, would
have V2
ring> 0.5 at 2 µm on baselines shorter than
13 meters and V2
ring< 0.1 on the 30 meter base-
line we used. High precision measurements would
still be necessary given the small flux contribution
from the ring.
The relationship between the characteristic
grain size and the physical grain radius depends
on the grain composition and the distribution of
sizes. One specific example of a grain material
which could be approximated by our δ = 2 emis-
sivity model is graphite, which is strongly absorb-
14

Page 15

ing for grain radii > 0.1 µm (Draine & Lee 1984).
To compute the physical grain size for graphite
we take λo/a ∼ 2π. The other factor is the dis-
tribution of grain radii. Following Backman et al.
(1992), one method of tying λo to the physical
radii is to find the radius which divides the grain
population into two equal halves of surface area.
For a distribution of n(a) ∝ a−3.5, this radius is
4 times the minimum radius, a ∼ 4amin. Putting
these two factors together, we have amin∼ λo/8π.
For our upper limit on the characteristic size in
the inner ring λo < 2µm, this corresponds to
amin< 0.08µm.
Grains that small are below the nominal radia-
tion blowout radius for these stars. However, ra-
diation pressure may not completely clear all the
grains from debris disks like these. Krivov et al.
(2000) modeled the β Pic debris disk, which has a
similar spectral type (A6 V) and mid-infrared ex-
cess to our targets. In their model, small grains are
constantly created through collisions, particularly
between particles on stable orbits and those being
blown out of the system. They found that grains
smaller than a few microns were depleted com-
pared to the a−3.5distribution of a collisionally
dominated disk, but that a significant population
remained. The resulting overall grain population
could be approximated by a more shallow slope
in the distribution, for example, fitting the result-
ing grain radii distribution with a single power-law
between 0.1 and 100 µm, results in n(a) ∼ a−2.8.
Thus the inner rings may contain grains smaller
than 1 µm, the nominal blowout radius.
Our model for both targets includes a much
larger characteristic grain size in the outer ring,
λo ∼ 35 − 50µm, which corresponds to a mini-
mum size of ∼ 1 − 2µm for graphite grains. This
is roughly the radiation blowout size. The conclu-
sion from our models that the inner ring contains
substantially smaller grain sizes than the outer
ring should be confirmed with more detailed grain
models, but as discussed more below, may suggest
either different origins for the grains or different
dynamics.
The dust sublimation temperature of 1600 K
used in these models may be plausible for amor-
phous grains such as those represented by the
δ = 1 model, but is higher than generally used
for crystalline grains (e.g. 1250 K; Bauer et al.
1997) as represented by the δ = 2 model.A
sublimation temperature of 1250 K results in
much poorer fits to the data.
temperatures, micron-sized grain lifetimes will be
short; for example, Lamy (1974) found lifetimes
of less than 104seconds for 1 µm radius grains at
1500-1600 K. However, once the grains are very
small, the grain temperature and lifetime may in-
crease. In a study of grains with radii < 0.01 µm
heated through interactions with a single photon,
Guhathakurta & Draine (1989) found a broad dis-
tribution of temperatures with excursions as high
as 2800 K for graphite grains and 2050 K for
silicate grains.Guhathakurta & Draine (1989)
calculated the sublimation rates for these grains
including a correction derived from fluctuation
theory for finite systems which decreases the sub-
limation rate by ∼ 104. The resulting lifetimes for
grains with radii from several to tens of Angstroms
are > 102yrs.
The combination of sublimation, radiation
pressure and collisions will result in a grain size
distribution substantially more complicated than
the simple power-law often used in debris disks
and assumed here. The result of all these pro-
cesses may be a population of small hot grains
which is constantly created through collisions and
depleted through sublimation and radiation pres-
sure. Alternatively, the presence of a significant
number of small grains may imply origin in a tran-
sient event, as discussed in the next section. Our
formulation of the grain temperature and emis-
sivity efficiency does not properly represent very
small grains and more detailed models than those
considered here are necessary to determine if the
temperatures and lifetimes of sub-micron sized
grains are consistent with a stable grain popula-
tion which could produce near-infrared flux ob-
served here. Emission from small, hot grains has
been invoked to fit the spectral energy distribu-
tions of debris disks (Sylvester et al. 1997) and
the much more massive primordial disks of Herbig
Ae/Be stars (Natta et al. 1993).
In the models presented here, the mid-infrared
excess for β Leo is produced by dust grains located
r ≈ 12±5 AU from the star. At the pixel scale of
our MICHELLE/Gemini data (0.′′1 pixel−1, §2.4)
and the distance of β Leo (d = 11.1 pc), the mid-
infrared dust emission is located of ∼8 pixels from
the central core of the stellar image (FWHM ∼ 5.4
pixels); thus, the Gemini observations of β Leo
At these high
15