北大低温研・研究集会

干渉計解析の概要
百瀬宗武（茨城大理）

1Interferometer in radio (1D):Each antenna obtains “E-field” information
when the delay is adjusted
to the zenith ...

Cross correlation for δφ direction|E(2)| exp +i2

D

Integrating all the directions
V (D) =T() exp +i2

Dd()

V(D) is a Fourier component of T(δφ)→Image can be reconstructed by data from many baselines2

Visibility and Brightness in 2D
Definition of “uv vector” for a projected baseline

D = (Du , Dv )(u, v)(u, v)は地面，(x, y)は天球。u ∥ xは東西，v ∥ yは南北
Fourier Transform between Visibility and Sky Brightness

V (u, v) =T (x, y) =+i2 (ux+vy)

dxdy

i2 (ux+vy)

dudv

T (x, y)eV (u, v)e3

In reality, (1) antenna power pattern and(2) uv-sampling also affect the obtained data …

(1) V (u, v)(得られるビジビリティ)

P (x, y)T (x, y)(パワーパターン)※パワーパターンは測定可能な量(2) P (x, y)Tdirty (x, y)=(真の輝度分布)(直接出るマップ= Dirty Map)

[S(u, v)V (u, v)] ei2 (ux+vy)

dudv(標本関数 or 重み関数)= B(x, y) [P (x, y)T (x, y)](求めたい輝度分布)(Dirty Beam)where S(u, v)

B(x, y)※関数積のフーリエ変換 = 各関数のフーリエ変換の畳み込み
4

Visibility vs. Image
Dirty Beam and Dirty Image(from NAASC Memo #104) ALMAb(x,y)

B(u,v)
S(u,v)

B(x,y)(dirty beam)

convolution

D(x,y)
TDirtyT(x,y)
T(x,y)

Image(dirty image)

ALMA ES Proposal Preparation Tutorial

5

Charlottesville, April 26, 201115

• Fourier transform of the measured V(u,v) to the image plane TD(x,y)
Visibility→Dirty
Image→DeconvolvedImage• But difficult
to do scienceon dirty image(fromNAASC
Memo#104)• To determine(modelof) T(x,y)
we need to deconvolve b(x,y)
from TD(x,y)  “clean” image
visibilities

dirty image

sky brightness

Deconvolution Method
CLEAN
ALMA ES Proposal Preparation Tutorial
Charlottesville, April 26, 2011
MEM
629

T (x, y, ), or T ( , , vlos )Data Cube v is measured in “Local Standard of Rest” (LSR).
losDeclination

Vlos, orFrequency

Right Ascension
7

Channel Maps
Declination

Right Ascension
8

PV Diagram
Declination

Vlos, orFrequency

Right Ascension
9

PV Diagram
Declination

Vlos, orFrequency

Right Ascension
10

PV Diagram
Declination

Vlos, orFrequency

Right Ascension
11

PV Diagram
Declination

Vlos, orFrequency

Right Ascension
12

Moment Maps※ 以下はすべて場所(α, δ)ごとで計算

0th: IT (v)dv

1st: v los
2nd:v
=

積分強度

v · T (v)dv
平均速度I
速度分散(v v los )2 · T (v)dv
局所的なものI(乱流・熱速度)+v 2 · T (v)dv
v los . ビーム内速度勾配
I
13

Examples: HD142527 in 13CO (J=3-2)
Briggs Wt (upper) and Uniform Wt (lower)0th

1st2nd14

Continuum Subtraction
F or TLine Free Channels

SimpleSpectroscopyνContinuumLine+15

Continuum Subtraction
Real partorImag. part

Complex Visibility

Line Free Channels

Vector Averaging inline free channels

Cont.
Visibility

νContinuum

Residual

LineVisibility

Line+
16

Peak Tb map of 13CO (J=3-2) around
13CO
peakinHD142527: Line only and Cont. added… in different color code

Continuum in grey lines

C18O が、少なくとも北側の明るい領域では両方とも光学的に厚く
17

Advanced Materials

18

CLEAN Algorithm
Dirty Map

Operation FlowID(x,y)

Data Input/Output
Start / Resume
放射ピーク ICC(x,y)
を見出す

ICC(x,y) を記録

I(x,y)*B(x,y) を引く
Residual Map
R(x,y)Yes

A list of ICC(x,y)(Clean Components)

Residualに拾うべきピークある？

No[Σ ICC(x,y)*BCL(x,y)]+ R(x,y)
CLEAN beam

Final Image

BCL(x,y)IFIN(x,y)19

Data Reduction in the case of
Fukagawa et al. (2013)
CLEAN (Uniform Wt)

Continuum
Visibility CLEAN

Raw
Data

Calib.
Data

Flux Scale(Titan & Neptune)

FinalContinuum
Image

ModelImage

SELF-CALx 6 Phase only↓x 1 Amp & Phase

Gain
Solution

Final Gain
Solution

FinalLineImages

Line
Visibility

Bandpass(3C279 & J1924-242)

Gain Table(contsubtracted)
CLEAN (Uniform Wt)(J1427-4206)
20(contsubtracted)

Self-Calibration: “Before-After”
HD142527 Continuum @ 330 GHz
Before Self-Cal.

after
1 iteration
solving phase only

Final Image
6 iterations
solving phase +1 iteration
solving amp. & phase(理想は熱雑音のみで計算される値と一致すること)

rms = 2.24 mJy/beam

0.68 mJy/beam

0.13 mJy/beam※ 少しずつ慎重に（やり過ぎると1箇所に放射を集める場合あり）※ 独立した偏波成分を取得する受信機同士でconsistentな解を示しているか21

明るくコンパクト

Self-Calibration
Nant < Nbaseline↓
アンテナベースの位相誤差(大気水蒸気の揺らぎ，装置の位相揺らぎ等)を解くことができる（モデルイメージの解との比較）

22

瞬時瞬時のVisibility Phaseを支配。予測可能。

Data Reduction in the case ofFukagawa et al. (2013)
CLEAN (Uniform Wt)

Continuum
Visibility CLEAN

Raw
Data

Calib.
Data

Flux Scale(Titan & Neptune)

Final
Continuum
Image

ModelImage

SELF-CALx 6 Phase only↓x 1 Amp & Phase

Gain
Solution

Final Gain
Solution

FinalLine
ImagesLine
VisibilityBandpass(3C279 & J1924-242)

Gain Table(contsubtracted)
CLEAN (Uniform Wt)(J1427-4206)
23(contsubtracted)

Self-Calibration: “Before-After”
HD142527 Continuum @ 330 GHz
Before Self-Cal.

after
1 iteration
solving phase only

Final Image
6 iterations
solving phase +1 iteration
solving amp. & phase(理想は熱雑音のみで計算される値と一致すること)

rms = 2.24 mJy/beam

0.68 mJy/beam

0.13 mJy/beam※ 少しずつ慎重に（やり過ぎると1箇所に放射を集める場合あり）※ 独立した偏波成分を取得する受信機同士でconsistentな解を示しているか24