# Features of Structure and Absorption in the Jet-Launching Region of M87
## 3 Results
### Spectral-Index Distribution  
In this part, we present the spectral-index distribution in the jet calculating between 22 and 43 GHz. To align the VLBI images at different frequencies, we employ an algorithm which is first introduced by [15]. It is based on the premise that the positions of optically thin regions to synchrotron radiation are not affected by absorption effects, then images can be aligned by finding out the maximum of 2D cross-correlation of the optically thin regions. This method is now commonly used [16; 17; 18] and was once employed by [19] to obtained the RM map of jet in M87.  
The practical steps are listed as follow: (1) First, we produce VLBI images using the fully self-calibrated dataset. A same \(u-v\) range is used to minimize the difference of baseline coverages between 22 and 43 GHz. Both images are restored with a circular beam with a diameter of 0.65 mas (the same as used in Section 3.1). (2) Second, we produce an initial spectral-index map by simply aligning the geometric center of two images. With the initial map, we roughly determined the optical thick region which will be masked in the next step. (3) Third, we import the two images into an interactive Python program, VIMAP [20], based on the algorithm introduced by [15]. A circular mask with a diameter of 1 mas is used to cover the core region, and a rectangle box with relative RA range from 2 to \(-4\) mas and DEC range from \(-2\) to 3 mas is used to choose a optically thin region. The VIMAP gives the values of 2D-correlation after shifting and find out the shifting corresponding to the maximum (see Figure 6). In our case, shifting corresponding to the maximum is zero. (4) The final spectral-index map is produced in AIPS with task COMB. Only regions with intensity \(>5\sigma\) are used to make sure the result is reliable and a noise map showing the distribution of errors is also produced. The 22-43 GHz spectral-index map superimposed on the intensity contour plots of 22 GHz are presented in panel (a) of Figure 7 with its noise map in the panel (b).  
We also produce a 43-86 GHz spectral-index map with the same procedure. Considering there is a gap of six days between the observations of two frequencies, this spectral-index map is produced only for reference. In calculating the 2D cross-correlation, a circular mask with a diameter of 0.5 mas and a rectangle box with a RA range from 1 to \(-2\) mas and a DEC range from \(-1\) to 0.5 mas are used. The 43-86 GHz spectral-index map with its noise map are presented in the lower panel of Figure 7.  
Figure 6: Panel (**a**) shows 2D cross-correlation of the optically thin regions of 22 and 43 GHz images. Panel (**b**) 2D cross-correlation of the optically thin regions of 43 and 86 GHz images.  
On the 22-43 GHz spectral-index map, inverted-spectrum with index \(\alpha>0\) (\(S_{obs}\propto\
u^{\alpha}\)) are detected between \(d=1.30\) and 1.60 mas on the edge of the northern limb, and between \(d=1.30\) and 1.80 mas on the edge of the southern limb. Some less confident inverted-spectrum are seen on the edge of northern limb around \(d=3.80\) mas. Strongly inverted spectrum with \(\alpha>2\) is detected in the northern limb between \(d=5.00\) and 7.00 mas where the limb is interrupted. These inverted-spectrum features indicate absorptions in the jet in M87. Interestingly, on the map, we see a pattern-like distribution along the jet, the regions with steep spectrum interlace with regions with flatter spectrum (the pattern-like distribution is also seen on the 43-86 GHz spectral-index map).