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MNRAS 000,1–11 (XXX) Preprint 21 August 2020 Compiled using MNRAS L
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Observability of Dusty Debris Discs around M-stars
Patricia Luppe,1?Alexander V. Krivov,1Mark Booth1and
Jean-Fran¸cois Lestrade2
1Astrophysikalisches Institut und Universit¨
atssternwarte, Friedrich-Schiller-Universit¨
at Jena, Schillerg¨
aßchen 2-3, D-07745 Jena,
Germany
2Observatoire de Paris, CNRS, 61 Av. de l’Observatoire, 75014 Paris, France
Accepted XXX. Received YYY; in original form 21 August 2020
ABSTRACT
Debris discs are second generation dusty discs formed by collisions of planetesimals.
Many debris discs have been found and resolved around hot and solar-type stars.
However, only a handful have been discovered around M-stars, and the reasons for
their paucity remain unclear. Here we check whether the sensitivity and wavelength
coverage of present-day telescopes are simply unfavourable for detection of these discs
or if they are truly rare. We approach this question by looking at the Herschel /DEBRIS
survey that has searched for debris discs including M-type stars. Assuming that these
cool-star discs are “similar” to those of the hotter stars in some sense (i.e., in terms
of dust location, temperature, fractional luminosity, or mass), we check whether this
survey should have found them. With our procedure we can reproduce the 2.1+4.5
−1.7%
detection rate of M-star debris discs of the DEBRIS survey, which implies that these
discs can indeed be similar to discs around hotter stars and just avoid detection. We
then apply this procedure to IRAM NIKA-2 and ALMA bands 3, 6 and 7 to predict
possible detection rates and give recommendations for future observations. We do
not favour observing with IRAM, since it leads to detection rates lower than for the
DEBRIS survey, with 0.6%–4.5% for a 15 min observation. ALMA observations, with
detection rates 0.9%–7.2%, do not offer a significant improvement either, and so we
conclude that more sensitive far-infrared and single dish sub-millimetre telescopes are
necessary to discover the missing population of M-star debris discs.
Key words: stars: low-mass, brown dwarfs – circumstellar matter – planetary systems
1 INTRODUCTION
In many respects, M-star planetary systems differ from sys-
tems around earlier-type stars. They seem to contain dif-
ferent populations of planets, as many M-stars host Earth-
to Neptune-mass planets but only a few harbor gas giants
(Bonfils et al. 2013;Mulders et al. 2015;Dressing & Char-
bonneau 2015;Winn 2018). Another peculiarity is related to
debris discs around these cool stars. Debris discs are second
generation, dusty circumstellar discs formed by collisions of
planetesimals that have formed in previously existing planet-
forming discs. While during the last few decades many debris
discs have been found and resolved around A to K-type stars
(Matthews et al. 2014;Hughes et al. 2018), only a handful
of them have been discovered around M-stars.
We start with a brief summary of previous searches of
debris discs around M-stars. Many of these surveys, such
?E-mail: patricia.luppe@posteo.net
as Plavchan et al. (2005,2009), Lestrade et al. (2009) and
Avenhaus et al. (2012), did not result in detections. Other
surveys have been more successful. Forbrich et al. (2008) de-
tected 11 M-star debris discs with Spitzer, including 9 new
disc candidates. Theissen & West (2014) used data released
by the Wide-field Infrared Survey Explorer (WISE) to look
for excesses around field M-stars. In their sample of 70841
M-stars they detected excesses around 175 stars. Binks &
Jeffries (2017) also used WISE data and found 7 debris disc
candidates in nearby young moving groups. More recently,
Zuckerman et al. (2019) used WISE data to demonstrate
infrared excesses around 9 stars in the χ1cluster, many
around stars that are in binary systems.Nibauer et al. (2020)
used Planck data to statistically determine that ∼10% of A-
M stars within 80 pc of the Sun (of which almost a half
are M stars) have detectable debris discs. Their method
cannot conclusively pinpoint which stars have debris discs,
but they do propose a number of specific M stars as likely
candidates. Three surveys searched for M-star debris discs
©
XXX The Authors
2P. Luppe et al.
with Herschel/PACS. The DEBRIS survey, which we use in
this paper, probed 94 M-stars at 100 and 160µm (Lestrade
et al., in prep.). It yielded the detection of a disc around
GJ 581 (Lestrade et al. 2012) and Fomalhaut C (Kennedy
et al. 2014). Another PACS survey presented by Kennedy
et al. (2018) examined 21 M-stars with known radial-velocity
planets and yielded two disc candidates around GJ 433 and
GJ 649. A PACS survey by Tanner et al. (2020) observed 20
stars between spectral types K5 and M5 and yielded three
disc candidates around CP-72 27131, GJ 784 and GJ 707.
In the sub-mm, 32 M-stars were observed at 850µm with
SCUBA at JCMT and at 1.2 mm with MAMBO at the
IRAM telescope (Lestrade et al. 2006), leading to one disc
candidate around GJ 842.2. At present, there are still only
five debris disc detections around M-stars based on more
than one independent observation: AU Mic (Kalas et al.
2004,Liu et al. 2004,Fitzgerald et al. 2007,MacGregor et al.
2013), GJ 581 (Lestrade et al. 2012;Kennedy et al. 2018),
Fomalhaut C (Kennedy et al. 2014; Cronin-Coltsmann et al.,
in prep.), TWA 7 (Matthews et al. 2007;Bayo et al. 2019;
Matr`a et al. 2019) and CP-72 27131(Mo´or et al. 2020;Tan-
ner et al. 2020). Apart from that, several other detections
(such as TWA 25; Choquet et al. 2016) were claimed, but
these are pending confirmation.
The reasons for the paucity of debris disc detection
around M-stars remain unclear. It might be possible that
M-star debris discs are no less common than those around
earlier-type hosts, and that the lack of detections is solely
caused by observational limitations. Morey & Lestrade
(2014) investigated this possibility with detailed modelling.
They assumed a debris disc created by a planetesimal belt
that is undergoing a steady-state collisional cascade and em-
ployed the analytic model of Wyatt et al. (2007) for the
disc collisional evolution. By using distributions of mean disc
radii and initial disc masses they generated a synthetic disc
population. They fit their model to the fractional luminosi-
ties of 34 A-star debris disc from Su et al. (2006) and 28
FGK-star debris discs from Trilling et al. (2008). All of these
stars were observed with Spitzer at 24 and 70µm. With the
assumption that the same disc population that was found
around AFGK-stars is located around M-stars they used
the best fit parameters of their models to simulate total disc
masses and mean disc radii of potential debris discs around
the M-stars listed in Gautier et al. (2007) and Lestrade et al.
(2006,2009). In this way they estimated the flux density of
these potential discs at 70µm. They concluded that a signif-
icant population of M-star discs could exist and would still
be consistent with the non-detections in preceding surveys.
Like Morey & Lestrade (2014), here we check whether
the sensitivity and wavelength coverage of present-day tele-
scopes are simply unfavourable for detection of these discs
or if they are truly rare. However, our approach is differ-
ent to theirs and can be referred to as more empirical. The
idea is to investigate whether a true population of debris
discs around M-stars could be “similar” in one or another
sense to a population of debris discs around stars of earlier
spectral types (AFGK-stars), and whether the paucity of
1Although there is debate over whether this has a spectral type
of K7 (Torres et al. 2006;Pecaut & Mamajek 2013) or M0 (Gaidos
et al. 2014).
M-star discs could be solely due to their limited detectabil-
ity with the instruments used previously to search for them.
That “similarity” can be formulated in terms of dust loca-
tion, temperature, fractional luminosity, or mass. We for-
mulate several hypotheses, with the simplest one being a
hypothesis that discs around stars of all spectral types have
comparable radii and dust masses. Alternatively, we can as-
sume some scalings of disc radii and/or masses with the
spectral type, for instance take an expectation that discs of
later-type stars are smaller and/or lighter than those around
earlier-type primaries. For each hypothesis, we generate an
expected population of M-star debris discs, check their de-
tectability with existing instruments, and compare the pre-
dicted detection rates with actual ones reported previously.
In this way, we select those hypotheses that are consistent
with the previous surveys and reject the others that are not.
Finally, we use the preferred hypotheses to make testable
predictions by calculating the expected detection probabil-
ity of M-star discs in future surveys with other instruments
(ALMA and IRAM instead of Herschel).
Section 2 describes important equations, our procedure,
and the different hypotheses we make. Section 3 presents the
results for our tests with the DEBRIS survey and our pre-
dictions for observations with IRAM and ALMA. Section 4
contains a discussion and section 5 lists our conclusions.
2 METHODS
2.1 Basic equations
Two basic quantities characterizing a debris disc are dust
temperature Tdand fractional luminosity fd. Both can be
inferred from the observed spectral energy distributions
(SEDs) by means of a straightforward SED modelling. The
fractional luminosity is related to the total cross section of
the dust material σtot and the disc radius Rd:
fd=σtot
4πR2
d
.(1)
In turn, σtot is directly proportional to the dust mass Md:
Md∝σtot,(2)
assuming that the dust size distribution is the same in all
discs considered. The dust temperature is determined by
stellar luminosity L∗and blackbody disc radius RBB:
Td=L∗
16πσB1/4
(RBB)−1/2,(3)
with σBbeing the Stefan-Boltzmann constant.
The blackbody radius is the one the disc of tempera-
ture Tdwould have if the grains were absorbing and emit-
ting as blackbodies. However, this is not the case since any
disc also includes dust particles that are smaller than the
typical wavelengths of their thermal emission. Such grains
are efficient absorbers of the incoming stellar radiation, but
inefficient emitters of thermal radiation (e.g., Krivov et al.
2008). This makes these grains warmer than the blackbody
approximation predicts, implying that the actual radius of
a disc of temperature Td, which we denote as Rd, must be
larger than RBB (Rodriguez & Zuckerman 2012;Booth et al.
2013;Pawellek et al. 2014;Pawellek & Krivov 2015). In this
MNRAS 000,1–11 (XXX)
Observability of Debris Discs around M-stars 3
paper, for the conversion of RBB to Rdwe use the relation
(see Fig. 4b in Pawellek et al. 2014)
Rd= Γ RBB ,with Γ = 2.0 (L∗/L)−0.16 .(4)
As shown by Pawellek et al. (2014), this relation holds over
a wide range of stellar luminosities, albeit with a large scat-
ter. It is also approximately valid for two prominent debris
discs around M-stars, AU Mic (Matthews et al. 2015) and
GJ 581 (Lestrade et al. 2012). We thus assume that Eq. (4)
is applicable to all M-star discs considered in our analysis.
Of course, this is just an assumption that we need to make
to be able to estimate the temperature of the as yet undis-
covered discs around M-stars from their expected “physical”
radii.
To calculate the detection limits of the telescopes, it is
sufficient to assume that dust emits as a blackbody. Denot-
ing by Fν,min the minimum specific thermal emission flux
detectable with a certain instrument at a frequency ν, it is
easy to express the minimum fractional luminosity of a disc
that this instrument is able to detect:
fd,min =2d2c2
hν3
σBT4
d
L∗
Fν,min exp hν
kTd−1,(5)
where dis the distance to the star, cthe speed of light, and
hthe Planck constant.
2.2 Hypotheses
We now formulate a set of hypotheses of what the population
of M-star debris discs may look like. To this end, we consider
possible trends of disc radius and mass with the spectral
type of disc host stars. In doing so, we include both the
relations reported for debris discs of AFGK-stars and those
found in millimetre surveys of protoplanetary discs, which
are progenitors to the debris discs in question.
We start with the disc radius, Rd. It is still debated if
a significant correlation between the spectral type and the
debris disc radius exists. Taking ALMA- and SMA-resolved
debris discs, Matr`a et al. (2018, their Fig. 1) find a slight
correlation between debris disc radius and stellar luminosity.
Others, such as Pawellek et al. (2014, based on Herschel-
resolved debris discs, see their Fig. 2) and Hughes et al.
(2018, their Fig. 4) do not find any correlation. Therefore we
state for hypotheses 1 to 5 that the disc radii of debris discs
around M-stars correspond to those around the reference
stars: Rd= const. If RBB of the reference star is 100 AU,
this leads to RM
d=Rref
d= Γ Rref
BB = Γ ·100 AU. For the
hypotheses 6 to 10 we use the radius-luminosity relation
proposed by Matr`a et al. (2018):
Rd∝L0.19
∗.(6)
As far as the scaling of the disc mass is concerned, we
used five different assumptions. For hypotheses 1 and 6, we
assumed the mass of the M-star discs to be the same as
for the reference stars. For hypotheses 2 and 7, the rela-
tion of Williams & Cieza (2011) for protoplanetary discs
(Md∝M∗), deduced from the (sub-)mm surveys between
2000 and 2009, was adopted. For the other hypotheses, we
took relations listed in Pascucci et al. (2016). They presented
an ALMA observation of the 2 Myr-old Chamaeleon I star-
forming region and found the relation Mdust ∝M1.3−1.9
∗. Re-
analysing all data of nearby star-forming regions that were
Table 1. Summary of hypotheses. The second and the third col-
umn list the assumed scalings of the disc radius and dust cross
section with the stellar mass. The fourth and the fifth column
show how the dust temperature and fractional luminosity change
from AFGK-star debris discs to M-star discs assuming these scal-
ings.
Hypothesis Rdσtot Tdfd
1 const const decrease const
2 const ∝M∗decrease decrease
3 const ∝M1.3
∗decrease decrease
4 const ∝M1.9
∗decrease decrease
5 const ∝M2.7
∗decrease decrease
6∝L0.19
∗const decrease increase
7∝L0.19
∗∝M∗decrease increase
8∝L0.19
∗∝M1.3
∗decrease slight in-/decrease
9∝L0.19
∗∝M1.9
∗decrease slight in-/decrease
10 ∝L0.19
∗∝M2.7
∗decrease decrease
available in the mm-range, they found that the 1-3 Myr-old
Taurus, Lupus and Chamaeleon I regions all show similar
Mdust −M∗dependence. We therefore used the slope of 1.3
for hypotheses 3 and 8 and the slope of 1.9 for hypotheses 4
and 9. A much higher slope, with Mdust ∝M2.7
∗, was found
by Pascucci et al. (2016) for the 10 Myr old Upper Sco as-
sociation. We used this slope for our hypotheses 5 and 10.
An overview of all hypotheses (i.e., combinations of the pa-
rameter scalings) is given in Table 1.
2.3 Procedure
To check a specific hypothesis, we proceed as follows (Fig. 1).
1. First, we consider a sample of reference stars of earlier
spectral types (we take AFGK-stars) with debris disc detec-
tions. All their discs have known dust fractional luminosities
fref
dand temperatures Tref
d.
2. Then we take one of the observational programs that
searched for debris discs around M-stars (numbered 1, 2, ...
N) with a certain instrument and reported both detections
and non-detections. For every M-star observed, we know
the stellar parameters. We then take fref
dand Tref
dand use
Eqs. (1)–(4) together with the scalings of Table 1to calcu-
late fM
dand TM
d, i.e. the fractional luminosity and temper-
ature that dust in the (scaled) reference discs would have if
their central star were the M-star considered. This yields a
population of debris discs around an M-star that we expect
in the framework of the selected hypothesis. We also know
the specifications of the instrument used in the M-star sur-
vey, and the integration time. This allows us to compute the
detection limit (Eq. 5), which will be a U-shaped curve in
the temperature – fractional luminosity plane (Fig. 1). For
all following calculations we used 3σdetection limits. Note
that the detection limit depends on the distance and lumi-
nosity of the M-star and so, the detectability curves will be
different for different M-stars in the survey.
3. Finally, we compare the position of the generated
cloud of fiducial M-star discs with that detectability curve
and count the fraction of discs that would be observable, pi
(where iis a number of the M-star in the M-star survey).
4. The entire procedure is executed for all M-stars in
the survey, yielding the detection probabilities pi(i= 1, 2,
MNRAS 000,1–11 (XXX)
4P. Luppe et al.
Figure 1. Schematic that illustrates how we generate a population of M-star debris discs, which is expected in the framework of a certain
hypothesis. This is the plot for a 15 min observation of GJ 447, an M4 star, at a distance of 3.37 pc, and for hypothesis 3. The brown
U-shaped curve gives the limit for the Herschel/PACS 100µm band, the black curve for the PACS 160µm band. The blue points represent
the reference debris discs of the AFGK-stars found by DEBRIS and the pink stars are the calculated M-star debris discs. The shaded
regions highlight the parameter space of the reference discs and the simulated discs in order to better visualise how this changes based
on the assumed hypothesis. Observable are discs that are located above the detectability curves. The reference discs do not necessarily
lie above these curves, because the curves were calculated for an M-star, GJ 447. Detectability curves change with L∗and distance and
have to be calculated independently for every star.
... N) and the average detection probability
¯pM=1
N
N
X
i=1
pi.
5. Finally, we multiply ¯pMwith the detection rate of
debris discs around reference stars, pref, to get the detection
rate of discs around M-stars that we predict with a given
hypothesis:
pM
pred = ¯pMpref.
This should be compared with the actual detection fraction
of discs in the M-star survey, pM
obs. The closer pM
pred and pM
obs,
the more viable the hypothesis we made about the true pop-
ulation of discs around M-stars.
3 RESULTS
In this section, we describe our sample of reference stars and
the M-star debris disc survey. We then apply the algorithm
described above to test the hypotheses. Finally, we make
predictions for future IRAM and ALMA surveys of M-star
debris discs.
3.1 The DEBRIS sample
The DEBRIS Open Time Key Program (Matthews et al.
2010) was an unbiased Herschel survey that searched for
debris discs around A-, F-, G-, K- and M-stars (Table 2).
In this survey 83 A-stars (Thureau et al. 2014), 92 F-stars,
90 G-stars, 91 K-stars (Sibthorpe et al. 2018) and 94 M-
stars (Lestrade et al., in prep.) were observed at 100µm and
160 µm. In the originally planned unbiased sample 86 A-
stars were listed (Phillips et al. 2010). Three of those stars
were excluded from the survey because they were guaran-
teed time Herschel targets: Vega (Sibthorpe et al. 2010),
Fomalhaut (Acke et al. 2012) and βPic (Vandenbussche
et al. 2010). They nevertheless were included in all DEBRIS
statistics, because leaving them out would have biased the
sample. The same applies to the GK-stars. In the original
unbiased sample 91 G-stars and 92 K-stars were listed. εEri-
dani (Greaves et al. 2014) and τCeti (Lawler et al. 2014)
were part of the Herschel guaranteed time program. Both
were included in all DEBRIS statistics for the same rea-
sons as for the A-stars. Therefore we use all 86 A-stars, 91
G-stars and 92 K-stars for our calculations and statistics,
too. The number of observed stars and disc detections that
we used for our analysis are listed in Table 2. Five A-stars
were identified to possess a two-component disc. For those
stars we only included the cold disc component (Thureau
et al. 2014, their Table 4) in our calculations. For three of
the FGK-star discs a two-component fit was necessary to
MNRAS 000,1–11 (XXX)
Observability of Debris Discs around M-stars 5
Figure 2. Test of AFG-to K-type stars. The black horizontal
line is the DEBRIS detection rate of debris discs around K-stars.
The gray-filled area is the binomial confidence interval of this
detection rate at a 95% confidence level (Jeffreys’ interval2). Each
hypothesis predicts individual detection rates for all 92 K-stars,
pi. The vertical bars show the 95% confidence level of the mean
of those pi. The dashed lines on these bars show the mean and
the dotted lines show the median for every hypothesis. For the
detection rate of the reference stars we use the combined detection
rate of A-, F- and G-stars, which is 20.8%.
reproduce the observed excesses (Sibthorpe et al. 2018). In
these cases Sibthorpe et al. (2018) used the cooler of the two
components for their analysis.
3.2 Tests of hypotheses with AFGK-stars
We first tested our method described in section 2by con-
sidering the A-, F-, and G-stars and using them to predict
detections of discs around K-stars in the DEBRIS sample.
Specifically, we took 56 AFG-stars with debris discs as ref-
erence ones and generated a population of debris discs ex-
pected around K-stars from Sibthorpe et al. (2018). We com-
pared the predicted detection rate with the actual one. In
Fig. 2, the predicted mean detection rate for every hypoth-
esis with its 95% confidence level is plotted. This has to be
compared to the DEBRIS detection rate of the K-star discs,
which is also shown, together with its binomial uncertainty
interval (at the 95% confidence level). A comparison sug-
gests that hypotheses 1, 7 and 8 provide the best matches.
However, all besides hypotheses 4 and 5 are statistically con-
sistent with the detection range of K-star debris discs. This
test shows that our method and all but two of our hypotheses
work for our purposes. For our further analyses we therefore
exclude hypotheses 4 and 5, but for completeness they are
still shown in our prediction plots.
1http://www.pas.rochester.edu/~emamajek/EEM_dwarf_
UBVIJHK_colors_Teff.txt
2http://epitools.ausvet.com.au
Figure 3. Detection rates of M-star debris discs. The structure
of this figure is analogous to Fig. 2. The black horizontal line is
the DEBRIS detection rate of debris discs around M-stars. Each
hypothesis predicts individual detection rates for all 94 M-stars,
pi. Plotted is the mean of these individual rates for every hypoth-
esis. The detection rate of the reference stars is now the combined
detection rate of A-, F-, G-, and K-stars, which is 18.8%. All hy-
potheses except for hypothesis 6 confirm the detection rate.
3.3 Application of hypotheses to the M-stars
Next, we applied our method described in section 2by con-
sidering the A-, F-, G-, K- and M-stars in the DEBRIS sam-
ple (Table 2). We took all 68 AFGK-stars with debris discs
as reference ones and generated a population of debris discs
expected around M-stars (Lestrade et al., in prep.). The pre-
dicted mean detection rate for every hypothesis along with
its 95% confidence level is plotted in Fig. 3. As before, we
overplot the actual DEBRIS detection rate with its 95% con-
fidence interval. A comparison demonstrates that all but hy-
pothesis 6 are consistent with the M-star DEBRIS detection
rate. We therefore also exclude hypothesis 6 from our fur-
ther analyses. We get a detection rate between 1.6%
±
0.6%
and 6.4%
±
1.1% for the M-stars in the DEBRIS sample. We
see that the mean and the median for most hypotheses dif-
fer widely. This shows the large asymmetric spread between
individual values.
This demonstrates that it is indeed possible that M-
stars harbour populations of debris discs similar to those
around earlier-type stars, and that many of these discs may
have just eluded detection due to occupying a different re-
gion in the Tdvs. fdplot compared to discs around earlier
type stars. Unfortunately, with the Herschel data we can
only rule out three of the hypotheses. Surveys with other fa-
cilities will be necessary to distinguish between the remain-
ing ones.
3.4 Prediction for IRAM surveys
Next, we predict the number of detections for the M-stars
from the DEBRIS survey that would be expected if this
sample were observed with the new NIKA-2 instrument on
MNRAS 000,1–11 (XXX)
6P. Luppe et al.
Table 2. DEBRIS sample by spectral classes. In the left part of the table the DEBRIS sample is listed. For five A-stars reported as
having two-component discs we only used the cold component for our calculation. The right part lists the sources we used for the stellar
parameters.
stellar type stars detections det. rate refs SpT L∗M∗distance Tdisc fdRBB
A 86 21 24.4%+9.8
−8.11 1 6,7,8,9,10,11 calc. from L∗- 1 1 1
F 92 22 23.9%+9.4
−7.82 2 6,13 13 or calc. from L∗- 2 2 2
G 91 13 14.3%+8.3
−6.02 2 6,13 13 or calc. from L∗- 2 2 2
K 92 12 13.0%+8.0
−5.72 2 6,13 13 or calc. from L∗6,12 2 2 2
M 94 2 2.1%+4.5
−1.72,3,4,5 3 13 13 6 - - -
References: 1: Thureau et al. (2014), 2: Sibthorpe et al. (2018), 3: Lestrade et al. (in prep.), 4: Kennedy et al. (2014), 5: Lestrade
et al. (2012), 6: Gaia Collaboration et al. (2018), 7: Kennedy et al. (2012), 8: Di Folco et al. (2004), 9: Anderson & Francis (2012),
10: Zorec & Royer (2012), 11: (Boyajian et al. 2012), 12: van Leeuwen (2007), 13: Mamajek (Version 2018.12.10)1
Note: The errors for the detection rates were calculated with the 95% confidence limit, Jeffreys interval2
Figure 4. Same as Fig. 1, but with the 15 min IRAM detection
limits overplotted. The light green curve represents the IRAM
1 mm band and the dark green curve shows the IRAM 2 mm
band.
Table 3. IRAM sensitivities in different bands for different ob-
servation times.
15 min 30 min 60 min
1 mm 3σ3.3 mJy 2.3 mJy 1.65 mJy
2 mm 3σ0.8 mJy 0.6 mJy 0.4 mJy
Note: The sensitivities are calculated from those in Kramer &
Sanchez Portal 20193
IRAM (Adam et al. 2018;Catalano et al. 2016,2018). To
this end, we do the same calculation as before, but change
the detection limit curves to those of the 1 mm and 2 mm
bands of NIKA-2. A comparison of the detection limit curves
is shown in Fig. 4. We assumed different observing times of
15 min, 30 min and 60 min and therefore different sensitiv-
ities, see Table 3.
In Fig. 5we plot the detection rates for the different ob-
serving times and hypotheses. The lowest and highest rates,
including 95% confidence limits, are listed in Table 4. As
3http://www.iram.fr/GENERAL/calls/s19/30mCapabilities.
pdf
Table 4. IRAM minimum and maximum detection probabilities
min [%] max [%]
15 min 0.6
±
0.2 4.5
±
0.5
30 min 0.8
±
0.3 5.4
±
0.5
60 min 1.0
±
0.3 6.5
±
0.6
expected, the minimum and maximum detection rates get
higher for longer observing times.
If we compare the 15 min observation of the DEBRIS
survey (Figure 3) with the 15 min observation of IRAM (Fig-
ure 5left), we see a slightly lower detection rate for the
observation with IRAM. The conclusion is that the NIKA-
2 IRAM instrument would not provide a better detection
rate than Herschel/PACS did. However, IRAM could detect
colder and fainter M-star discs than PACS, which would re-
sult in partly different disc detections (see Fig. 4).
A potential issue with these proposed IRAM obser-
vations is the possibility of contamination by background
galaxies. To calculate the number of galaxies above a given
flux density expected within a certain area we used the
Schecter function as shown in Booth et al. (2017):
n(Fν)d log Fν=Aφ∗Fν
S∗α+1
exp −Fν
S∗ln 10 d log Fν.
(7)
Parameter Ais the area of sky in deg2and φ∗,S∗and α
are parameters that depend on the wavelength of the obser-
vations. These parameters are determined through surveys
for galaxies. Here we will use the values provided by Carni-
ani et al. (2015) for observations at 1.3 mm. We therefore
need to adjust our flux densities to the appropriate wave-
length. A galaxy SED slope is Fν∝λ−2modified by an
additional factor (λ/λ0)−β, with β=−1.6 (Casey 2012).
Accordingly, we assumed a slope of λ−3.6±0.38 to translate
our 1 and 2 mm flux densities to flux densities at 1.3 mm.
Using this we then considered the case of a 60 min
IRAM observation assuming a >3σdetection. We determine
that there is a 6.6% and 1.8% chance of finding a galaxy
within 1 beam of a star for IRAM 1 mm and 2 mm, respec-
tively. These probabilities are similar to the probabilities of
detecting a debris disc around an M-star thus meaning that
an IRAM survey for debris discs around M-stars is unfortu-
nately likely to detect as many galaxies coincident with the
stars as it does detect debris discs around them.
MNRAS 000,1–11 (XXX)
Observability of Debris Discs around M-stars 7
Figure 5. Prediction for an observation with IRAM with three different observing times.
Figure 6. Same as Fig. 1, but with the 15 min ALMA detection
limits overplotted. The light brown curve represents ALMA band
3, brown curve ALMA band 6 and the dark brown curve shows
ALMA band 7.
Table 5. ALMA sensitivities in different bands for different ob-
servation times.
15 min 30 min 60 min
Band 3 3σ78.3 µJy 55.4 µJy 39.2 µJy
Band 6 3σ92.4 µJy 65.4 µJy 46.2 µJy
Band 7 3σ147.4 µJy 104.2 µJy 73.7 µJy
Note: The sensitivities were calculated with the ALMA
sensitivity calculator4
3.5 Prediction for future ALMA surveys
We now assume that the same DEBRIS sample of M-stars
(Lestrade et al., in prep.) is observed with ALMA in a dedi-
cated program. We consider observations with the 12m array
in its most compact configuration in bands 3 (111 GHz), 6
(226 GHz) and 7 (340 GHz). A comparison of the detection
limits from DEBRIS and ALMA is shown in Figure 6. The
sensitivities are listed in Table 5.
The lowest and highest detection rates for bands 3, 6
4https://almascience.eso.org/proposing/
sensitivity-calculator
and 7 for observing times of 15 min, 30 min and 60 min,
including 95% confidence limits, are given in the left part of
Table 6. The detection rates in bands 6 and 7 do not differ
much.
3.5.1 Resolved discs
In the previous section we have made the assumption that
the discs are unresolved. However, ALMA not only has a
superior sensitivity to previous millimetre observatories, but
also a superior resolution. This makes it an excellent obser-
vatory for detailed studies of the structure of debris discs,
but may prove to be problematic for our attempts to detect
discs around the nearest M-stars. Their proximity means
the discs have a large angular size, which will result in a
decreased flux density per beam. In order to understand
how strongly this affects our results we have checked how
many of our calculated debris discs have a diameter larger
than the resolution for the different bands, see Table 7. Our
calculation shows that the number of discs that would be
resolved increases with increasing frequency. For the cases
where R∝L0.19
∗(hypotheses 6 to 10) the number of resolved
discs increases from band 3 with 20.4% to 54.4% for band
7. For R= const (hypotheses 1 to 5) the numbers get even
higher, with 55.5% for band 3 to 90.6% for band 7.
To get an estimate of how the detection rates may be
affected, we calculated for every band the mean number of
beams that are necessary to cover the complete disc by di-
viding the disc circumference by the beam size. The values
were determined by calculating the number of beams nec-
essary for every single disc and dividing the fractional lu-
minosity by the number of beams. For R∝L0.19
∗we get
means of 2.4 beams for band 3, 5.0 beams for band 6 and
7.5 beams for band 7. The number increases for hypotheses
with R= const to 6.7 beams for band 3, 13.7 beams for band
6 and 20.6 beams for band 7. These large sizes, especially
for the hypotheses with R= const, result in a reduction to
the detection rates calculated in the previous section.
Since ALMA is an interferometer we also have to con-
sider the Maximum Recoverable Scale (MRS)5. The MRS is
the largest scale size that can be detected by the smallest
baseline of an interferometer. This scale has to be compared
with the Largest Angular Scale (LAS) of the disc. If the
LAS of the disc is bigger than the MRS, then some of the
emission will not be detectable. Calculating the exact effect
would require time consuming simulations for every disc.
MNRAS 000,1–11 (XXX)
8P. Luppe et al.
Figure 7. Prediction for an observation with ALMA Band 3, 6 and 7 with three different observing times each.
Table 6. ALMA minimum and maximum detection probabilities firstly considering only the total flux of the disc and secondly taking
into account the reduction of the flux per beam due to many of the discs being resolved.
using total disc flux correcting for the resolution
min [%] max [%] min [%] max [%]
Band 3 15 min 2.1
±
0.6 10.5
±
0.7 0.9
±
0.2 5.2
±
0.6
30 min 2.6
±
0.6 12.0
±
0.7 1.2
±
0.3 6.6
±
0.7
60 min 3.1
±
0.7 13.4
±
0.7 1.7
±
0.4 8.6
±
0.8
Band 6 15 min 3.9
±
0.8 15.1
±
0.6 1.3
±
0.3 7.2
±
0.7
30 min 4.5
±
0.9 16.1
±
0.5 1.7
±
0.4 9.1
±
0.7
60 min 5.3
±
1.0 17.1
±
0.4 2.3
±
0.5 11.0
±
0.7
Band 7 15 min 4.3
±
0.9 15.8
±
0.5 1.0
±
0.2 6.3
±
0.6
30 min 5.1
±
1.0 16.8
±
0.4 1.3
±
0.2 8.1
±
0.7
60 min 5.9
±
1.1 17.6
±
0.3 1.8
±
0.3 10.0
±
0.7
We, therefore, decided to use a conservative approach and
assumed that any disc with a diameter larger than the MRS
is not observable and set their fractional luminosity to zero.
In this way we got a “corrected” fractional luminosity for
every disc, which decreased our detection rates. The mini-
mum and maximum detection rates after accounting for this
correction are compared to those simply using the total flux
in Table 6. The minimum detection rates are now all ≤2.3%
5https://almascience.eso.org/proposing/proposers-guide
and the maximum detection rates are roughly half as high as
before. Due to the high resolution of band 7, the detection
rates of this band become even lower than those of band 6.
Including the MRS only had a minor effect on the detection
rates since discs that are larger than the MRS will already
be difficult to detect due to their low flux density per beam.
For band 3 the detection rates did not change, for band 6
they decreased by 0.1–0.2% and for band 7 they decreased
by 0.2–0.5%.
In Fig. 7we plot the detection rates for the resolved
discs for bands 3, 6 and 7 for observing times of 15 min,
30 min and 60 min. In contrast to Herschel and IRAM,
MNRAS 000,1–11 (XXX)
Observability of Debris Discs around M-stars 9
Table 7. The resolution of each band and the fraction of discs
that would be spatially resolved for two different assumptions
about the disc radii.
resolution R=const R∝L0.19
∗
Band 3 2.9900 55.5% 20.4%
Band 6 1.4700 81.9% 40.6%
Band 7 0.9700 90.6% 54.4%
Note:
The angular resolution in Table 7was calculated with
θres = 0.574 λ/L80 (ALMA Partnership et al. 2019) with λ
being the observing wavelength and L80 the 80th percentile of
the (u, v) distance.
with ALMA it not possible to observe different bands at the
same time. For that reason we show a separate plot for every
ALMA band. We see that the detection rate increases from
band 3 to band 6 and, quite as expected, with observation
time. From band 6 to band 7 the detection rates decrease
again, because the high resolution of band 7 leads to a large
number of resolved discs and thus to a decrease of flux per
beam. The exact detection rate varies significantly from one
hypothesis to another. For the hypotheses with R= const,
the detection rates for all but hypothesis 1 are very small.
For the hypotheses with R∝L0.19
∗(7 to 10) the detection
rates look a bit more promising.
We reiterate here that the sample we have chosen to
work with in this paper is a sample of the closest M-stars.
In order to avoid resolving discs to an extent that they are
no longer detectable we could consider a more distant pop-
ulation of M-stars since the angular diameter of the disc is
inversely proportional to the distance. Unfortunately, since
the flux density is inversely proportional to the distance
squared, the flux density per beam will still reduce with
distance, thus showing that a more distant population of M
star discs would also not benefit from ALMA observations.
3.5.2 Extragalactic confusion
In the same way as for IRAM (subsection 3.4) we extrapo-
lated the number of galaxies from band 6 to band 3 and 7.
This was done for a 60 min ALMA observation assuming a
>3σdetection. There is a 0.8%, 4.0% and 5.6% chance of a
detectable galaxy within 1 beam of a star for band 3, 6 and
7, respectively. For band 7 the expected number of galax-
ies is higher than for bands 3 and 6. This could be a larger
problem for observations. However, the number of resolved
discs is also much higher for band 7 than for the other bands,
which can help distinguish between a disc and a galaxy.
3.5.3 Confusion due to stellar emission
When considering unresolved observations we must also con-
sider the contribution of the stellar flux density to the pho-
tometric flux density. Whilst the flux from the photosphere
falls off with wavelength at millimetre wavelengths, similar
to a blackbody, contributions from the chromosphere and
corona can exceed the photosphere at such long wavelengths
(see e.g. Liseau et al. 2015). This is particularly a prob-
lem for late type stars and can be hard to distinguish from
disc emission. For instance, when AU Mic was first observed
by ALMA, in addition to the clearly resolved debris disc,
the central emission was found to be about six times higher
than predicted for the photosphere. It initially was not clear
whether this excess was emission from an asteroid belt or
from the star (MacGregor et al. 2013), although further in-
vestigation clearly favours coronal emission (Cranmer et al.
2013;Daley et al. 2019). Similarly, Anglada et al. (2017) an-
nounced a tentative discovery of multiple dust rings around
Proxima Centauri, however this star has long been known to
be a flare star (Shapley 1951) and MacGregor et al. (2018)
have since shown that the flares are responsible for the unre-
solved excess. Any sensitive surveys of M-stars would need to
take this into consideration, although it is currently hard to
quantify how large of an effect this would have since further
research is necessary to better understand the flux distribu-
tion of M-stars at these wavelengths.
3.5.4 Prospects for ALMA Observations
Overall, our calculations show that a population of discs
around M-stars similar to that around hotter stars could
be detected with ALMA much more easily than with Her-
schel when simply comparing the sensitivity of the obser-
vations with the expected total disc flux (Table 6). How-
ever, the higher sensitivity of the observations does intro-
duce potential issues due to extragalactic confusion (which
is more severe for the shorter wavelengths) and stellar con-
fusion (which is more severe for the longer wavelengths).
Resolving the discs would help alleviate these confusion is-
sues, but also reduces the flux density per beam of the disc
and thus the disc detectability. We also note that a limita-
tion of our model is that it is reliant on the AFGK discs
detected by Herschel. As seen in Figure 6, ALMA extends
the parameter space of detectable debris discs beyond what
could be detected by Herschel and so the detection rates
reported in Table 6should be considered lower limits.
In conclusion, the high resolution of ALMA observa-
tions proves to be a hindrance to an unbiased survey aimed
at detecting debris discs around M-stars.
4 DISCUSSION
In view of these results, we argue that future observatories
are necessary to help us find the missing population of de-
bris discs around M-stars. One possibility would be to wait
for far-infrared space missions of the next generation, such
as Space Infrared Telescope for Cosmology and Astrophysics
(SPICA;Roelfsema et al. 2018) and Origins Space Telescope
(OST ;The OST mission concept study team 2018). Offer-
ing sensitivity by up to two orders of magnitude higher than
Herschel, these facilities would be able to probe the major-
ity of the M-star debris discs predicted in the framework of
our hypotheses, while also going much deeper in detecting
debris discs of solar-type stars, catching the discs at least as
tenuous as the dust disc in the Kuiper-belt region of our own
Solar System. This would allow one to better determine how
debris discs around M-stars relate to those around earlier
type stars. Another possibility would be to use large sub-
mm single-dish telescopes like the proposed Atacama Large
Aperture Submillimeter Telescope (AtLAST; Holland et al.
2019). Being more sensitive than ALMA, this instrument
MNRAS 000,1–11 (XXX)
10 P. Luppe et al.
would offer a resolution of ≈1.400 at 350 µm, comparable to
ALMA’s in a compact configuration.
Observations with future observatories of this kind
would not only promise a higher detection rate of discs
around M-type stars. The chances to distinguish between
different hypotheses would also be better. Should detection
rates be on the lower side, this would favour steeper dust
mass – stellar luminosity relations (hypotheses 5 or 10).
Conversely, higher detection rates would be indicative of a
weaker dependence of the dust mass on the stellar luminos-
ity (hypotheses 1–2 or 6–7). Very high detection rates, above
20% or so, would be suggestive of a population of large and
cold debris discs around M-stars that do not have counter-
parts around earlier-type stars.
Obviously, the value of the future searches of M-star
debris discs would extend far beyond pure statistics. Should
they yield a set of new detections, and should some of the
newly discovered discs of M-stars get resolved, this would
allow one to infer disc radii and dust masses. This, in turn,
would shed light onto the physics of debris discs around cool
stars, which is not well understood at present. Further, com-
bined with information (even observational upper limits) on
planets in the same systems, disc parameters would enable
deeper insights into the architecture of planetary systems
of low-mass stars and into the history of their formation.
For instance, should the majority of the newly discovered
M-star discs be more compact than those around earlier-
type primaries, this might be consistent with the fact that
many M-stars host close-in low-mass planets, but only a few
more distant planets in the giant mass range. Indeed, Gai-
dos (2017) suggested that the density profiles on protoplan-
etary discs of M-stars may be steeper than those of solar
type stars. This was deduced from the differences in planet
masses and planet distributions. Effectively this would lead
to more compact planet-forming discs around M-stars. Since
debris discs are successors of protoplanetary ones, this could
also be inherited by M-star debris discs. This would then be
in line with hypotheses 6 through 10 in our analysis.
Another aspect is the dust mass. Should debris dust
masses inferred from the new detections be abnormally low,
this might be an indication that either the dust production
rate in the discs of M-stars is low or the dust removal mech-
anisms in these discs are more efficient than in their coun-
terparts of solar-type stars. The former could be attributed,
e.g., to the lower stirring level of the M-star discs (Th´ebault
& Wu 2008), which might be consistent with the lack of
massive planets as stirrers — or with the absence of large
(Pluto-sized) embedded planetesimals (Pawellek & Krivov
2015), as was specifically inferred for the AU Mic disc from
its low vertical thickness (Daley et al. 2019). The latter can
be indicative of the role of the stellar winds (e.g. Plavchan
et al. 2005;Sch¨
uppler et al. 2015) or intensive sweep-up of
dust by coronal mass ejections (Misconi 1993;Misconi &
Pettera 1995;Osten et al. 2013).
5 CONCLUSIONS
In an attempt to clarify the reasons for the paucity of debris
discs around M-type stars, in this paper we check whether
in reality these discs could be as common as those around
earlier-type stars, but could have just escaped frequent de-
tection because of the observational limitations of the in-
struments used to search for them. We assume that M-star
discs are “similar” to those around AFGK-stars, by formu-
lating several hypotheses of what exactly that “similarity”
may mean. Specifically, we speculate that the discs may ei-
ther preserve nearly the same sizes around host stars from
A to M — or may become somewhat more compact from
earlier towards later-type primaries. In the same style, we
assume that the dust mass in the discs may be either inde-
pendent of the spectral type or gently decrease from more
luminous to less luminous hosts. For each of the hypotheses,
we compute the expected detection rates and compare them
with those actually reported.
As a test of our method, we first predicted the detection
rates of discs around K-type stars, based on the AFG-star
disc observations done by the Herschel Open Time Key Pro-
gram DEBRIS (Thureau et al. 2014;Sibthorpe et al. 2018).
We found the predicted rates to be consistent, within sta-
tistical uncertainties, with those found by the DEBRIS pro-
gram (Sibthorpe et al. 2018), except for two hypotheses thus
allowing us to rule out hypotheses 4 and 5 from the K-star
data alone. With this test, we have validated the procedure
before applying it to the M-star discs.
We computed the expected detection rates for M-stars,
using the AFGK-stars reported by the DEBRIS program,
and compared them with actual detection statistics (2 detec-
tions out of 94 stars probed, or 2.1+4.5
−1.7at a 95% confidence
level) of that survey (Lestrade et al., in prep.). We found es-
sentially the same result. The rates (1.6%–6.4%) are consis-
tent with nearly all of the hypotheses, and we were not able
to favour any of them. We identified only one hypothesis out
of 10 considered that should formally be rejected. This was
the assumption that discs of M-stars are more compact than,
but as dusty as, those of earlier-type hosts (hypothesis 6 in
Table 1). For all other hypotheses, the assumption that discs
of M-stars are as frequent as those of AFGK ones, would be
fully consistent with the low incidence rates found by DE-
BRIS. This is in agreement with the conclusions drawn by
Morey & Lestrade (2014). Our method confirms their results
with a completely different approach.
We tested which of the current-day instruments could
be able to detect those M-star debris discs that we calculated
before. First, we considered the new NIKA-2 instrument on
IRAM, which operates at 1 mm and 2 mm. We tested it for
three different observing times: 15 min, 30 min and 60 min.
For the 15 min observation we obtained a detection prob-
ability between 0.6%–4.5%, depending on which hypothesis
is assumed. These detection rates are slightly lower than
those we estimated for the Herschel/PACS DEBRIS survey.
The minimum and maximum detection rates increase for a
60 min observation to 1.0%–6.5%. We determine that there
is a 6.6% and 1.8% change of finding a galaxy within 1 beam
of a star for IRAM 1 mm and 2 mm, respectively, which is
similar to the probabilities of detecting a debris disc. With
these results we do not favour IRAM NIKA-2 for the de-
tection of a larger number of M-star discs. Nevertheless, one
must consider that IRAM would partly detect different discs
than Herschel/PACS did, see Fig. 4.
Second, we made a prediction for a possible observation
with ALMA in bands 3, 6 and 7. Analogous to the IRAM
prediction, we tested three different observing times: 15 min,
30 min and 60 min. The detection rate in band 3 ranges from
MNRAS 000,1–11 (XXX)
Observability of Debris Discs around M-stars 11
2.1%–10.5% for the 15 min observation to 3.1%–13 .4% for
the 60 min observation. For band 6 the detection rate ranges
from 3.9%–15.1% for 15 min integration time to 5.3%–17.1%
for the 60 min observation. For band 7 the detection rate for
15 min is 4.3%–15.8% and increases to 4.5%–17.6% for the
60 min observation. Therefore, when only considering the
sensitivity, our results imply that ALMA observations have
the potential to discover many debris discs around M-stars,
particularly with observations in bands 6 and 7. Unfortu-
nately, there are a number of caveats that need to be taken
into account with ALMA observations.
Firstly, we considered the number of discs that would
be resolved, as this decreases the amount of flux per beam
and so the detectability of the discs. For hypotheses 1 to 5,
where R=const, the number of resolved discs for band 3, 6
and 7 are 55.5%, 81.9% and 90.6%, respectively. So for all
bands, the majority of the discs would be resolved. The mean
number of beams that are necessary to cover the complete
discs are in this case 6.7 beams for band 3, 13.7 beams for
band 6 and 20.6 beams for band 7. For hypotheses 6 to 10,
where R∝L0.19
∗, the number of resolved discs for band
3, 6 and 7 are 20.4%, 40.6% and 54.4%, respectively. With
a mean of 2.4 beams (band 3), 5.0 beams (band 6) and 7.5
beams (band 7) to cover the complete discs. Considering the
exact number of beams necessary to completely cover each
disc and the MRS, we get the following detection rates for
ALMA: band 3 ranges from 0.9%–5.2% (15 min) to 1.7%–
8.6% (60 min), band 6 from 1.3%–7.3% for (15 min) to 2.3%–
11.0% (60 min) and band 7 from 1.0%–6.3% (15 min) to
1.8%–10.0% (60 min).
Another aspect is the expected probability of having
contaminating galaxies in the beam. It increases from 0.8%
in band 3 through 4.0% in band 6 to 5.6% in band 7. Es-
pecially in the latter case, given a maximum predicted de-
tection rate of 10.0% in band 7, background galaxies could
lead to a number of false disc candidates. Since for band 7
also the number of resolved discs is the highest, it could,
however, be possible to identify those galaxies.
In summary, we find that current observatories and in-
strumentation are not able to help us answer the question
of how debris discs around M-stars relate to those around
earlier type stars. Future observatories are necessary to help
us find the missing population of debris discs around M-
stars such as the far-infrared space missions SPICA and
OST or large sub-mm single-dish telescopes like AtLAST.
Future surveys with facilities like these will show whether
debris discs around M-stars are rare or rather ubiquitous.
They may also allow one to distinguish between several hy-
potheses made here, enabling valuable constraints on salient
features and physics of M-star debris discs. Combined with
the results of planet searches, this would lead to better un-
derstanding of the architecture and formation circumstances
of planetary systems around M-type stars.
ACKNOWLEDGEMENTS
We are grateful to Grant Kennedy, Luca Matr`a, and Mark
Wyatt for enlightening discussions. Useful comments by the
anonymous reviewer that helped to improve the manuscript
are very much appreciated. This research was supported
by the Deutsche Forschungsgemeinschaft (DFG), grants
Kr 2164/13-2, Kr 2164/14-2 and Kr 2164/15-2.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable
request to the corresponding author.
This paper has been typeset from a T
E
X/L
A
T
E
X file prepared by
the author.
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