Good good study, day day up

# Section2: The Milky Way Galaxy (p. 1044)

First, the structure of the Galaxy.

Different views:
Face-on: usually no jet.
Edge-on
Pole-on: view along with the jet.
the difference of face and pole on depends on the object and celestial.
Alpha Centauri---有大项目计划send explorer去
We are in Globular Cluster? right?

## Galactic coordinate 银道坐标

Use the Galactic structure as a reference frame.
The distance of solar system to GC is about 8.0 kpc --> Galactocentric distance (银心距)
Galactic longitude 银经 (l)
Galactic latitude 银纬 (b)

- unit usually use degree.
- 银道坐标系中心应当是地心！！图里的center只是表示银心～

### The direction of th Galactic rotation (银河自转)

Galactic bulge(星系核球)
What is Stellar halo(恒星晕)?

### Conversion between galactic and equatorial coordinates

--> Just change the direction of the axis:
The Galactic Center (G.C.) in J2000.0 G.C. is at $l=0\degree, b=0\degree$, for equatorial system are: $\alpha_0=17^h45^m 37.20^s, \delta_0= -28\degree 56' 9.6''$

double stars
binary stars

#### Conversion tools:

• HEASARC Coordinate Converter
• Astropy.Coordinates

# Section3: Conclusion of the last class and so on.. --> Time systems used in astronomy

what is sidereal time? -->恒星时
diurnal[daɪˈɜːrnl] motion-->周日运动
G.C. : Sgr $A^*$

Q: What is the angular separation of the closest possible double star?

• This depends on the angular resolution of the telescope
• Two main factors determing the angular resolution of the telescope (p. 168)
• Seeing (视宁度): Sgaroness of a telescopic image. Seeing depends on the degree of turbulence in the Earth's atmosphere at telescope sites. ~0.1'' in the best case (in the case of optical)
• Diffraction limit (衍射极限): Theoretical limit of angular resolution due to wave nature of light. $\theta=1.22\frac{\lambda}{D} [rad]$, more detail plz see here

Radial velocity $v_r$: 径向速度(视向速度)
Transverse velosity $v_{\theta}$: 切向速度

Boltzmann ditribution: $\frac{N_2}{N_1}=e^{\frac{\Delta E}{kT}}$

## Additional note on transitions

1. Electronic transition (电子跃迁)
• Energy: optical, ultraviolet, shorter wavelength
2. Vibrational transition (振动跃迁)
• Energy (hv): intrared
3. Rotation transition (转动跃迁)
4. Hyperfine transition (超精细跃迁)
• Energy: Radio (very long $\lambda$)

### 2021.9.18

$R_0\omega_0^2=a_0$
$R\omega^2=a$
$\omega(R)=\frac{v}{R_0sin l}+\omega_0$
---- Good question on the beginning😀

## Solar time (太阳时)

• The time sytem based on the Solar position
• Noon (12pm) in the Solar time is
• the annually averaged time of the Solar meridian passage
• Note: meridian passage = 中天

## Sidereal time (恒星时)

• The time system based on the positions of stars
• Noon (12pm) ....

Common year 平年 --- 1/4 day shorts of a complete orbit
Leap year 闰年 +3/4 day over a complete orbit
Tropical year 回归年 ---One complete orbit around the Sun

## Local standard time (LST)

Universal time (UT, GMT)
-The mean solar time at the Greenwich meridian(LONGITUDE = $0\degree$)

## Julian day (儒略日)

Time-variation plot with date (年月日, i.e., Gregorian calendar) on horizontal axis
---not ez for observer
Why dont use 年月日？

1. Numerical analysis is not ez
• Difficult to read the number of days
• Leap year
So in astronomy, we use Julian day

### What is Julian day?

• Julian day (JD) is the total number of days from JD 0.0
• JD 0.0 is defined as
• the noon (UT) on $1^{st}$ Jan, B.C. 4713, why Noon??
• bcz Day doesnt change in the night
• Examples
• Noon (UT) on 1st Jan 2000 is JD 2,451,545.0
• 6pm (UT) on 1st Jan 2000 is JD 2,451,454,25
• Today now is (Sep 2021, 11h40m0s CST) JD 2459476.31944

### Modified Julian Day (MJD) MJD=JD-2400000.5

Note: MJD starts at midnight

## Practice a bit

(1) Calculate Julian Day for $10^{th}$ September 2021 1pm CST
(2) What is the corresponding modified Julian day (MJD)?

## Homework next week:

UT
UTC
GMT
CST

ST clock
UT clock
and how to transform from UT to ST?

☘️☘️☘️☘️☘️☘️☘️☘️☘️☘️☘️Split line☘️☘️☘️☘️☘️☘️☘️☘️☘️☘️☘️

### Assignment:

1. 什么是太阳时和恒星时
2. 为何天文台里需要使用恒星时ST
3. 如何转换UT to ST?

Trying to explain :

1. What is Solar Time and Sidereal Time (st)?
2. Why observatory use ST time?
3. How to transform from UT to ST?

#### 在开始之前(Before the beginning)：

Sidereal time 是基于恒星的， 这个恒星不是太阳的意思，而是除太阳外的恒星，或是一些深空天体，比如星云或者类星体等(详情可见here)
Sidereal time bases on STAR, and this "STAR" is not the Sun, but some other stars, or some deep space objects, like Nebula or Quasar (see more detail at here)

##### 一些常用缩写的定义 (Definitions of some frequently used acronyms):

UT: Universal Time (reflect the Solar time, the duration from one solar noon to next)
UTC: Coordinated Universal Time (leap second, subset of UT)
GMT: Greenwich Mean Time (time zone)
CST: China Standard Time (UTC/GMT+8)

#### 第一个问题(The $1^{st}$ question)：

Just follow the illustration is enough ~
💁该动画原链接
💁The url of this animation is Link

#### 第二个问题(The $2^{nd}$ question)：

1. 当某一物体经过中天点的时候，当地的LST时间即是该物体的赤经，贼方便。
2. 太阳时是根据太阳来的，而恒星时是根据(star)来的，更准直，可以更好地抵消地球公转带来的影响

1. When a celestial object passes the meridian passage point, the time of LST is the RA of the object, its really convenient for astronomer.
2. Solar time is based on our Sun, and Sidereal time is based on star, the latter is more collimated, can moderately omit the influence of the Earth revolution.

#### 第三个问题(The $3^{rd}$ question)：

💁该图片原网页
💁The source page of this picture Web

# Section3~4 (follow-up 2021.9.22): Distance unites used in astronomy

Julian Day Calculator: https://www.aavso.org/jd-calculator

-->See, 3. The continuous spectrum of light in the textbook

1. Light year
$1 [ly] =9.46 \times 10^{15} [m]$
1 tropical year!!!

2. Astronomical Unit (AU, 天文单位)
Averaged distance between the Sun and the Earth
$1 [AU]=1.49597 \times 10^{11} [m]$
(Note: $1 [ly] ~ 100,000 [AU]=10^5 [AU]$)
Fundamental of the distance unit used in astronomy

3. Trigonometry
Trigonometry (三角测量) is the distance measurement method using the trigonometric parallax (三角视差)
Concepts: B: base line (基线), p: parallax angle (视差角)

4. Trigonometry in Space
In priciple, we can measure the distance to stars with trigonometry, but stars are very far away.
So, we need a long baseline to see the parallax angle.

5. Definition of parsec
Parallax angle is very small, $p << 1['']$
Therefore, $tan p\approx p$
So, the formula can be approximated as follows:

$d \approx \frac{1}{p [rad]} [AU]$

Since, $1 [rad]=206,264.806 ['']$
We can convert the angle from rad to arcsec as follows:

$\frac{p''[arcsec]}{206264.806}=p [rad]$

So, the distance formular can be changed as follows:

$d\approx \frac{206264.806}{p''[arcsec]} [AU]$

Therefore, when p''=206,264.806 [AU}, the $d=206,264.806 [AU]$
With this definition, the distance formula can be changed as follows:

$d=\frac{1 [AU]}{p''} [parsec]$

6. Summary of parsec

• The distance at which 1 AU subtends (对着) an angle of 1''
• Relation with "light year", 1 [pc]=3.26[ly]
• Examples:
$\alpha$ Centauri (The nearest star to us) ~ 1.3 [pc]

7. Relation between parallax angle and parsec
The relationship between parallax angle (p'') and distance (d) is beautifully obcvious.

8. Exercise a bit

solution:

$tan \theta=\frac{1 [AU]}{8.0 [kpc]}$

$\theta=1.22\frac{\lambda}{D}$

so we have, $D=1.22\frac{\lambda}{\theta}$=6039.43351968km

Nakashima's answer: annular parallax angle is $\frac{1}{8000 [pc]}=0.0001235 ['']$

### Little small discussion:

How to measure 1 AU?

## Brief introduction of Maser:

Boltzmann distribution: $\frac{N_2}{N_1}=e^{\frac{E_1-E_2}{kT}}$

## 关于利用金星测量太阳距离：

$\lambda ~ 10$ cm is used due to the good reflection rate.
$\lambda ~ 3-4$ cm used to investigate the Venus atmosphere

## 星系没有一个确切的边界，边界是由Scale height来定的

Scale height H: $\frac{P}{P_0}=e^{-\frac{z}{H}}$
z: distance from the disk
$P_0$: number density of stars at z=0
P: number density of stars at z

## Continue on ..

Elongation(距角): The angular distance(这是一个角度，而不是距离) separating the Moon or a planet from the Sun
Venus at the greatest eastern elongation (东大距)----evening
At the greatest western elongation (西大距) ---- in the morning

## How to obtain the Galactic rotation curve from gas?

We can derive the Galactocentric distance and radial velocity of the gas component at tangential point D.
VLBI --- 限于看小angular distance，高亮的源，可参考Fourier Transform.

# Section 5: The Magnitude Scale

(see, "3. The Continuous Spectrum of Light" in the textbook)

## Apparent magnitude (视星等)

--> Basic idea formed in ancient Greek
(1 mag difference ($\Delta$m=1))=(a factor 2.512 times in brightness)

$100^{\frac{1}{5}}\approx 2.512$

for instance, $\Delta$m=2, factor is $10^{\frac{2}{5}}=2.512^2=6.310$....
How many times different above two magnitudes in brightness?
answer: $10^{\frac{30-(-26.83)}{5}}\approx 100^{\frac{57}{5}}\approx 10^{23}$

import numpy as np
L=3.839*10**26
r=1.496*10**11
S=4*np.pi*r**2
flux=L/S
print(flux)


minor次要的

## Qualitative introduction of radio interferomerry EHT

• Short baselines are sensitive to large stucture
• Long base lines are sensitive to small structure

### THe interferometer is mathematically the Fourier transform.

$\frac{a_0}{2}+\sum^{\infin}_{k=1}(a_k cos kx+b_k sin kx)$
Limited number of antennas cannot completely reproduce the source structure

fringe pattern-->条纹图案
deconvolve-->去卷积

• The true emission should not change in shape if we use different CLEAN box sizes.

Jansky
1 Jy = $10^{-26} W m^{-2} Hz^{-1}$

## 2021.10.11

### Distance modulus

--> 距离模数:
m-M is mathematically equivalent to the distance:
$d=10^{\frac{(m-M+5)}{5}}$
so far, we have assumed that there is no interstellar extinction!!

### Interstellar extinction (星际消光)

Here we define interstellar extinction $A_{\lambda}$:

$A_{\lambda}=-2.5log_{10}\frac{F}{F_0}=m-m_0$

interstellar extinction depends on the wavelength! 河内和外都会有，主要集中在光学波段

## Some more details on radio interferometry

visibility=brightness $\times$ fringe pattern

point spread function: 点扩散函数

## CLEAN algorithm

• CLEAN is the dominant deconvolution algorithm in radio astronomy.

Initial step:

1. Identify the highest peak in the dirty map as a point source.
2. Subtract a fraction of this peak from the residual map using a scaled dirty beam
3. Add this point source location and amplitude to Clean Component list (https://2199k.cn/post-images/1633922962928.JPG)
4. Go to step 1 (an iteration) unless stopping criterion reached
• Stopping criteria:
• Residual map maximum < threshold =several X rms

## 2021.10.13

References for further study of radio astronomy.

• Essential Radio Astronomy
• Interferometry and Synthesis in Radio Astronomy
• NRAO Systhesis ...
• CASA ...

## Black body radiation

jargon:行话

Neutrinos doenst belong to** Cosmic rays**---> because Cosmic rays are electrul charged.

Stefan-Boltzmann law:

$B(T)=\int^{\infin}_{0}B_{\nu}(\nu,T)=\sigma T_e^4$

$\;$where $T_e$ is effective temperature.

Plank's law 波长到频率的转换参考:

Key procedure:

$\int 1 d\lambda=-\int \frac{c}{\nu^2} d\nu$

Revelation:

• If we know the equation $f(y)=f(x)$, then we can:

$df(y)=df(x),$

and

$\frac{df(y)}{dy}\cdot dy=\frac{df(x)}{dx}\cdot dx \to y'\cdot dy=x'\cdot dx.$

So, in the case $y=\frac{c}{x}$, this form becomes like:

$1\cdot dy=-\frac{c}{x^2}\cdot dx,$

so we have finally:

$\lvert \frac{dy}{dx} \lvert=\frac{c}{x^2}$

Rayleigh-Jeans --> 右边
Wien --> 左边

Effective temperature: T calculated under the assumption of black body radiation.
Note that effective temperature not always represent the True temperrature.

## Maser pumping mechanism

• Collisional pumping
• Collision with ambient molecules
• This often happens in interaction regions between two kinematic componnets
• Radiation from a star etc. (SiO maser in evoleved starts)
• Line over lapping

# Photometric bands and color index #1

Section 7, (see, "3. The continuous spectrum of light" in textbook)

## Photometry and photometric band

In the reality, the $\lambda$ usually the intensity $I_{\lambda}$ is measured in a certain range of $\lambda$.

The wavelength ranges used for photometry are called photometric bands

We use a "filter" to limit the wavelength range.

Filter can pass the radiation only a certain range of $\lambda$.

Johnson-Cousin UBVRI system (3000$\overset{\circ}{A}$-9000$\overset{\circ}{A}$):

• CCD arrays enable us to do imaging and photometry at one time

## Color index

The apparent and absolution magnitudes can be defined for each photometric band.

U, B, V: apparent magnitudes at U, B, V bands

$M_U, M_B, M_V: absolute magnitudes$

U, B, V can be measured separately

$U=-2.5log_{10}(F_U)+C_U$

$F_U=\int^{\infin}_{0} F_{\lambda}S_Ud\lambda$

$S_U$: passband characteristics of the U band filter.

The difference in magnitude like U-B and B-V is called "color index"

$U-B=M_U-M_B$

$B-V=M_B-M_V$

As we see above, color is independent of distance !!!

## Color temperature

The color index U-B equals to a ratio of flux values at U band B-bands:

$U-B=-2.5log_{10}\frac{F_U}{F_B}$

So, the color index uniquely determines the effective temperature.
--> color is relying on effective temperature

## Color index #2

HR (Hertzsprung-Russell) diagram.

# Bolometric correction

Section 9

• Definition of the bolometric correction
• Meaning of the bolometric correction
• Possibly, could be a useful tool to obtain the total energy emitted from source
• Limitation of the bolometric correction

The bolometric correction BC is defined as follows:

$BC=m_{bol}-V$

$m_{bol}$: apparent bolometric magnitude
$V$: apparent V-band magnitude

"bolometric" means "total energy" over "all" wavelength

### Meaning of the bolometric correction

If we know BC, we can convert $V$ to $m_{bol}$

For instance, for bright(nearby) stars, $m_{bol}$ and V are measure, and BC is determined.

Then, the obtained BC is applied for similar stars, which possibly show the similar SEDs.

BC always taks a negative value, because flux for $m_{bol}$ is larger than that for $V$

### Limitation of the bolometric correction

1. It is often impossible to properly measure $m_{bol}$, because the photometric system doesnt cover entire SED.
2. SEDs are not completely the same even among the stars with the same spectral type.

## Color excess

"color" is altered by extinction, and this effect on color is called "color excess" or "reddening"

The color excess isimportant from 2 viewpoints:

1. Calculating the distance of a certain object

$m-M=5log_{10}(d)-5+A_{\lambda}.$

1. Investigating the properties of interstellar materials.

The color excess isquantitively defined as follows:

$E(B-V)=(B-V)-(B_0-V_0),$

where $E(B-C)$ is not minus, but a funtion !!

Using $A_{\lambda}=m-m_0$, the above formula can be written as follows:

$E(B-V)=A_B-A_V$

## Intensity

Prove that specific intensity $I_\nu$ is independent of distance:

## Defining flux and luminosity

• Section 12, Based on Section 2.1 in ERA(Essential Radio Astronomy)

Specific flux=Flux density=Spectral flux
1 [Jy]= $10^{-26}$ [$W\,m^{-2}\,Hz^{-1}$]

$I_\nu$通常用于Structure is resolved;
$F_\nu$通常用于Structure is not resolved.

## Optical depth and Radiative trasfer equation

• Section 13, Based on Section 2.2 in ERA(Essential Radio Astronomy)

$\kappa=\frac{dP}{\text{ds}}$--> linear absorption coefficient

Only consider absorption:

Key formula:

$\frac{dI_\nu}{I_\nu}=-\kappa\,ds$

$\int^{s_{out}}_{s_{in}}\frac{dI_\nu}{I_\nu}=lnI_\nu|^{s_{out}}_{s_{in}}=-\int^{s_{out}}_{s_{in}}\kappa(s')ds'$

Optical depth $\tau$:

$\tau\equiv -\int^{s_{in}}_{s_{out}}\kappa(s')ds'$

Contribution from emission:

$j_\nu$: emission coefficient

$\frac{dI_\nu}{ds}=-\kappa_\nu$

$+$

$\frac{dI_\nu}{ds}=j_\nu$

$\downarrow$

$\frac{dI_\nu}{ds}=-\kappa I_\nu+j_\nu \rightarrow \text{Radiative transform equation}$

## Intro to thermal equilibrium (TE)

Under TE, $I_\nu$ is the black body radiation: $I_\nu=B_\nu(T)$
Kirchhoff's law":

$B_\nu(T)=\frac{j_\nu}{\kappa}$

$\quad$(此处关系的推导可参阅中岛的ppt)
The Planck function does not depend on material, so under TE, the dependency on material is cancelled out.

## Simple applications of radiative transfer:

Section 15
(Based on Section 2.2.3 in ERA)

### Brightness temperature

When $h\nu<, the black body radiation is approximated by the Rayleigh-Jeans law,

$B_\nu \approx \frac{2kT\nu^2}{c^2}$

Then the temperature T is calculated to be :

$T\approx \frac{B_\nu c^2}{2k\nu^2}$

Even if we replace $B_\nu$ with $I_\nu$ mathematically T can be calculated:

$\rm T_b=\frac{I_\nu c^2}{2k\nu^2}$

$T_b$ is called "brightness temperature"
$\quad$ not a real physical temperature, but often the indicator of $I_\nu$

### System temperature

The sum of all noises, which is converted to the temperature scale [K], is called the system temperature ($T_s$)

### Observation to determine $\tau$

Source function:

$\rm S_\nu=\frac{j_\nu}{\kappa}$

## Emission or absorption

bg: background
fg: foreground
if $\rm T_{bg}< T_{fg}$, emission line is formed
if $T_{bg}> T_{\text{fg}}$, absorption line is formed

## Line radiative transfer fundamental

Section 17, see e.g., ERA 7.3
......
Q&M: How can we describe the microscopic phycsical processes in the equation of RT?

New concepts:

1. Profile function: $\psi(\nu)$
2. Radiation energy corresponding to the line: $\bar{j}$
3. Einstein coefficients: $A_{12}, B_{12}, B_{21}$
$\downarrow$
***Only need to know one Einstein coefficient, and another two can be derived.

$\rm \int_{0}^{\infin} \psi(\nu) d\nu = 1$

$\bar{j}=\frac{1}{4\pi}\int\{\int_0^{\infin}I_{\nu}(\theta,\phi)\psi(\nu)d\nu\}d\Omega$

## Excitation temperature

Section 18
ERA 7.4
Defination: T dericed from Boltzman equation, not real temperature.
Q: what is Boltzman distribution?
$\rightarrow$point at LTE

## Polarization fundamental

Section 19
See. e.g., ERA 2.3