We used the MPFIT package in Interactive Data Language (IDL) to find the best-fit values of the above parameters in Eq.3.2. MPFIT uses a non-linear least squares fitting and iterative calculation method to get the minimal χ2 (Eq.3.9) and find the best fitting result (Markwardt, 2009).
χ2 is defined by :
4χ2 = represents the spectrum of Spitzer-IRS; the N is degrees of freedom.
MPFIT can be used to find the best-fit model parameters to fit any 1-dimensional data with errors or uncertainties, including our study.
We describe here the successive steps used in our fitting pro-cedure:
1. Read the raw data which is from Spitzer-IRS
2. Shift the module data to match at band edge
There is a jump at 14µm between SL and LL modules and it is so hard to fit the two channels. Therefore, we need to shift the data to match in the flux at the joining point before fitting.
Source SL2 SL1 LL2 LL1
Table 3.1: This table shows the factors of every sources and modules for shifting in this study.
3. Remove overlapping wavelength range:
We set the boundaries at the edge of every modules redundant data.
The boundaries are set at 7.5 µm, 14 µm, and 20.6µm. (see section 2.2)
4. Convert to rest wavelength:
Using the redshift equation (see Eq.2.2), we convert the observed data to rest frame data.
5. Fitting
(a) Read the Q-values from laboratory measurements:
There are many different species for which Q value is measured in laboratory: Alumina(Al2O3), Periclase(MgO), Fe-rich amorphous olivine, Fe-poor amorphous olivine, and Forsterite(Mg2SiO4) (Markwick-Kemper et al., 2007). We use both Q-values of spherical(mie) and non-spherical(CDE) in three of those components: Fe-rich amor-phous olivine, Fe-poor amoramor-phous olivine, and Alumina(Al2O3), then we separate the Q-values of spherical and non-spherical and use all compositions for fitting; Polycyclic aromatic hydrocarbons (PAHs) are included. For PAHs, Forsteite, and Periclase(MgO),we use opacities directly because they do not have grain size or shape is assumed.
The power-law continuum part of spectrum will be determined by our model, therefore, we need to use the continuum-divided Q-value for fitting with the quasars we select for analyzing.
(b) Call our model function
Using the model we described at Sec.3.1 (Eq.3.2) (c) Choose a starting guess:
We use the MPFITFUN function in the MPFIT package. The MPFITFUN need to set seed values as starting point. The starting guess value allows MPFITFUN to find a local minimum for chi-square (χ2) value. If we set a suitable starting value, we will find not only a local but also a global minimum for χ2, and this minimal χ2value will not have any change when we alter the starting guess.
In this study, we set the starting values for all parameters to 1.
for all the sources: PG 0043+039, PG 1351+640, PG 0050+124, PG 1211+143, and PG 2112+059.
(d) Call MPFIT then plot the result
The built-in function MPFITFUN solves for the best fit param-eters and then optimizes for the 1-D curve, therefore, we used this function to fit and find the minimal χ2.
Chapter 4
Results and Discussion
4.1 PG 2112+059 with 2 different methods
Markwick-Kemper et al. (2007) selected PG 2112+059, therefore, we included it in our sample as consistency check for our new model. The Table.4.2 shows the results induced by the two methods. Although the χ2 value (about 5) in our method is larger than the value in the Markwick-Kemper et al. (2007), the χ2 values of other sources we selected are much small, therefore, we still can use this method to examine the difference between the two methods for PG 2112+059.
The two different methods imply the similar power-law index, α. Markwick-Kemper et al. (2007), used the photometric and spectroscopic data, and got α values of -0.617±0.004 and -0.674± 0.003 respectively, for the model Fν = Fν,0να. Using our model, we get α=0.555±0.003 and α=0.559±0.004 (see the Table.4.1), for CDE and Mie models respectively . (As our model Fmod[λi] = (Aλαi)(1 +P
jajqji), is in the λ frame, the α values are positive in
our case). When comparing, our α values are slightly smaller than the values in Markwick-Kemper et al. (2007), our continuum model fitting being more slack and Markwick-Kemper et al.’s being more steep (Fig. 4.1). Fitting method determines the continuum model and an optically thin dust compo-nents for the different species simultaneously, then our continuum model are determined by dust specious we used, therefore, even it the results of the continuum model is not very close to the spectrum, they also provide a new direction for figuring out the contribution from various dust species in these sources.
Table 4.1: This table shows the results for power-law index, α, of all the quasars we used in this study. The lowermost shows the power-law index in the results of Markwick-Kemper et al. (2007).
Fig.4.1 and Fig.4.2 show the spectrum fitted with our different mod-els compared with Markwick-Kemper et al. (2007). Table. 4.2 summarizes the fraction of every dust species in PG 2112+059 for our two models and
Markwick-Kemper et al.’s two models. The fits of the spectrum with the dif-ferent methods yield similar; results with large amount of amorphous olivine and alumina, as well as a small quantity of MgO. We determine the compo-sition of PG 2112+059 to be 51.5%±3.5% amorphous olivine, 39.9%±1.27%
alumina, and 8.3%±4.4% periclase (those values are average of CDE model and Mie); When Markwick-Kemper et al. (2007) measured values of alumina and periclase of 38 ± 3% alumina and 5.9 ± 2.6%, and 56.55 ± 1.4% amor-phous olivine. Although slightly different the values for amoramor-phous olivine are consistent at the 2σ level.