Silicate Extinction Profile Based on the Stellar Spectrum by Spitzer/IRS

The 9.7$\mu m$ and 18$\mu m$ interstellar spectral features, arising from the Si--O stretching and O--Si--O bending mode of amorphous silicate dust, are the strongest extinction feature in the infrared. Here we use the"pair method"to determine the silicate extinction profile by comparing the \emph{Spitzer}/IRS spectra of 49 target stars with obvious extinction with that of un-reddened star of the same spectral type. The 9.7$\mu m$ extinction profile is determined from all the 49 stars and the 18$\mu m$ profile is determined from six stars. It is found that the profile has the peak wavelength around $\sim$9.2- 9.8$\mu m$ and $\sim$18-22$\mu m$ respectively. The peak wavelength of the 9.7$\mu m$ feature seems to become shorter from the stars of late spectral type, meanwhile the FWHM seems irrelevant to the spectral type, which may be related to circumstellar silicate emission. The silicate optical depth at 9.7$\mu m$, $\Delta\tau_{9.7}$, mostly increases with the color excess in $J-K_S$ ($E_{\rm JK_S}$). The mean ratio of the visual extinction to the 9.7$\mu m$ silicate absorption optical depth is $A_{\rm V}/\Delta\tau_{9.7}\approx 17.8$, in close agreement with that of the solar neighborhood diffuse ISM. When $E_{\rm JK_S}$>4, this proportionality changes. The correlation coefficient between the peak wavelength and FWHM of the 9.7$\mu m$ feature is 0.4, which indicates a positive correlation considering the uncertainties of the parameters. The method is compared with replacing the reference star by an atmospheric model SED and no significant difference is present.


INTRODUCTION
Silicate as well as carbon dust is the major component of interstellar dust.It can be generally divided into two types, amorphous and crystalline by their structure, while the highly dominant type in the ISM is amorphous (Li & Draine 2001) that shows two prominent broad absorption features around about 9.7 and 18 µm respectively (e.g.Gillett et al. 1975;Kemper et al. 2004;Chiar & Tielens 2006;Min et al. 2007) due to the Si-O stretching and O-Si-O bending vibration.In comparison, crystalline silicate exhibits multiple distinct narrow bands from ∼ 10 − 60 µm (Molster & Kemper 2005;Henning 2010;Liu et al. 2017) in almost all circumstellar environments.
The extinction profile caused by interstellar silicate has been analysed with different samples, which lead to inconsistent results on the peak wavelength λ peak and the FWHM.Kemper et al. (2004) analyzed the ∼ 8-13 µm spectrum of the Galactic center source Sgr A ⋆ obtained with the Short Wavelength Spectrometer (SWS) on board the Infrared Space Observatory (ISO).They derived the 9.7 µm silicate absorption profile of Sgr A ⋆ with the peak at ∼ 9.8 µm and the FWHM of ∼ 1.73 µm by subtracting a fourth-order polynomial continuum from the observed spectrum.They suggested that the narrow profile could be caused by the contamination of the silicate emission intrinsic to the Sgr A ⋆ region.Chiar & Tielens (2006) studied the ∼ 2.38-40 µm ISO/SWS spectra of the diffuse ISM along the lines of sight towards four heavily extinguished WC-type Wolf-Rayet stars.They found that the 9.7 µm silicate absorption features of the four sources all peak at ∼ 9.8 µm, but their widths vary from ∼ 2.35 µm for WR98a to ∼ 2.7 µm for WR104, apparently wider than ∼ 1.73 µm of Sgr A ⋆ .McClure (2009) derived the ∼ 5-20 µm extinction curves from 28 G0-M4 III stars lying behind the Taurus, Chameleon I, Serpens, Barnard 59, Barnard 68, and IC5146 molecular clouds by comparing the observed spectrum by the Infrared Spectrograph (IRS) on board the Spitzer Space Telescope with the stellar photospheric model spectrum of Castelli et al. (1997).She found that the silicate extinction profile in the regions of A K S < 1mag peaks at ∼ 9.63 µm and has a FWHM of ∼ 2.15 µm whereas, in regions of 1 <A K S < 7mag, it peaks at ∼ 9.82 µm and has a FWHM of ∼ 2.72 µm.This means that the 9.7 µm silicate extinction feature appears to be broadened in more heavily extinguished regions.Olofsson & Olofsson (2011) derived the mid-IR extinction curve for a highly obscured M giant (# 947) behind the dark globule B335 (R V ≈ 4.9, A V ∼ 10 mag), using the ∼ 7-14 µm Spitzer/IRS spectrum complemented by the ∼ 5-16 µm spectrum obtained with the ISOCAM/CVF instrument on board ISO.Their result is even more special in that the 9.7 µm feature peaks at ∼ 9.2 µm and has a FWHM of ∼ 3.80 µm, i.e. the ever shortest peak wavelength and the largest width of the silicate profile.Also based on the Spitzer/IRS data, Van Breemen et al. (2011) investigated the silicate absorption spectra of three sightlines towards diffuse clouds and four sightlines specifically towards the Serpens, Taurus, and ρOphiuchi molecular clouds.They found that the 9.7 µm silicate absorption bands in the diffuse sightlines show a strikingly similar band shape and all closely resemble that of Sgr A * (Kemper et al. 2004), while the 9.7 µm features in the molecular cloud sightlines differ considerably from that of Sgr A * by peaking at ∼ 9.72 µm and having a FWHM of ∼ 2.4 µm.More recently, Fogerty et al. (2016) analyzed the Spitzer/IRS spectra of the 9.7 µm silicate optical depths of the diffuse ISM along the line of sight towards Cyg OB2-12, a heavily extinguished luminous B5 hypergiant with A V ≈ 10.2 mag, and towards ζ Ophiuchi, a lightly extinguished bright O9.5 star with A V ≈ 1mag.They found appreciable differences between the spectral profile of the 9.7 µm silicate absorption towards Cyg OB2-12 and ζ Ophiuchi; while the former peaks at ∼ 9.74 µm and has a FWHM of ∼ 2.28 µm, the latter peaks at ∼ 9.64 µm and has a FWHM of ∼ 2.34 µm.Moreover, the contrast between the feature and the absorption continuum of the former exceeds that of the latter by ∼ 30 per cent.Recently Hensley & Draine (2020) show a new analysis of archival ISO-SWS and Spitzer IRS observations of Cyg OB2-12 using a model of the emission from the star and its stellar wind to determine the total extinction A λ from 2.4-37 µm.They derived the FWHM of the 9.7 µm silicate absorption profile being 2.23 µm, consistent with the result of Fogerty et al. (2016).
We investigated the 9.7 µm silicate absorption profile using the Spitzer/IRS observations of five early-type stars and found that their peak wavelength concentrates at 9.7 µm, and the FWHM is divided into two groups, 2 µm for three stars and 3 µm for two stars (Shao et al. 2018).The possible reasons for the diversity were discussed but no solid conclusions could be drawn.
In summary, previous studies about the peak and FWHM of silicate dust extinction profile yield inconsistent conclusions.The peak varies from ∼ 9.2 to ∼ 9.82 µm, and the range of FWHM fluctuates even more from ∼ 1.73 to ∼ 3.8 µm.Moreover, no pattern of variation could be found.From the technical side, various works use different data, including the dispersion of stellar type, sightline direction, and correspondingly interstellar environment.In addition, the method to derive the extinction profile is different.In this work, we try to investigate the systematic variation of the silicate extinction profile by including a significantly large sample of tracing stars which span a wide range of spectral type and sightlines.The method will be described in Section 2, and in Section 3 the data is introduced.Then the results and discussion will be shown in Section 4.

METHOD
The interstellar UV/optical extinction curve is often determined by comparing the spectrum of a reddened star with that of an unreddened star of the same spectral type, which is the so-called "pair method".There are additional methods to derive the intrinsic spectrum, including the atmosphere model and polynomial fitting (McClure 2009;Kemper et al. 2004).In this work, we continue to use the "pair method"to calculate the ratio of the color excess E(λ − K S )/E(J − K S ) where J and K S are the two near-infrared bands centering at 1.2 and 2.2µm respectively.The details of this method were described by Shao et al. (2018).In brief, the spectrum of the target source that has obvious silicate extinction is compared with a reference source that is of the same spectral type and the same luminosity class but with negligible extinction.However, such reference star cannot always be found because the Spitzer/IRS spectrum is available only for a limited number of stars.In such case, spectral type has the priority over luminosity class since it is the primary factor to decide the intrinsic spectral energy distribution.The one-to-one pair is listed in Table 1 with their spectral type, which shows that most of the targets are well paired with the reference.
The apparent stellar spectrum F λ is the intrinsic spectrum F 0 λ dimmed by the interstellar extinction A λ and the geometrical distance d: where R is stellar radius.If the reference star has the same intrinsic spectrum as the target star, then the intrinsic flux ratio of the star in the K S and λ band can be replaced by that of the reference star, i.e., F 0 where "ref" refers to the reference star.By comparing the observed spectrum of the source star with the reference star, the color excess between the λ and K S band, E(λ − K S ), is obtained as: where m K S and m ref K S are respectively the apparent K S -band magnitudes of the source star and the reference star.Since we are considering the color excess E(λ − K S ) instead of the absolute extinction A λ , the distance d to the source star (or the reference star) is cancelled out in Eq. 2.
After the extinction curve is determined, the peak wavelength and the FWHM of the 9.7 µm and 18 µm features are calculated in the same way as Shao et al. (2018), i.e. by subtracting the continuum.The continuum is fitted by a power law to the brightness in the JHK S bands and 6-7.5 µm extinction with an index of -1.06 (Gordon et al. 2023).It should be mentioned that the method to determine the continuum would influence the results.Linear fitting has also been adopted in other works, which works well for high quality spectrum.A constant index implies that the continuum extinction law does not change with the source, which may not be true.However, as we are interested in the silicate profile, this can reflect its relative variation.

THE TARGETS
The data are taken from the ∼16,900 low-resolution Spitzer/IRS spectra (Houck et al. 2004;Werner et al. 2004).These spectra were merged from four slits: SL2 (∼5.21-7.56µm), SL1 (∼7.57-14.28µm), LL2 (∼14.29-20.66µm), and LL1 (∼20.67-38.00µm), and some spectra are available only in the first two or three slits.Chen et al. (2016) cross-identified the types of these objects in the SIMBAD database, supplemented with the photometry by the 2MASS and WISE all-sky surveys.This cross-identification resulted in a database of 126 O stars, 414 B stars, 806 A stars, 453 F stars, 543 G stars, 1397 K stars, and 1260 M stars, which forms the initial catalog of our selection of target and reference stars.We take the following approaches to select our sample: 2. Require the signal-to-noise ratio (S/N) of the Spitzer/IRS spectrum in SL1 and SL2 to be ≥ 25.The signalto-noise ratio is quoted from the IRS spectrum, and for more specific information see (Houck et al. 2004).For multiple observations, we select the observation with the highest SNR and remove other repeated observations.Stacking the repeated observation may increase the S/N, but taking one single spectrum can have the highest accuracy because different observations can have systematic error due to, e.g.instrument instability and calibration process.Indeed, the three sources with repeated observations, specifically Elia 3-3, BD+43 3770 and HD 147889, all had the highest SNR > 70, and no stacking process is performed.Nevertheless, both stacking and single spectrum is used by various works.This removes 5 O stars, 11 B stars, 47 A stars, 60 F stars, 97 G stars, 209 K stars, and 51 M stars.
3. Require the color excess E(J − K S ) > 0.3 mag, corresponding to a line-of-sight extinction of A K S > 0.2 mag, which is comparable to three times of the photometric error.While the peak extinction of the 9.7 µm silicate feature is comparable to A K S , the extinction of the blue and red wings of the 9.7 µm silicate feature drops by a factor of >2 (e.g., see Xue et al. 2016).With A K S > 0.2mag, the entire 9.7 µm silicate extinction profile should be measurable.The color excess E JKS (the short for E(J − K S )) is calculated by using the intrinsic color index C 0 JKS according to the object's spectral type listed in the Allen's Astrophysical Quantities (Allen & Cox 2000).Figure 1 displays the distribution of E JKS , and it can be seen that only a third of the sources satisfy this criterion.The E JKS of some sources is less than 0, which may be caused by the photometric error as most negative E JKS is within the range of three times the photometric error.The median of the negative E JKS is -0.04, which can be taken as the uncertainty of this color excess.This uncertainty would be transferred to 0.02 mag in A K S ∼ 0.5E JKS .This criterion E(J − K S ) > 0.3 corresponds to a significance level of 7 sigma.This restriction removes more sources, leaving only 13 O-type stars, 21 B-type stars, 4 A-type stars, 5 F-type stars, 4 G-type stars, 12 K-type stars, and 44 M-type stars.
In addition, silicate dust exists in circumstellar envelop as well as in interstellar medium.The presence of circumstellar silicate can deform interstellar extinction mostly by emission.With the geometric distance from the Gaia/EDR3 catalog and correcting the extinction A K S converted from E JKS , the color-magnitude diagram, J − K S vs. M KS of the above selected stars is displayed in Figure 2. Apparently, some stars are red (super)giants that may have circumstellar dust resulted from stellar wind, which should be removed.Xue et al. (2016) found that the stars with circumstellar silicate emission appear to display apparent color excess at K S − W 3 (W 3 is the WISE band centering around 12 µm) in the J − K S vs. K S − W 3 diagram (see Fig. 19 in Xue et al. 2016) due to the silicate broad features around 9.7 µm.Following this criterion, the K S − W 3 vs.J − K S diagram is plotted in Figure 3 to identify the emission of the circumstellar silicate.In general, the color index K S − W 3 is proportional to J − K S as expected from interstellar extinction.However, the existence of circumstellar silicate emission would increase the brightness in W 3 and correspondingly the color index K S − W 3. According to the reference stars, the threshold line is determined by connecting the two points with the largest color index K S − W 3, which also coincides with the trend line of Xue et al. (2016) to determine whether there is circumstellar dust emission (see Figure 3).The stars above the line are considered to have circumstellar dust emission and removed.It should be kept in mind that this operation cannot remove all the sources with silicate emission since an interstellar absorption can weaken the emission and lead to a normal color index K S − W 3.
Finally, we select 49 stars as the target sources for studying silicate dust extinction, consisted of five O stars, thirteen B stars, three F stars, four G stars, nine K stars, and fifteen M stars.Their infrared photometric magnitudes are listed in Table 2 together with the reference stars.Their spectrum flux reliability is further examined by the WISE photometry (c.fTable 2).For each target star, the photometric fluxes in the four WISE bands at 3. 35, 4.60, 11.56, and 22.08 µm agree with the Spitzer/IRS spectra for most stars.A few stars' photometric fluxes in the W4 band deviate from the spectra as shown in Figure 4 and 5 for those ending at 38 µm, 21 µm and 14 µm separately, which is attributed to the photometric error in the W4 band due to its relatively poor quality.The derived 49 extinction curves are displayed in Figure 6 for the 9.7 µm feature only and Figure 7 for those including the 18 µm feature.All the 49 curves cover the 9.7 µm feature with reliable results, meanwhile only six of them show a relatively evident 18 µm profile because of the much smaller extinction and the limitation of data quality at longer wavelength.The peak wavelength (λ peak ), full width at half maximum (F W HM ), and optical depth at 9.7 µm (∆τ 9.7 ) of the silicate feature are listed in Table 3.It is worth noting that the optical depth ∆τ 9.7 and FWHM here are those of the silicate features after subtraction of the continuum spectrum and the optical depths ∆τ 9.7 refers to the value at the actual peak.The parameter definitions for the 18 µm feature are alike.
The uncertainties of these feature parameters are calculated by the bootstrap method which repeats the analysis 1000 times by re-sampling, i.e. by adding synthetic noise to the previously-taken observations and reanalyzing.The synthetic noise follows the prior that the noise amplitudes obey the Gaussian distribution around the observed IRS flux with the sigma being the flux error.The standard deviation of the 1000 results is taken as the uncertainty (σ later) of the curve.The left panel of Figure 8 shows two extinction curves with the error of TIC 345429046 and HD 229238.Depending on the spectral and photometric data quality of the target and reference star, the error of the extinction curve differs significantly.The error for HD 229238 is relatively small, on the order of 0.05 in A λ /A K S .However, for the case that both the target and reference stars have large error, the error of A λ /A K S can be up to 0.4.The two examples in left panel also show that the extinction curves of different line of sight directions are different, although taking into account the existence of errors, the actual difference may be smaller than it looks, but this difference cannot be fully explained by the error.The right panel of Figure 8 shows the average extinction curve from 49 curves weighted by the uncertainty, the shaded area represents 1σ uncertainty in the weighted average, where the sigma is the median sigma of the corresponding 49 sources at the same wavelength.It can be seen that the effect of error on the extinction curve is apparent.The error also affects the silicate extinction profile parameters, which will be discussed in section 4.2.Considering that the extinction of silicate is closely related to the total extinction of stars, these sources are divided into two groups according to whether the E(J − K S ) value is greater than 1.5.The average extinction curve weighted by the error is calculated respectively, and shown on the right panel of Figure 8.The group with E(J − K S ) < 1.5 has smaller extinction with bigger error than that with E(J − K S ) > 1.5.When weighted by the uncertainty, sources with E(J − K S ) > 1.5 have a larger average extinction in most infrared bands, but are slightly lower around the peak of 9.7 µm.Because the sources with smaller E(J − K S ) have larger errors, their weights and influence are reduced in the average extinction curves so that the average curve closely resembles the case for E(J − K S ) > 1.5.
Figure 9 shows the average extinction curves from the 49 and 6 curves respectively, weighted by 1/σ 2 (The red and blue lines in Figure 9).The global trend agrees with the previously established curve.From 2 µm on, the extinction decreases sharply with the wavelength until the local minimum at ∼ 7 µm, then it begins to increase because of the silicate feature and reaches the peak at ∼ 9.7 µm, and the peak extinction of the silicate profile is comparable to that in the K S band.In general, the peak of the extinction curve is below our former result (Shao et al. 2018) and Hensley & Draine (2020), but is consistent with Gordon et al. (2021), which also used the Spitzer/IRS spectrum with pair method.While Gordon et al. (2021) choose 16 O-type or B-type stars, our sample include all spectral-types.Early-type stars generally have a spectrum with fewer lines than late-type stars, but their continuum includes the free-free emission from ionized wind which needs to be treated very carefully (Hensley & Draine 2020).Between 5-7.5 µm, the extinction curve is higher than Gordon et al. (2021)'s and consistent with the extinction value calculated from photometric data (Lutz 1999;Indebetouw et al. 2005;Xue et al. 2016) and Hensley & Draine (2020).On the other hand, the curve is below Hensley & Draine (2020)'s, especially around the peak region.It may be understood since the Hensley & Draine (2020) curve is derived from the observation of Cyg OB12-2, a highly buried star in Cygnus while our extinction law is the average of the 49 extinction curves.This difference may indicate the variation of the silicate extinction profile with sightline.

The Profile around 18 µm
The extinction at the peak of the 18 µm feature is about half of the 9.7 µm feature.This significant decrease in the intensity strength as well as the instrument sensitivity leads to that only six extinction curves exhibit prominent 18 µm feature displayed in Figure 7.As can be seen from this figure, the extinction profile of silicate dust around 18 µm varies apparently.The peak wavelength fluctuates from 18 µm to 22 µm.The extinction profile is more complicated, where the wavy shape is possibly caused by the low S/N.The average extinction curve from 6 curves in Figure 9 weighted by 1/σ 2 shows that the peak wavelength and the profile are similar to that of Hensley & Draine (2020), but the peak value is a little bit lower.

Comparison with the models
As some works take the stellar model spectrum as the reference, we compared the results by using the observational spectrum of non-reddening star as the reference with using the theoretical spectrum by the ATLAS9 atmosphere model (Kurucz 2014).The comparison is displayed in the right panel of Figure 9.The extinction curves derived with stellar atmosphere model and with the observational spectrum are highly consistent at the 9.7 µm and lower around 18 µm.A visible difference is that the curve from stellar model shows a more prominent feature at ∼ 13.2 µm.The agreement indicates that there is no significant difference in taking the stellar model or the observed spectrum as the reference.
The extinction curves are also compared with that derived from the dust model by Weingartner & Draine (2001).As shown in the right panel of Figure 9, the curve is basically consistent with the WD01 curve with R V =5.5 at 9.7 µm and higher around 18 µm, but much higher than that with R V =3.1.This agrees with the conclusions of previous studies of the average infrared extinction law (e.g.Xue et al. 2016).
Regardless of whether the pair method or an atmospheric model method, we can find a tiny bump feature in the range of 12.7-14.1 µm.van Dishoeck et al. ( 2023) reported two broad bumps centered at 7.7 and 13.7 µm from protoplanetary disk by JWST, which are due to the ν4 + ν5 and ν5 bands of C 2 H 2 (Tabone et al. 2023).It seems that this tiny bump coincides with the C 2 H 2 feature around 13.7 µm, but the lack of the 7.7 µm feature challenges such identification.The JWST/MIRI spectra in McClure et al (2023) show a broad absorption feature near 11.5 µm, which can plausibly be attributed to libration in H 2 O ice.The extension of this feature to long wavelength may partially explain the tiny bump feature at 12.7-14.1 µm, but the spectra have low S/N and this feature can hardly extend to 14 µm.The nature of this feature needs further study.4.2.Relation of the Peak and the FWHM of the 9.7 µm Feature The dependence of the silicate profile on the color excess is examined since the silicate grains may be different with the environment density or temperature or radiation field.Figure 10 presents the silicate optical depth at 9.7 µm and the color excess E JKS .The silicate optical depth increases with E JKS as expected since both are proportional to the total extinction.The whole trend follows the relation of the diffuse medium, which coincides with the result of Chiar et al. (2007) for diffuse medium (The green dashed line in Figure 10).However, when E JKS becomes larger than 4, the silicate optical depth at 9.7 µm does not seem to increase with the color excess.This may imply that the stars with large color excess have circumstellar silicate emission to reduce the 9.7 µm extinction in spite of that we already tried to remove the stars with excess emission in the WISE/W3 band.Chiar et al. (2007) found similar bifurcating, and they attributed it to the effect of dust grain growth in the high extinction regions.The dust grain growth increases the near-IR extinction while has no effect on the silicate feature.
The optical extinction in the V band was found to be proportional to the optical depth at 9.7 µm for the diffuse interstellar medium, quantitatively A V /∆τ 9.7 ≈ 18.The value of A V of the targets is calculated by the relation E JKS /A V ≈ 0.165 (Chiar et al. 2007) and labeled by the upper abscissa in Figure 10.Consequently, this line implies the ratio A V /∆τ 9.7 ≈ 17.83, consistent with previous results.It should be mentioned that this ratio depends on the conversion relation between E JKS and A V , and can differ with the conversion factor.On the other hand, this ratio increases when the extinction at V band A V is particularly large.One possible reason is mentioned above, i.e. this group is consisted of late-type stars with circumstellar silicate emission that fills up the absorption.Another possible reason is the dust size effect as suggested by Chiar et al. (2007).The A V /∆τ 9.7 ratio for pure silicate dust is ∼ 1 for very small (a∼ 0.01 µm) particles, rising to ∼ 5 for classical (a∼ 0.15 µm) grains (e.g.Stephens 1980;Gillett et al. 1975;Shao et al. 2018).By adding carbonaceous dust with M carb /M sil >0.15, this ratio can rise to 18 (Shao et al. 2017).So another possible reason for the larger ratio A V /∆τ 9.7 is that the medium in these sightlines contains more carbon dust.
On the correlation between the peak wavelength and the FWHM of the feature, Gordon et al. (2021) thought they are well correlated.In Figure 11, our data are compared with theirs, which present similar scattering.The correlation coefficient is 0.4, which indicates a positive correlation considering the uncertainties of the parameters.It should be noted that our peaks and FWHMs are measured directly from the curve after subtracting the continuum extinction (mentioned at the end of Section 2), and no Drude or Gaussian profile fitting is performed.This study has 6 stars in common with Gordon et al. (2021), namely VI CYG 1, HD229238, VI CYG 2, HD147889, HD147701 and HD283809.In comparing the profile parameters, the peak wavelengths are essentially similar after accounting for the errors, while the difference in FWHM is obvious.With the exception of VI CYG 1 for which the FWHM is consistent, there is a difference of 1-2 µm in the FWHM of the other five sources.This is partly due to the fact that the peak and FWHM themselves have some errors, whether measured by us or Gordon et al. (2021), and partly probably due to the spectral data limitations, including the choice of different reference stars, and the differences in the method of subtracting continuum extinction and measuring the profile widths.It is worth noting that the FWHM for Gordon et al. (2021) in our own measurement is based on the extinction curve provided by Gordon et al. (2021), not the γ 0 given in Table 6 of Gordon et al. (2021).We found the difference between the FWHM and γ 0 to be very small.
Considering correlation between E JKS and ∆τ 9.7 , the effect of E JKS on the peak and FWHM of the extinction profile at 9.7 µm is further investigated and is shown in Figure 12, and no significant correlation is present.

Dependence of the Silicate Profile Features on Stellar Spectral Type
Figure 13 shows that the peak wavelength of the 9.7 µm profile may be divided into two groups.For OB-to even early F-type stars, the peak concentrates at longer wavelength around 9.7-9.8µm with the maximum being 9.9 µm, while it moves to about 9.6 µm for late F-to GKM-type stars with the minimum being 9.5 µm.It may be caused by that the crystalline silicate around late-type stars shifts the peak wavelength since the crystalline silicate emits stronger at λ > 9.7 µm (Chen et al. 2016).Do-Duy et al. (2020) found a minor absorption band around 11.1 µm whose carrier is attributed to crystalline forsterite.This weak feature is also found in a small fraction of our sources, such as HD 283809, VI CYG 1, HD 29647, CI*IC 348 LRL 11.This drop is not so obvious after considering the error, and whether this trend is real needs further study.On the other hand, the FWHM shows no systematic dependence on spectral type as exhibited in the same figure that the variation of FWHM with spectral type is irregular.The most narrow FWHM is about 1.2 µm, occurring in an O9-type star, and the widest is about 5 µm seen in a few stars, meanwhile, most FWHM are between 2.0 and 3 µm.
Detailed experimental studies have revealed the existence of tight correlations of silicate profile properties with the dust physical and chemical parameters.Koike & Hasegawa (1987) investigated the infrared spectra of grains of natural and synthesized silicate glasses covering a wide range of SiO 2 .They found that the peak wavelength and the width of the 10 µm band decrease with increasing SiO 2 , whereas the peak absorption of the band increases with growing SiO 2 .Dust extinction originates from molecular vibration, especially the chemical bond stretching or bending.The influence factors of the dust extinction profile include dust composition, size distribution, and shape.Based on our research as well as others (e.g.Shao et al. 2018), the variety of dust composition may account for the above mentioned variation of the profile.

SUMMARY
The Spitzer/IRS spectra as well as the 2MASS and WISE photometry of 49 stars with obvious extinction are analyzed to obtain the extinction curves in the infrared by using the "pair" method.The silicate extinction profile around 9.7 µm and 18 µm are determined after subtracting the continuum by a constant power-law function.The characteristic parameters of the 9.7 µm profile are calculated and investigated.The following results are obtained: (i) The general 9.7 µm and 18 µm silicate extinction features peak around ∼ 9.2-9.8µm and ∼ 18-22 µm respectively.
(ii) The wavelength of the peak of the silicate extinction profile may decreases with the spectral type, but this trend is not so clear due to the presence of errors and needs to be verified with more data support in the future, meanwhile the FWHM is independent of the spectral type.
(iii) For most sources, the silicate optical depth increases with E JKS , consistent with the proportional relation found in diffuse medium.However, when E JKS >4, this relationship is not applicable.Some late-type stars may contain circumstellar emission, weakening the silicate optical depth at 9.7 µm.The mean ratio of the visual extinction to the 9.7 µm silicate absorption optical depth for diffuse medium is 17.83, in close agreement with that of the solar neighborhood diffuse ISM.Still, for some late-type stars, the ratio is larger than that of the neighborhood diffuse ISM, which may be caused by the silicate dust emission from the circumstellar dust.

1.
Exclude the sources showing silicate and/or polycyclic aromatic hydrocarbon (PAH) emission features, which indicates the presence of circumstellar dust.This reduces the sample to be consisted of 55 O stars, 203 B stars, 806 A stars, 453 F stars, 543 G stars, 1397 K stars, and 1260 M stars.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.The histogram of EJK S of the stars in the initial sample.The EJK S = 0.3 is labeled by the red line for selecting the targets with visible silicate absorption feature.

Figure 10 .Figure 11 .Figure 12 .Figure 13 .
Figure10.The relation between the color excess E(J − KS), the optical extinction at V band AV and the optical depth at 9.7 µm ∆τ9.7.The color denotes the spectral type of stars.The green dash line is the relation in diffuse ISM.The gray dash line is the reference line at E(J − KS)=1.5.

Table 1 .
Stellar parameters for the target and reference stars

Table 1 .
Stellar parameters for the target and reference stars