Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths
aa r X i v : . [ a s t r o - ph . E P ] A ug Astronomy&Astrophysicsmanuscript no. 37858 © ESO 2020August 19, 2020
Relating grain size distributions in circumstellar discs to thespectral index at millimetre wavelengths
T. L ¨ohne
Astrophysikalisches Institut und Universit¨atssternwarte, Friedrich-Schiller-Universit¨at Jena, Schillerg¨aßchen 2–3, 07745 Jena,Germany, e-mail: [email protected]
Preprint online version: August 19, 2020
ABSTRACT
The excess emission seen in spectral energy distributions (SEDs) is commonly used to infer the properties of the emitting circumstellardust in protoplanetary and debris discs. Most notably, dust size distributions and details of the collision physics are derived from SEDslopes at long wavelengths. This paper reviews the approximations that are commonly used and contrasts them with numerical resultsfor the thermal emission. The inferred size distribution indexes p are shown to be greater and more sensitive to the observed sub(mm)spectral indexes, α mm , than previously considered. This e ff ect results from aspects of the transition from small grains with volumetricabsorption to bigger grains that absorb and emit near to their surface, controlled by both the real and the imaginary part of the refractiveindex. The steeper size distributions indicate stronger size-dependence of material strengths or impact velocities or, otherwise, lesse ffi cient transport or erosion processes. Strong uncertainties remain because of insu ffi cient knowledge of the material composition,porosity, and optical properties at long wavelengths. Key words. circumstellar matter – planetary systems – opacity
1. Introduction
The excess emission commonly observed in spectral energy dis-tributions (SEDs) reflects the properties and the evolution of theemitting circumstellar dust. Material from a range of grain sizes,radial distances, grain compositions, etc. contributes to the ob-served flux density. In turn, the distributions across these grainproperties result from the past and ongoing processes that createand remove the dust.The shape of an observed disc-integrated SED is a convolu-tion of several dust properties and processes. A prominent exam-ple for the degeneracies that occur is the small grains at greaterdistances that can have the same temperatures as bigger grainscloser to the star, where big and small are relative to the stellarspectrum. In a similar way, the temperature distribution and theresulting SED can be broadened by both a range of grain sizesand a range of distances. A narrow ring containing small and biggrains can mimic a wide ring with a narrow size distribution.The analysis of long-wavelength emission su ff ers less froma temperature degeneracy. Grains are ine ffi cient emitters in theRayleigh regime, that is, at wavelengths much longer than theirown scale. At long wavelengths, the contribution from smallergrains to the SED is small. With these warmer grains out of theequation, the range of temperatures involved is reduced; temper-atures and grain size become less entangled. This allowed, forexample, Draine (2006) to derive an analytic approximation thatrelates the spectral index in the mm regime, α mm , to the power-law index of the underlying grain size distribution, p . The linearrelation that results is: p = α mm − α Pl β s + , (1)where α Pl is the slope of the Planck function at the appropriatetemperature and β p is the power-law index of the wavelength-dependent opacity. Knowledge of the size distribution index provides deepinsights into the physical environment of the dust. In pro-toplanetary discs, the processes of agglomeration and com-paction result in net grain growth and shape the size distri-bution (e. g. Dominik et al. 2007). In debris discs, disruptionand erosion steadily transfer mass from bigger to smaller ob-jects in a collisional cascade (e. g. Krivov 2010; Matthews et al.2014). On the one hand, the resulting dust size distributionis steepened by the material strength decreasing with grainsize (O’Brien & Greenberg 2003; Wyatt et al. 2011) and by suc-cessive collisional damping as material is moving down thecollisional cascade towards smaller sizes (Pan & Schlichting2012). On the other hand, an interplay of radiation pressureand dynamical excitation (Th´ebault & Wu 2008) and transportprocesses such as Poynting–Robertson or stellar wind drag(Reidemeister et al. 2011; Wyatt et al. 2011) flatten the size dis-tribution. The canonical value of p = . . p . ff due to radiation pressure blowout in-duces a wavy deviation where slopes can exceed this rangelocally (Campo Bagatin et al. 1994). That waviness partiallytranscends to the observed SEDs (Th´ebault & Augereau 2007).However, Eq. (1) is derived under certain assumptions, with al-ternative approximations available for other assumptions. Theaim of this work is, therefore, to delimit the parameter ranges inwhich these approximations apply and when it is best to use afull numerical modelling and fitting of thermal emission, whichcan be considered the benchmark.Section 2 provides a review of the basic equations that gov-ern all thermal emission models. Possible approximations arediscussed, including those that lead to Eq. (1). Section 3 containsa comparison of numerical and analytic results, revealing and ex-
1. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths plaining discrepancies that significantly a ff ect the desired directlink between observables and dust physics. The immediate im-pact on the analysis of literature data is discussed in Sub-section3.7. Sections 4 and 5 discuss the implications more broadly andsummarize the findings.
2. Analytic approximations
An optically thin circumstellar dust ring at a distance d from theobserver produces a total thermal flux density, F ν ( λ ) = s max Z s min π B ν [ T ( s ) , λ ] Q abs ( T ( s ) , s , λ ) s d N ( s ) d s , (2)with B ν the Planck function, T ( s ) the grain Temperature, λ thewavelength, Q abs the absorption and emission e ffi ciency, N ( s )the di ff erential size distribution, and s min and s max the minimumand maximum grain radii. In general, the absorption e ffi ciencydepends on wavelength, material, and temperature as well asgrain size and shape. In thermal equilibrium, grain temperature T and distance r from the star are related through (Krivov et al.2008): r = R ∗ vt R C abs ( T , s , λ ) F λ, ∗ d λ R C abs ( T , s , λ ) B λ ( T , λ )d λ , (3)where C abs ≡ π s Q abs is the e ff ective absorption cross-sectionper grain. Whenever the temperature can be considered constantover the relevant range of grain sizes, Eq. (2) can be simplifiedto F ν ( λ ) ≈ B ν ( T , λ ) d s max Z s min C abs ( T , s , λ ) N ( s ) d s . (4)Grain temperature is discussed in greater detail in Sect. 3.5.Defining further a total cross-section, C abs,total ( λ ) ≡ Z C abs ( s , λ ) N ( s )d s (5)and disc mass M d = Z M ( s ) N ( s )d s , (6)where M ( s ) is the mass of a single grain with radius s , thewavelength-dependent total opacity is given by κ ( λ ) ≡ C abs,total ( λ ) M d . (7)After inserting Equations (5)–(7) into (4), the resulting flux den-sity is F ν ( λ ) ≈ B ν ( T , λ ) d κ ( λ ) M d . (8)The measured flux density can thus be decomposed into the un-derlying blackbody and the opacity. If κ follows a power law, κ ∝ λ − β , (9)the opacity index β is related to the spectral index vialog F ν B ν = − β log λ + const , (10) which is equivalent tolog F ν, / F ν, log λ /λ | {z } = − α mm − log B ν, / B ν, log λ /λ | {z } = − α Pl = − β (11)for any given pair of wavelengths λ and λ . Assuming that thetemperature is known and constant over the range of relevantgrain sizes, the blackbody component can thus be removed fromobserved fluxes. The spectral index of the opacity, and hence thesize distribution, can be derived (e. g. Ricci et al. 2012, 2015;MacGregor et al. 2016; Marshall et al. 2017).While the details of C abs ( s , λ ) depend on grain morphology,structure, and material, two asymptotic cases and an intermedi-ate case can be identified for a given wavelength:(s) grains that are geometrically small,(t) grains that are large but optically thin and(o) grains that are optically thick and geometrically large,corresponding to cases 1, 2, 3 in Kataoka et al. (2014), respec-tively. Grains in categories (s) and (t) absorb with all their trans-parent volume, thus having C abs ∝ s and Q abs ∝ s . Theopaque grains in category (o) absorb all radiation that is notreflected o ff of their surface. They absorb (and emit) only intheir surface layers, corresponding to C abs ∝ s and Q abs ∼ ff man (1983) and Kataoka et al. (2014) give thefollowing explicit absorption e ffi ciencies for the three regimes: Q abs,s ≈ x Im m − m + ! = nkx ( n − k + + (2 nk ) ∝ s , (12) Q abs,t ≈ kx n (cid:16) n − ( n − / (cid:17) ∝ s , (13) Q abs,o ∼ , (14)where m ≡ n + i k is the complex refractive index and x ≡ π s /λ the size parameter. The transition from (s) to (t) occurs aroundgrain sizes where 2 ns ≈ λ , corresponding to a characteristic size, s ′ c = λ n . (15)Regime (o) is reached when the grains become opaque, that is,around 8 kx ∼ n , corresponding to s c = max n λ π k , s ′ c ! . (16)Two extremes are plausible:(a) a wide transparent regime (t), that is, s c ≫ s ′ c (b) a vanishing transparent regime (t) if Q abs ( s ′ c ) is alreadyclose to unity because of a high A , in which case s c ≈ s ′ c Figure 1 illustrates cases (a) and (b) for Q abs as a functionof grain size. Grains becoming opaque while still being smallwould imply Q abs,s & n /λ . A / n . π k / Q abs,s and Q abs,t , the depen-dence on wavelength can be subsumed in the volume absorptioncoe ffi cient A : Q abs,s = sA , (17) Q abs,t = sAy , (18)
2. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths −2 −1 A b s o r p ti on e ff i c i e n c y , Q a b s Grain size, s [µm] s m a ll s m a ll t r a n s p a r e n t opaque (a) l = 2 cm(b) l = 0.5 mm Fig. 1.
Schematic comparison of the two di ff erent cases (a) and(b) for the transition from small grains that absorb by volumeto big, opaque grains that absorb by cross-section. Case (a) hasan intermediate size interval where the grains are already bigcompared to the wavelength but still transparent. In case (b) thetransition from small to opaque is direct. The values assumed inequations (17), (18), and (28) are β s = . A = .
63 mm − , λ = n = y = n .where y ≡ Q abs,t / Q abs,s . A commonly used alternative symbol for A is α λ . The characteristic size for the transition from (t) to (o)then becomes s c = max Q abs,o Ay , s ′ c ! . (19)As y or A increase, s c approaches s ′ c and the transparent regimevanishes.For low absorption, k ≪ n , the ratio between the two regimesis given by y ≈ ( n + n (cid:16) n − ( n − / (cid:17) , (20)which is then independent from k . Over the range 1 < n < y can be very roughly approximated with 0 . n . For materialswith low refractive index n ≈
1, the di ff erence between (s) and (t)vanishes and A becomes identical with the linear attenuation co-e ffi cient, which is related to the imaginary part of the refractiveindex (Bohren & Hu ff man 1983): A ≈ π k λ , (21)which is independent from n .Assuming a power-law for the grain size distribution, N ( s ) = N ss ! − p , (22)and inserting equations (17), (18) and (14) into (5) and (6), thedust mass and total absorption cross-section are M d = πρ N s − p ) s max s ! − p − s min s ! − p (23) and C abs,total ( λ ) = π AN s − p ) s c’ s ! − p − s min s ! − p + y s c s ! − p − y s c’ s ! − p + π N s (3 − p ) Q abs,o s max s ! − p − s c s ! − p , (24)for s min < s ′ c < s c < s max . The opacity is then κ ( λ ) = A ρ (1 − y ) s ′ c s c ! − p − s min s c ! − p + y − − pp − Q abs,o As c s max s c ! − p − × s max s c ! − p − s min s c ! − p − , (25)where s = s c is assumed. For a top-heavy size distribution,where big grains dominate both total mass and total cross-section, κ ( λ ) p ≪ −→ Q abs,o ρ s max − p − p , (26)which inherits only the dependence of Q abs,o on wavelength,roughly leading to β → p ≪
3. For a bottom-heavy sizedistribution, where small grains dominate both total mass andtotal cross-section, κ ( λ ) p ≫ −→ A ρ , (27)which is independent of p . A power-law approximation to thevolume absorption coe ffi cient, A = A λλ ! − β s , (28)then leads to β → β s for p ≫ < p <
4, where smallgrains dominate total cross-section and big grains the total mass, κ can be simplified to κ ( λ ) ≈ A ρ y + (1 − y ) s ′ c s c ! − p + − pp − Q abs,o As c s max s c ! p − . (29)With Eq. (19), the two cases (a) and (b) imply the opacities κ a ( λ ) ≈ yA ρ ( + − pp − ) s max s c ! p − ∝ Q abs,o × s − p c ∝ Q abs,o × λ − β s ( p − (30)for a wide transparent regime and κ b ( λ ) ≈ A ρ ( + − pp − Q abs,o As ′ c ) s max s ′ c ! p − ∝ Q abs,o × ( s ′ c ) − p ∝ Q abs,o × λ − p (31)for a direct transition from small to opaque, respectively. For y ≈ Q abs,o ≈ const, Eq. (30) is equivalent to Eq. (8) inDraine (2006).
3. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths
Table 1.
Alphabetic list of recurring symbols.
Symbol Meaning(a), a wide transparent regime A volume absorption coe ffi cient A normalization constant for A ( λ ) α mm measured spectral index at mm wavelengths ( F ν ) α Pl spectral index of underlying Planck function ( B ν )(b), b no transparent regime β spectral index of κβ e ff coe ffi cient of linear fit to α mm ( p ) β s spectral index of AB ν Planck function C abs absorption cross-section per grain C abs,total total, disc-integrated absorption cross-section d distance to observer F ν spectral flux density (per frequency interval) k imaginary part of complex refractive index κ mass absorption coe ffi cient (or opacity) λ wavelength λ reference wavelength for A ( λ ) M grain mass M d total disc mass n real part of complex refractive index N grain number per size interval N normalization constant for N ( s ) ν frequency(o), o opaque (grains) p size distribution index ∆ p vertical intercept of linear fit to p ( α mm ) − Q abs absorption (and emission) e ffi ciency r distance from star to grain ρ bulk grain density(s), s small (grains) s grain radius s reference grain radius for N ( s ) s ′ c radius for transition from small to transparent s c radius for transition from transparent to opaque(t), t transparent (grains) T grain temperature x grain size parameter y ratio between Q abs for (t) and (s) grains The combination of equations (30) and (11) results in Eq. (1),a relation between observed spectral index α mm and size distribu-tion index p that depends on the material through β s . For case (b),the combination of Eqs. (31) and (11) results in p = + α mm − α Pl , (32)which depends only weakly on the material through α Pl and thegrain temperature.
3. Comparison with numerical results
In this section, the assumptions made for the basic analytic ap-proximations are tested against numerical models for di ff erentmaterials. Absorption and emission are calculated assuming ho-mogeneous, compact spheres. For grains where 2 π sn < λ ,Mie theory as implemented by Wolf & Voshchinnikov (2004)is used. For larger grains the geometrical optics algorithm byBohren & Hu ff man (1983) is used. A set of five di ff erent materials is assumed: compact astro-nomical silicate (Draine 2006) with assumed bulk density −1 R ea l p a r t o f r e fr ac ti v e i nd e x , n no temperature dependenceAmorphous carbonAstronomical silicatePyroxeneCrystalline water icePorous astron. sil.Wavelength, l [µm]10 −3 −2 −1 no temperature dependenceAmorphous carbon (cid:181) l − . Astronomical silicate (cid:181) l − . Porous astron. sil. (cid:181) l − . −5 −4 −3 −2 −1 −1 measured temp.−dep. I m a g i n a r y p a r t o f r e fr ac ti v e i nd e x , k Pyroxene (10−300 K)Crystalline water ice (10−250 K)Wavelength, l [µm] Fig. 2.
Refractive indexes of the five materials discussed: (top)real and (middle, bottom) imaginary parts. The legends are or-dered vertically according to the order at long wavelengths. Thevertical dashed lines delimit the wavelength range over whichthe labelled slopes are fitted. ρ = . / cm , porous astronomical silicate (mixing rule:Bruggeman 1935) with a filling factor f =
10 % and ρ = .
33 g / cm , amorphous carbon (Zubko et al. 1996) with ρ = / cm , pyroxene (H. Mutschke and P. Mohr, private comm.,manuscript in preparation) with ρ = . / cm , and crystallinewater ice (Reinert et al. 2015) with ρ = . / cm . Only the datasets for the latter two materials include a dependence on graintemperature. Figure 2 shows the real and imaginary parts of therefractive indexes of these materials. All materials exhibit moreor less pronounced absorption features for wavelengths λ . µ m and smoother behaviours for λ & µ m. Amorphouscarbon is exceptional in that absorption is high across all consid-ered wavelengths, without a significant decline in the (sub)mmrange.All numerical calculations were done for a grid of parame-ters. Size distribution indexes are varied from 2.6 to 4.2 in stepsof 0.1. Maximum grain radii are s max =
4. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths silicates and s max =
100 km for water ice. Minimum grain radiiare determined from the radiation pressure blowout limits formain-sequence stars of six di ff erent luminosities: 0.0625, 0.25,1, 4, 16, along with 64 solar luminosities. In between s max and s min , the grid of grain radii is spaced logarithmically with factorsof 1 . / between neighbouring points. Distances between dustand star are set to 1.5625, 3.125, 6.25, 12.5, 25, 50, 100, and 200au. Two computations were performed for each of the 4080 com-binations of size distribution index, luminosity, distance, and ma-terial: one with absorption and emission directly defined by theoptical properties of the materials and another one were actualmaterials are only used for determining grain temperatures (for s > α Pl as required in Eqs. (1) and (32). The resulting absorption e ffi ciencies are plotted in Fig. 3 fora range of wavelengths and grain sizes, illustrating the occur-rences of the regimes (s), (t), and (o) as defined in Sect. 2. Asexpected, the opaque regime (o) with Q abs ∼ k is high and wavelengths λ are short compared to grain radii s .The transparent regime (t) is found for s & λ/ (2 n ) but low ab-sorption. As soon as s . λ/ (2 n ), the grains can be consideredsmall, corresponding to regime (s). The transition radius and theratio y = Q abs,t / Q abs,s depend, notably, on n .Figure 3 shows that, for silicates and carbon, the transpar-ent regime is missing for λ . λ & Q abs curves. In the transparent regime,the numerical values for Q abs exceed the values predicted byEq. (13) significantly, showing strong oscillations. This excessamounts on average to an additional factor ≈ n . This is due to themonochromatic calculations and the idealized shape resultingin internal resonances, which increase absorption and emission.For more irregular grains and integration over a broader rangeof wavelengths, the curves are expected to become smoother. Inthe opaque regime, Q abs of amorphous carbon grains convergesto ≈ .
4, that is, a value significantly below unity. This is causedby the assumed even surfaces and the resulting Fresnel reflec-tion on the surface of this high- n material. Surface structure onsub-wavelength scales could reduce reflection and increase ab-sorption. The total mass absorption coe ffi cient or opacity, κ , is an interme-diary between absorption e ffi ciencies and SEDs. Figure 4 showsresults for four di ff erent materials as functions of size distribu-tion indexes and wavelengths. These curves partly inherit theabsorption features seen for k , but they are smoother becausethey tend to be dominated by grains that are rather opaque. Theopacities of astronomical silicate and amorphous carbon followsmooth power-laws already for λ & µ m. The data sets forpyroxene and water ice have measured changes in slope for λ & µ m, which lead to a corresponding significant depar-ture of κ from a simple power law. All four materials show near-constant opacities for shallowsize distributions around p = s max not only dominate the total mass butalso contribute significantly to the total cross-section. The discthen has the same e ff ective cross-section at all wavelength. Theremaining slope for carbon at p = . k , and n increasing with λ . As a result, reflectance increases and Q abs,o decreases. For the silicates and water ice, k is lower and n con-stant.Towards size distribution slopes p ≥
4, smaller grains be-come more and more important. Given the same total mass M d ,the total cross-section C abs,total increases, leading to higher valuesfor κ overall. However, the small grains’ e ff ective cross-sectionmore strongly depends on wavelength, leading to steeper opac-ity slopes for steeper size distributions, as expected from bothEqs. (30) and (31). Around wavelengths of a few tens of microns, k is comparably high for all materials, and so is κ .The wavelengths at which the κ curves for di ff erent sizedistribution slopes converge are the wavelengths at which thebiggest assumed grains become transparent, that is, the wave-lengths at which s c ( λ ) = s max . At yet longer wavelengths, κ would become insensitive to p . For carbon and the silicates, theextrapolated curves converge at wavelengths of several metres oreven kilometres. For water ice at ∼
10 K they converge alreadyaround λ ∼ s max for this material.Figure 5 compares the β ( p ) dependencies expected fromthe Mie calculations with the analytic approximations in equa-tions (30) and (31). Neither of the two approximations can repro-duce the numerical results over the full range of size distributionslopes for all materials. While the solid curve for amorphous car-bon approaches the grey line that follows Eq. (31), the resultsfor pyroxene, water ice, and astronomical silicate are closer tothe dashed lines that represent Eq. (30). The slopes of the re-gression lines for the two silicates are only halfway between thatof Eqs. (30) and (31), meaning that the sensitivity of β to dif-ferences in p is also halfway between cases (a) and (b). Thesefindings for the four materials are consistent with their refractiveindexes in the mm wavelength regime. The high n and k of car-bon puts it in category (b), the low absorption of pure water iceleads to case (a), with the two silicates remaining in between.Taking Astronomical Silicate as an example, Fig. 6 showshow the results shift more towards case (b) for shorter, submmwavelengths. In this wavelength range absorption is higher forall materials, resulting in a narrower transparent regime (t).Figure 7 shows that the general shape of the β ( p ) relation iswell-reproduced by Eq. (25). The slopes of the near-linear sec-tions around p = . n . Around p ≈ p ≈ n , β can exceed β s for p >
4, whilestill converging to β s for p → ∞ . Figure 8 illustrates further de-pendencies of Eq. (25) on the material properties and grain sizeranges. A lower absorption e ffi ciency Q abs , o , caused by a morereflective surface due to a greater n , reduces the step factor y ,narrows the transparent regime and brings the results closer toEq. (32). A wavelength dependence Q abs,o ( λ ) ∝ λ − ξ , const inthe mm wavelength range, caused by n ( λ ) , const, results in anon-zero β → ξ for p <
3. This e ff ect is also seen in the numer-ical results for amorphous carbon in the top-left panel of Fig. 5.If the maximum grain radius is comparable to or less than thecharacteristic grain size s c , the opaque regime vanishes and β ( p )becomes shallower. If s max ≤ s ′ c , only the small-grain regime re-mains and β = β s = const. This is relevant for observed young
5. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths −4 −3 −2 −1 A b s o r p ti on e ff i c i e n c y , Q a b s l = m m Amorphous carbon MieSmall−grain asymptote l = c m Astronomical silicate10 −4 −3 −2 −1 l = m m Pyroxene 10 Grain size, s [µm] l = c m Water ice
Fig. 3.
Absorption e ffi ciencies for four materials and a range of wavelengths and grain sizes. Wavelength are separated by factors of10, the longest wavelengths are labelled. Solid lines indicate Mie results, dashed lines the small-grain asymptote given in Eq. (12).The upper boundaries of the light-shaded regions are defined by Eq. (13), the left boundaries by Eq. (15). Black dotted lines followEq. (13) multiplied by an additional factor n to roughly trace the actual smoothed Q abs . Values are given for stellar luminosity L = L Sun and a distance r =
100 au.protoplanetary discs where grain growth may not have reached s max > s c yet (e. g. Kr¨ugel & Siebenmorgen 1994; Natta & Testi2004; Draine 2006). It is also relevant for modelled discs, wheresystematic errors are introduced if the assumed s max is too small.The greater the minimum grain radius s min , the fewer smallgrains are present and the steeper is the size distribution requiredfor the small grains to dominate. With increasing s min , β ( p ) flat-tens (and p ( b ) steepens) around p ≈ β s is approached only for higher p . This dependence of β ( p ) on s min translates to a dependence on stellar luminosity ifthe latter is high enough to set a lower size cuto ff via radiationpressure blowout of the smallest grains. For materials with high n the e ff ect of di ff erent s min is smaller, as long as s min < s ′ c .Because of the jump by a great factor y from Q abs,s to Q abs,t , thesmall grains play a minor role and the exact location of theirlower size cut-o ff is not as important. Only as s min approaches orexceeds s ′ c or s c does the β relation react by converging quicklytowards a constant β ≈
0. This is the case for carbon grainsaround high-luminosity stars, where radiation pressure blowoutcan eliminate small and transparent grains completely. For ma-terials with low n , small grains and their lower size cuto ff aremore important. The lower s min , the more abundant are the smallgrains and the quicker is the convergence β ( p ) → β s aroundp ≈ ffi cient, as represented by A in the bottom row of Fig. 8, brings β ( p ) closer to Eq. (1) in therange 3 < p < n and low- n materials. For ma-terials with high n , this is due to a lower A causing a greater characteristic grain radius, s c , and a wider transparent regime.This e ff ect does not depend on s min or s max . For materials withlow n , where the transparent regime is less important, a lower A brings s c closer to s max and narrows the opaque regime. Thise ff ect for low n is similar to that shown in the top-right panel ofFig. 8. It disappears for s max ≫ s c . For a direct link between observed spectral index and size dis-tribution index, SEDs are computed for the range of parameters.As long as temperatures do not vary strongly across the relevantgrain sizes, SEDs and their slopes simply reflect the opacity in-dex and the slope of the blackbody function at that temperature.In Fig. 9 the analytic predictions for size distribution index p asa function of the observable spectral index in the mm regime, α mm , are compared to the numerical results. As expected fromthe close relation between β and α mm , the findings for the p ( α mm )relation reflect those for β ( p ) and the curves shown in Fig. 5.The full numerical solution typically falls in between cases (a)and (b), represented by Eqs. (32) and (1), respectively. With α mm (and β ) being less sensitive to p than predicted for case (b), p isin turn more sensitive to di ff erences or uncertainties in α mm . Allelse being equal, the size distribution indexes for systems withdi ff erent α mm di ff er more strongly than expected from Eq. (1).
6. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths −6 −4 −2 Amorph. carbon Astronomical silicate10 −6 −4 −2 PyroxeneWavelength, l [µm] 10 Water ice O p ac it y , k [ c m g − ] Wavelength, l [µm] p = 2.6 p = 3.0 p = 3.4 p = 3.8 p = 4.2fits Fig. 4.
Opacities for four di ff erent materials for a range of sizedistribution slopes and wavelengths. Values are given for stel-lar luminosity L = L Sun , a distance r =
100 au, and a totaldisc mass M d = . M Earth . The maximum object radius is set to s max =
100 km for water ice and 1 km for carbon and the silicates.Mie results are plotted with solid lines for five size distributionindexes from (top) p = . p = .
6. Sloped dashedlines trace power-law fits to wavelengths from λ = λ = λ and λ .The curves for p ( α mm ) are roughly linear for 3 . ≤ p ≤ . p ≈ + α mm − α Pl β e ff + ∆ p (33)in that range, underlining the similarity with Eqs. (32) and (1).The coe ffi cients β e ff and ∆ p for all assumed combinations ofmaterials, distances, and stellar luminosities can be read fromFig. 10. The dependence on stellar luminosity is mostly weakbecause grains near to s min , as defined by radiation pressureblowout, contribute only slightly to mm emission. One excep-tion is carbon with its high absorption e ffi ciency, which shifts thecharacteristic grain size close to or below the lower size limit. At L = L Sun , the values for β e ff are around 0.4, outside the plot-ted range. The other exception is the porous silicate, where n and ρ are low, the blowout limit higher, the small-grain regimemore important, and the dependence on stellar luminosity morepronounced.The following recipe can then be used for determining thesize distribution indexes from observed spectral energy distribu-tions: 1) pick a material (mix) that is appropriate for your system;2) derive the temperature of the big grains; 3) derive α Pl for thattemperature and your pair of wavelengths; 4) read β e ff and ∆ p o ff of Fig. 10; and 5) calculate p from Eq. (33).Figure 10 also shows a notable increase of β e ff for tempera-tures below ≈
30 K for all materials. This dependence on tem-perature is where the behaviours of α mm ( p ) and β ( p ) di ff er. Thefollowing section has a more detailed discussion. b » p − 3) + 0.09 b = 1.13 ( p − 3) b = p − 3 b » p − 3) + 0.12 b = 1.60 ( p − 3) b = p − 3 b » p − 3) + 0.04 b = 1.17 ( p − 3) b = p − 3 b » p − 3) + 0.28 b = 3.77 ( p − 3) b = p − 3 Size distribution index, p O p ac it y i nd e x , b Fig. 5.
Relation between the power-law indices of the grain sizedistributions and the total opacities between λ = λ = r =
100 au around a star with L = L Sun . Solid coloured lines show numerical results. Dottedlines of the same colours show linear fits for 3 . ≤ p ≤ .
8, withfit coe ffi cients indicated in the figure keys. Dashed lines showthe analytic approximation according to Eq. (30). The solid greylines trace the approximate Eq. (31). l = 0.25 mm, l = 0.85 mm b » p − 3) + 0.14 b = 1.98 ( p − 3) b = p − 3 Astron. silicate 2.8 3 3.2 3.4 3.6 3.8 4 00.511.522.5Size distribution index, p O p ac it y i nd e x , b l = 0.5 mm, l = 3 mm b » p − 3) + 0.11 b = 1.61 ( p − 3) b = p − 3 Fig. 6.
Same as the top-right panel in Fig. 5 (AstronomicalSilicate) but for two combinations of shorter wavelengths, as in-dicated.
Grain temperature influences the relation between size distribu-tion and SED in two ways. First, the optical properties can varywith temperature. The relation between opacity index and sizedistribution index thus depends on stellar luminosity and disc ra-dius. This variability in β e ff is visible in Fig. 10. The temperature-dependent water ice shows stronger variability in β s , β e ff , and ∆ p .The slope β s of pure water ice at low temperatures is so steep thateven a very steep SED could correspond to a moderate p . For car-bon, the two variants of Astronomical Silicate, and pyroxene, β s does not or only weakly depend on temperature, and β e ff is lessvariable. Second, the Planck function in Eq. (2) and the corre-sponding power-law index α Pl depend on temperature. Figure 11
7. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths O p ac it y i nd e x , b Size distribution index, p b fi b s b = b s ( p − ) b = p − b fi n = 8.50 n = 3.42 n = 1.1400.511.5 2.5 3 3.5 4 4.5 5 5.5 Fig. 7.
Slope of the opacity as derived from Eq. (25) for the fol-lowing parameters: λ = β s = . A = / cm, λ = ρ = . − , s min = . µ m, s max =
100 m, Q abs,o = y = . n , with (black) n = .
5, (red) n = .
4, and (blue) n = .
14. In the range 3 < p <
4, this order of colours reflectsthe vertical order of the corresponding lines, bottom to top. Thedashed black and solid grey lines follow Eqs. (30) and (31), re-spectively. O p ac it y i nd e x , b Size distribution index, p n = 8.50 b fi Q abs,o = 1 Q abs,o (cid:181) l −0.2 Q abs,o = 0.35 n = 1.14 s c ’ = 1.3 mm s c = 22 mm s max = 100 µm s max = 100 mm s max = 100 m n = 8.50 s c ’ = 0.18 mm s c = 0.4 mm s min = 100 µm s min = 10 µm s min = 1 µm n = 1.14 s c ’ = 1.3 mm s c = 22 mm s min = 100 µm s min = 10 µm s min = 1 µm n = 8.50 s c ’ = 0.18 mm a = 2/mm, s c = 0.18 mm a = 0.2/mm, s c = 0.4 mm a = 20/m, s c = 4 mm n = 1.14 s c ’ = 1.3 mm a = 2/mm, s c = 2.2 mm a = 0.2/mm, s c = 22 mm a = 20/m, s c = 0.22 m Fig. 8.
Same as Fig. 7, but with (top left) Q abs,o , (top right) s max ,(middle) s min , and (bottom) A varied. The left panels show re-sults for high n , the right panels for low n . The legends are or-dered vertically according to the order at p ≈ α Pl varies with observing wavelengths and tempera-ture. The colder the grains and the shorter the wavelengths, thestronger is the deviation of α Pl from the value of 2 found in theRayleigh limit. Any uncertainty in the physical grain tempera-ture therefore translates to an uncertainty in the derived size dis-tribution index. p = 3 + ( a mm − a Pl ) p » a mm − a Pl )/0.87 + 0.02 p = 3 + ( a mm − a Pl )/1.13 p = 3 + ( a mm − a Pl ) p » a mm − a Pl )/1.30 − 0.08 p = 3 + ( a mm − a Pl )/1.60 p = 3 + ( a mm − a Pl ) p » a mm − a Pl )/1.08 − 0.01 p = 3 + ( a mm − a Pl )/1.17 a mm S i ze d i s t r i bu ti on i nd e x , p Water ice p = 3 + ( a mm − a Pl ) p » a mm − a Pl )/3.23 − 0.07 p = 3 + ( a mm − a Pl )/3.77 Fig. 9.
Relation between the power-law indexes of the grainsize distribution and the observed spectrum for four materials.Parameters and line meanings are the same as in Fig. 5. Rangesfor the horizontal axes are di ff erent for the four panels, indicatedabove the top panels and below the bottom panels.Figure 12 shows the equilibrium temperatures resulting fromEq. (3). For grains with radii s & µ m temperature is nearlyconstant. These grains are large enough to emit e ffi ciently be-cause their Q abs is close to unity in the far infrared and submm.Their temperature is no longer a function of size. With theexception of very transparent materials, such as pure waterice, this temperature is close to the blackbody value (see, e. g.Pawellek et al. 2014). However, the temperature, which is as-sumed constant from Eq. 4 onwards, actually depends on grainsize. Smaller grains, which cannot re-emit as e ffi ciently, can besignificantly hotter. The top-left panel of Fig. 12 shows an ex-treme case, where the smallest carbon grains at 1000 au are hot-ter by up to a factor of five. When the size distribution is steepenough, the contribution from small ( . µ m) grains to themm emission is significant. The e ff ective blackbody spectral in-dex, determined from the weighted average over all grain sizes,increases for higher p because of the increased abundance ofsmaller, hotter grains. The α mm ( p ) relation inherits that depen-dence and steepens, while p ( α mm ) flattens, corresponding to ahigher β e ff . This e ff ects becomes more pronounced for distantdiscs or less luminous stars, where temperature and spectral in-dex are lower in general and the relative spread in T is higher.Hence, the β e ff curves in Fig. 10 rise as temperatures drop below ∼
30 K.For spatially resolved circumstellar rings, the temperature ofbig grains can be computed from the distance to the star and thestellar luminosity. In the more common case, where the disc isonly barely resolved or not at all, distance and temperature canonly be assessed through the SED. However, this procedure ispartly degenerate because the SED is already a convolute of sizeand temperature distributions. Determining grain temperature in-dependently from the size distribution can be di ffi cult. Near tothe flux density peak, which is around λ . µ m for cold
8. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths O ff s e t , D p E ff ec ti v e i nd e x , b e ff Temperature, T [K]Amorphous carbon1.251.31.35 Astronomical silicate l = mm , l = mm l = . m m , l = m m L Sun L Sun L Sun L Sun L Sun L Sun
10 100 −0.1−0.05 0 0.05
Fig. 10.
Regression coe ffi cients for linear fits to the modelled re-lation between observable spectral slope α mm and the size dis-tribution index p as defined in Eq. (33). Coloured lines indi-cate values for longer wavelengths, grey lines for short wave-lengths. Line width increases with stellar luminosity from L = . L Sun to L = L Sun .debris discs, grains with s < µ m or even s ≪ µ m con-tribute or dominate as long as p &
3. These warmer grains shiftthe emission peak to shorter wavelengths, masking the black-body curve of the bigger grains that dominate at longer wave-lengths. The overestimated temperatures result in overestimated α Pl and underestimated p for given α mm . As shown in Fig. 11 forobservations at 1 mm and 7 mm, assumed temperatures of 20 Kor 50 K instead of 10 K would result in α Pl = .
83 or α Pl = . α Pl = .
64. For β e ff ≈ . α mm , this would . . . . . . . . . . l / l m a x l / l max l = 1 mm, l = 7 mm l = 0.25 mm, l = 1.2 mm K K K K K K K K K K K K K K K K K K K Fig. 11.
Blackbody spectral indices as a function of combi-nations of wavelengths, normalized to temperature-dependentblackbody peak wavelengths λ max . Astronomical silicate10 Water ice10 Pyroxene102050100200500 1 au10 au100 au1000 au T e m p e r a t u r e , T [ K ] Grain radius, s [µm]Amorphous carbon102050100200500 Fig. 12.
Equilibrium grain temperature as a function of grain sizefor four materials and four distances around a star of solar lumi-nosity.correspond to errors in p of − .
15 or − .
22, respectively. TheSED analysis can be further complicated by the di ff erent temper-atures involved for debris or protoplanetary discs that are veryextended or composed of two or more separate rings of compa-rable luminosity.
9. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths −3 −2 −1 A b s o r p ti on e ff i c i e n c y , Q a b s Wavelength, l [µm] f = 1.0 f = 0.75 f = 0.5 f = 0.25 Fig. 13.
Absorption e ffi ciencies for porous spheres ofAstronomical Silicate with radii s = µ m and filling fac-tors f =
1, 0.75, 0.5, 0.25 (top to bottom at λ ≈ µ m). To illustrate the general impact of porosity, ddscat in version7.3.2 (Draine & Flatau 1994, 2013) is used to model sphericalgrains of Astronomical Silicate with filling factors f =
1, 0.75,0.5, and 0.25, corresponding to porosities of 0, 25, 50, and 75 %.The grain radius is fixed at s = µ m. The size of the individualdipoles was is to 0.5 µ m, much smaller than the wavelengths inthe considered range 30 µ m < λ < Q abs ( λ )curves in Fig. 13 show a trend towards lower Q abs values forlower filling factors over a wide range of wavelengths, consis-tent with lower absorption for less dense material. At long wave-lengths, the combined dependence of Q abs on n and k given inEq. 12 is reflected. At shorter wavelengths, in the opaque regime, Q abs is more a ff ected by n . There, lower filling factors result inlower n , reduced reflection, and hence, increased absorption.The transition from (s) small and transparent to (o) large andopaque is also a ff ected by porosity. For f = n is high enoughfor a direct transition from regime (o) to (s). For lower f andlower n the intermediate regime (t) becomes notable. With de-creasing f , the transition from (s) to (t) occurs at decreasingwavelengths because λ ∝ n for constant s ′ c (see Eq. 15). Thetransition from (t) to (o) occurs at λ ∝ k / n , i. e. λ decreases withdecreasing k , despite n levelling o ff at unity.For f →
0, as n →
1, regime (t) becomes indistinguishablefrom regime (s) because refraction becomes unimportant and theo ff -setting factor y approaches unity. The β ( p ) relation for porousmaterials is therefore expected to follow Eq. (30) more closelycompared to their compact counterparts, in line with the steepercurves for lower n seen in Fig. 7. As a result, porous grains ex-hibit higher values of β e ff and α mm when compared to compactgrains with the same size distribution index p . This e ff ect hasbeen shown by Brunngr¨aber et al. (2017) and is visible in Fig. 10.At the same time, the lower bulk density of porous material in-creases the blowout size, and hence, strengthens the dependenceon disc temperature and stellar luminosity. The coe ffi cient β e ff does therefore not converge towards β s , but keeps increasing forincreasing porosity (cf. Brunngr¨aber et al. 2017). Figure 14 shows the size distribution indexes resulting for a setof debris discs with observed α mm from Eq. (1) and from full Mie AU Mic e EriHD 61005HD 107146iHD 107146oHD 377HD 105q EriHD 104860AK ScoHD 15115iHD 15115oHD 181327 h CrvHD 95086 b PicHD 131835Fomalhaut49 CetHR 4796 A 3 3.5 4 4.5 O b j ec t Size distribution index, p b s = 1.8 (eq. 1): < p > = 3.36 b s = 1.6 (eq. 1): < p > = 3.40Amorph. carbon: < p > = 3.64Astron. Silicate: < p > = 3.42Pyroxene: < p > = 3.55Water ice: < p > = 3.16Porous ast. sil.: < p > = 3.39 Fig. 14.
Size distribution indexes as derived from observed spec-tral indices for the discs presented in Table 2. Grey plusses andcrosses mark the results obtained with Eq. (1) for β s = . β s = .
6, respectively. Black squares, filled and open redcircles, green upward and blue downward triangles indicate re-sults obtained with numerical Mie modelling for amorphous car-bon, compact and porous Astronomical Silicate, pyroxene, andamorphous water ice, respectively. Where symbols are missing,the observed α mm could not be reproduced within the range2 . < p < . α mm , L , disc radius r , wavelengths λ and λ , togetherwith the references. Temperatures, T , and black-body spectralindexes, α Pl , are computed for mm-sized grains of astronomicalsilicate. This comparison shows three main e ff ects.First, the spread in size distribution indexes for a set of discsis wider for the more accurate numerical model than for Eq. (1).Because typically 1 < β e ff < β s , Eq. (1) tends to underestimatethe sensitivity of p on α mm , while Eq. (32) overestimates it. Thenumerical model falls in between. Extreme values near to or be-yond the boundaries of the range 3 < p < ff ected evenmore because there the p ( α mm ) relation deviates more stronglyfrom linearity. The wider spread not only can make outliers moresignificant but also any potential trend, such as p ( L ∗ ) or p ( r ).Second, there is a systematic o ff set towards higher p valuesin some of the new results. The lines defined by Eq. (1) and thenumerical models typically cross around p = . . . . . h p i = . ± .
03 for Eq. (1)and β s = . h p i = . ± .
03 for β s = . h p i = . ± .
04 for astronomi-cal silicate and h p i = . ± .
05 for pyroxene. For the poroussilicate and the given set of discs, the mean (3.39) value is veryclose to that obtained for the simple β e ff = β s = . p values.When pyroxene is assumed instead of astronomical silicate, forexample, h p i is higher by 0.15 because the slope of the imagi-nary part of the refractive index of pyroxene is shallower.
10. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths
Table 2.
Observed spectral indexes for a set of debris discs.
System L ∗ Ref. λ λ α mm Ref. r Ref. T α Pl [ L ⊙ ] [mm] [mm] [au] [K]AU Mic 0.1 1 1.3 9.0 < .
46 1 40 11 26 1.87 ǫ Eri 0.3 1 1.3 7.0 > .
39 7 69 12 26 1.87HD 61005 0.5 1 1.3 9.0 2 . ± . ± . ± > .
39 1 50 15 42 1.92HD 105 1.2 2 0.87 9.0 2 . ± Eri 1.2 1 0.87 7.0 2 . ± . ± . ± . ± . ± . ± η Crv 6.6 3 0.85 9.0 2 . ± . ± β Pic 8.7 1 0.87 7.0 2 . ± . ± . ± . ± > . ± References. (1) MacGregor et al. (2016), (2) Marshall et al. (2018), (3) Marino et al. (2017), (4) Hung et al. (2015b), (5) Gerbaldi et al. (1999),(6) Gaia Collaboration et al. (2016, 2018, parallaxes from GAIA Data Release 2), (7) MacGregor et al. (2015), (8) Ricci et al. (2015), (9)Marshall et al. (2017), (10) Marino et al. (2017), (11) Matthews et al. (2015), (12) Booth et al. (2017), (13) MacGregor et al. (2018), (14)Marino et al. (2018), (15) Liseau et al. (2010), (16) Steele et al. (2016), (17) MacGregor et al. (2019), (18) Marino et al. (2016), (19) Su et al.(2017), (20) Matr`a et al. (2019), (21) Hung et al. (2015a), (22) MacGregor et al. (2017), (23) Hughes et al. (2017), (24) Kennedy et al. (2018).
4. Discussion
Most of the results from the previous sections concern the dis-crepancies between a full numerical model and an analytic ap-proximation based on power laws. The approximations leadingto Eq. 1 tend to underestimate the mean slope and the spreadof slopes of the grain size distributions for given observed spec-tral indexes. Assuming compact grains, the true size distributionslopes are somewhat steeper on average, which has implicationsfor the processes that shape the size distribution. Values for p > . Q ∗ D decreasing and / or collision ve-locities increasing with grain size (O’Brien & Greenberg 2003;Pan & Schlichting 2012; G´asp´ar et al. 2012). Values for p < . ff ect on the size distribution (Reidemeister et al. 2011;Wyatt et al. 2011). The same is true for erosion by sublimation orUV sputtering (e.g. Grigorieva et al. 2007; Potapov et al. 2018),and for a stronger dynamical excitation of small grains by ra-diation pressure (Th´ebault & Wu 2008). A correction towardshigher values of p could thus mean that collisions are more im-portant, the according timescales shorter than that of drag anderosion.A steeper size distribution would automatically imply a re-duced overall disc mass when extrapolating from dust to plan-etesimals. If p were higher by 0.1, the mass ratio between km-sized objects and mm-sized dust would be reduced by a factor of4. This could be one of several aspects that help reduce the sever-ity of the problem with the unrealistically high extrapolated discmasses that are derived for most of the more luminous debrisdiscs (e.g. Krivov et al. 2018).Additional uncertainties arise from a poor understanding ofthe dust composition and, hence, of its optical properties. On theone hand, the real part of the refractive index, n , determines the slope 1 /β e ff of the p ( α mm ) dependence by defining the absorp-tion and emission behaviour around the transition from transpar-ent to opaque. For materials with higher n , such as non-porousamorphous carbon or silicates, absorption of mid-sized grains ishigher. The transition from small to opaque is more abrupt, cor-responding to the two characteristic radii, s ′ c and s c , being closerto one another. The transparent regime is more distinguishedfrom the small-grain regime, but narrower. The coe ffi cient β e ff is closer to unity. For materials with n closer to unity, such aswater ice or porous silicate, the transparent intermediate regimeis less pronounced, but wider, and hence β e ff closer to β s , thepower-law index of the volume absorption coe ffi cient A . The fi-delity of Eq. 1 thus depends on the material. Closer to the stars,where high- n silicates are expected to dominate, the deviationthat comes with approximation (1) should be higher. Far fromthe stars, where low- n volatile ices can exist in abundance, thedeviation should be lower. Figure 10 shows that the parameter re-gion where Eq. 1 (with constants β s and α Pl ) is fully applicableis, however, very small.On the other hand, β s is itself a material property andthus depends directly on, for example, the silicate-to-ice ratio.Measurement and usage of better optical data in the mm regimeis required. Some of the commonly used data sets have neverbeen designed to be used at such long wavelengths. The icedata set that was derived from a model fit to the β Pictorisdisc (Li & Greenberg 1998), for example, inherits the uncer-tainties in the assumed size distribution. Figure 2 shows thatthe adopted k ∝ λ − . is flatter than the slope actually mea-sured for ice at a few tens of K, leading to an overestimationof the size distribution index p . Temperature dependence inoptical data at mm wavelengths has been discussed for sometime (e.g. Mishima et al. 1983) but only recent observations athigher resolution and sensitivity have brought the issue backinto focus. Consequently, temperature-dependent data for more
11. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths and more materials are becoming available (Reinert et al. 2015;H¨aßner et al. 2018; Mutschke & Mohr 2019).Strong uncertainties remain even if a full Mie model is usedand the grain composition is known. Two of the very basic as-sumptions, that of solid, spherical grains and that of a power-lawsize distribution, can be too crude. As shown in Sect. 3.6, thetransition from absorption by cross-section to absorption by vol-ume is smoother for filling factors f . .
5, which is more inline with the assumptions that lead to Eq. (1). The lower bulkdensity and the lower refractive index, however, cause smallergrains and the lower size cuto ff imposed by radiation pressureto become more important. The resulting relation between ob-served α mm and deduced p flattens (corresponding to a higher β e ff ) for lower disc temperatures and stellar luminosities. If aconstant β e ff = β s were imposed for porous grains, the sizedistribution index would be underestimated for less luminousstars or distant discs and overestimated for more luminous starsor closer discs, causing an artificial trend in disc statistics. If,indeed, porosity is significant, all results based on Mie theoryof compact spheres end up su ff ering from a systematic error.Brunngr¨aber et al. (2017, their Fig. 3) show that the size distri-bution index for a filling factor of f = . , derived based on theassumption of solid grains, is already too high by 10 to 15 %, or,in absolute terms, 0.35 to 0.5 higher than their reference valueof 3.5. However, their curves do not converge for even higherporosities (lower filling factors), indicating that there is no sim-ple porosity threshold above which Eq. (1) is applicable and be-low which it is not.Dust size distributions may deviate strongly from powerlaws when radiation pressure blowout introduces a dis-continuity in the collisional cascade, resulting in a wavypattern (Campo Bagatin et al. 1994; Durda & Dermott 1997;Th´ebault et al. 2003; Krivov et al. 2006). Parts of that wavinessare carried over to the total opacity and the observable SED(Th´ebault & Augereau 2007). The narrower the range of grainsizes that e ff ectively contribute to emission at any given wave-length, the more of the waviness is preserved. This means thata wavier, more expository SED is produced by the more abrupttransition from surface absorption to volume absorption that isseen for materials of higher optical density. For materials orwavelengths with a lower opacity, the convolution over a widerrange of grain sizes reduces the waviness. In addition, in dy-namically cold debris discs, smaller grains are under-abundantbecause they su ff er from dynamical excitation due to radiationpressure (Th´ebault & Wu 2008). If the transition from a ff ectedsmall to una ff ected big grains occurs in the (sub)mm size range,the average slope derived from the SED analysis will be shal-lower than the background power-law slope of an infinite, unper-turbed collisional cascade.Models ought to take into account that the characteristicgrain size s c can easily exceed λ by orders of magnitude (cf.Draine 2006): s c ≈ λ/ (16 k ) for n ≈ s ′ c < s c in Eq. 16.Assuming a moderate k ∼ − for water ice at λ close to 1 cm, s c is already in the order of 600 λ , that is, several metres. Forthe more opaque compact carbon and silicates, where k & . s c . λ . A porous blend of these materialcan be expected to fall in between these extremes. With the SEDbeing dominated by grains with radii around s c for 3 < p < s max in numerical and analytic modelsshould always be set such that s max ≫ s c or reflect a physicallymotivated maximum. If the assumed s max is too small, say 1 mm,a lack of flux results at the longer wavelengths. The resulting syn-thetic SED will be too steep, requiring a flatter-than-actual sizedistribution when fit to observational data. On the same note, any dust mass derived from mm or cm observations is a mass in cmor dm-sized grains (if these are present), which sets the startingpoint from which to extrapolate towards a total disc mass. Theoften quoted mass in mm grains and the total disc mass wouldthen be lower by a factor of (0 . . . . . p − ≈ . . . . . p = . s c and s max imply that observations in the mm-cm wavelength range are sensitive to much bigger, dm-sizedgrains. Any derived size distribution index would also coverthat size range, including potential changes in slope or cuto ff s.Models for planetesimal formation through pebble aggregation,such as the streaming instability scenario (e. g. Johansen et al.2015), predict a transition from stronger individual pebbles toweaker and, hence, less abundant pebble piles around cm sizes(cf. Krivov et al. 2018). This transition would translate to a steep-ening of the SED in the wavelength range of interest. Below andabove the transition size from pebbles to piles, where collisionphysics are assumed continuous, the size distribution would beflatter. At earlier phases, when the pebbles are still forming fromsmaller grains, s max is a measure of the actual maximum grainsize. The opacity curves and observed SEDs are steeper early on,flattening as s max grows with time (e. g. Kr¨ugel & Siebenmorgen1994; Natta & Testi 2004; Draine 2006). At wavelengths forwhich optical depths & p ( α mm ) derived in Sects. 2 and 3 areinvalidated.
5. Summary
When analysing SEDs of individual systems, a full numericalapproach is the most flexible in terms of materials and assumedsize distributions. The computational cost of Mie calculations islow. If only a quick estimate of the size distribution index and itsuncertainties is sought, Eq. (33) can be used in combination withthe coe ffi cients provided in Fig. 10. These coe ffi cients are fit-ted to the numerical results for moderate size distributions with3 . < p < . Acknowledgements.
TL is grateful to H. Mutschke for fruitful discussions ondust optical properties. Parts of this work have been supported by the
DeutscheForschungsgemeinschaft (grants LO 1715 / / References
Bohren, C. F. & Hu ff man, D. R. 1983 (New York: Wiley)Booth, M., Dent, W. R. F., Jord´an, A., et al. 2017, MNRAS, 469, 3200Bruggeman, D. A. G. 1935, Annalen der Physik, 416, 636Brunngr¨aber, R., Wolf, S., Kirchschlager, F., & Ertel, S. 2017, MNRAS, 464,4383Campo Bagatin, A., Cellino, A., Davis, D. R., Farinella, P., & Paolicchi, P. 1994,Planet. Space Sci., 42, 1079Dohnanyi, J. S. 1969, J. Geophys. Res., 74, 2531Dominik, C., Blum, J., Cuzzi, J. N., & Wurm, G. 2007, in Protostars and PlanetsV, ed. B. Reipurth, D. Jewitt, & K. Keil, 783
12. L¨ohne: Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths