Study of spectral analysis and Doppler effect

Motivation

Taking pictures of celestial bodies' spectrum, we get information about their spectral class, stellar atmosphere, temperature, magnetism, etc. Also, we could find out its radial velocity and rotation al velocity from Doppler shift. Which would be the point results in my interest of Spectroscopy.

Purpose

Find out selected celestial bodies' rotational velocity.

P.S. At first, I think if the resolution is precise enough, I can get such a picture like the left:

Then I can even figure out the relation between mass and radius compare to the center of M31. But finally I found it's almost impossible to do that because of instrument problems. Even the best instruments in Taiwan may not able to do this.

Methods and steps and schedule

This plan is going to separate into several parts:

To design observations and instruments in need.

For related affairs, see 4. Estimation.

Putting observations into practice and collecting previous reference datum.

The part of observation is going to be putted into practice by graduate students in NCU, thanks for their help.

This plan has three mainly kinds of observe objects (First, they all need to have enough brightness):

Bright fixed stars with fast rotational velocity

I choose stars that would be observed during plan period, apparent magnitude is lighter than 4, and rotational velocity (Found in "The BRIGHT STAR CATALOGUE") is faster than usual.

The table below is all selected fixed stars from the book "The BRIGHT STAR CATALOGUE"

HD	star name	RA(h m s)			DEL (° ' ")			V	Spec	RV(k/s)	Vsini(k/s)
5394	CAS gam	0	54	35.2	+19	11	18	2.47	B0Ive	-7SB	300:
8538	CAS del	1	25	48.9	+60	14	7	2.68	A5III-Ivv	+7SB	113
6882	PHE zet	1	8	23.0	-55	14	45	3.92	B6V+B9V	+15SB2O	127
10144	ERI alp	1	37	42.9	-57	14	12	0.46	B3Vpe	+16V	251
14228	ERI phi	2	16	30.6	-51	30	44	3.56	B8V-IV	+10	247
11443	TRI alp	1	53	48	+29	34	44	3.41	F6IV	-13SB1O	93
12311	HYI alp	1	58	46.2	-61	34	12	2.86	F0V	+1V	153
16970	CET gam	2	43	18	+3	14	9	3.47	A3V	-5V	183
17573	ARI 41	2	49	59.0	+27	15	38	3.63	B8Vn	+4SB	180
22928	PER del	3	42	55.4	+47	47	15	3.01	B5III	+4SB	259
23302	TAU 17	3	44	52.5	+24	6	48	3.7	B6IIIe	+12SB1O	215
23630	TAU eta	3	47	29.0	+24	6	18	2.87	B7IIIe	+10V?	215
23850	TAU 27	3	49	9.7	+24	3	12	3.63	B8III	+9SB1O	212
166014	HER omi	18	7	32.5	+28	45	45	3.83	B9.5V	-30SB	134
165024	ARA the	18	6	37.7	-50	5	30	3.66	B2Ib	+3	117
169022	SGR eps	18	24	10.3	-34	23	5	1.85	B9.5III	-15	140
175191	SGR sig	18	55	15.8	-26	17	48	2.02	B2.5V	-11V	201
177724	AQL zet	19	5	24.5	+13	51	48	2.99	A0Vn	-25SB	331
177756	AQL lam	19	6	14.8	-4	52	57	3.44	B9Vn	-12V	176
187642	AQL alp	19	50	46.9	+8	52	6	0.77	A7V	-26	242
184006	CYG iot2	19	29	42.2	+51	43	47	3.79	A5Vn	-20V?	226
186882	CYG del	19	44	58.4	+45	7	51	2.87	B9.5IV+F1V	-20SB	149
199629	CYG nu	20	57	10.3	+41	10	2	3.94	A1Vn	-28SB	241
202444	CYG tau	21	14	47.4	+38	2	44	3.72	F2IV	-21SB	89
196867	DEL alp	20	39	38.2	+15	54	43	3.77	B9IV	-3SB	162
198001	AQR eps	20	47	40.5	-9	29	45	3.77	A1V	-16V?	98
203280	CEP alp	21	18	34.7	+62	35	8	2.44	A7V	-10V	246
209952	GRU alp	22	8	13.9	-46	57	40	1.74	B7IV	+12	236
210418	PEG the	22	10	11.9	+6	11	52	3.53	A2V	-6SB2	117
214923	PEG zet	22	41	27.6	+10	49	53	3.40	B8V	+7V?	194
218045	PEG alp	23	4	45.6	+15	12	19	2.49	B9V	-4SB	148
213558	LAC alp	22	31	17.4	+50	16	57	3.77	A1V	-4V?	146
215789	GRU eps	22	48	33.2	-51	19	1	3.49	A3V	+0V	236
216956	PSA alp	22	57	39.0	-29	37	20	1.16	A3V	+7	100
217675	AND omi	23	1	55.2	+42	19	34	3.62	B6IIIpe+A2p	-14SB	330

RA & DEC => A.D.2000
V => magnitude
RV: SB => spectroscopic binary, O => orbit available V => variable star, ? => variable star that suspected

Galaxy-M31

When choose a galaxy, I considered about its brightness. M31 would be the only good choice of all. (mag=4.8, but it has a wide area=2°40' * 35') But there's still a big problem about its unit magnitude. For solution, see 4.Estimation.

Planets with atmosphere

Jupiter and Saturn are chosen because they are bright and rotate fast. And they are visible during the period.

The table below shows data about selected planets. (The R.A. and DEC is given as 1998/7/31, and rotational velocity is figured out by 2π*radius / rotational per iod)

Planet name	Magnitude	Rotational velocity(km/s)	RA(h m s)			DEC(° ' ")
Jupiter	-2.5	58.903	23	54	07.466	-2	10	41.99
Saturn	0.5	46.270	2	08	59.540	+10	17	05.39

P.S. Venus is bright enough, but it rotates only 8m/s, and the "wind velocity" is 10m/s~100m/s.

Data processing and examining disadvantages.

Get a digital image.

Find out a minimum Δλ in the spectra by PC.

Figured out the wavelength that shifted (also, I have to correct the radial velocity that shifted by comparing with the comparison spectra), then put the number into the formula of Doppler effect, we get the rotational velocity of the star.

P.s. In the NCU, the comparison spectra that settled are Hg/Ne.

Examining if the exposure time is enough, which estimated previously.

Examining if the gratings are properly, if not, change a different one.

To improve observations.

To integrate all results and coming to a conclusion with a report.

Estimation

Rotational velocity that measured before by Doppler shift

Looking at the star table above, almost selected stars have a rotational velocity that larger than 100km/s. From Doppler effect we know that their spectra would shift, and the wavelength shifted is o bey to the formula below:

V=C*Δλ/λ, so when the star rotates at the velocity of 100km/s, the wavelength that shifted would be:
10⁵ (m) * 6.563*10^-7 (m)/ 3x10*⁸ (m) = 2.187*10^-10(m) (Take the Hα wave for example.)

And the object has two sides, one is toward us while the other is backward, so the total shift would be 2.187*10^-10x2=4.375*10^-10 (m)

　P.S. "Shift" in this project means the quantity of certain beam's broading compare to the center of that wavelength (Rotational velocity). Not the shift of stellar spectra compares to standard lamp (Radial velocity).

Design a spectroscope

Resolution of dispersion instruments.

As a result, the resolution of a spectrograph that need must be so smaller than 2.187 angstrom that we can identify the shift.

Diffraction gratings

The spectral resolution, R, is defined as

R=λ/Δλ

That is to say, when the wavelength is 6563Å, and the shifted wavelength is 2.187 A, the resolution is

6563/2.187=3000 (apertures).

So the total number of apertures that I need is 3000 at least.

Prism

Resolution

For a prism, the resolution is given by

R≒ABL{1-0.25[A+B(λ-C)]²}^1/2/(λ-C)²

Where L is the length of the face of the prism, and A, B, C, is Hartmann constants of different material. So the size of prism that need is larger than

L=3000*(6.563*10^-7+2.5*10^-7)²/ 1.5x3.5*10^-8{< FONT FACE="System,新細明體" LANG="ZH-TW">1-0.25[1.5+3.5*10^-8(6.563*10^-7+2.5*10^-7)]²}^1/2=2.366*10^-5(m)=0.023mm (Take the Crown glass for example.)

Such a small size seems very easily available. But the material isn't usually found in a store. So let's take the Fluorite for example:

L=3000*(6.563*10^-7-3.6*10^-7)²/ 1.429x5.3*10^-10{1-0.25[1.429+5.3*10^-10(6.563*10^-7-3.6*10^-7)]²}^1/2
=2.306(m)

It's so large a size and even an educational institute wouldn't be able to afford.

We found that when using different material, the size that need changes a lot. I should make sure of what material can I available.

Angular

For the estimation of prism's angular, see CYCHEN's paper. (CYCHEN, a student of NCUO)

CYCHEN considered about resolution and refraction, and he figured it's most suitable angular would be 30 degree. (He took the BK-7 material for example, 10cm diameter.) But the resolution he requests isn't enough for my plan. So we can't order a prism in common. But Prof. Tsay already had a prism settled in a T-ring of Nikon. Maybe we could try it.

a. Diameter and focal length that need

i.Diameter:

The basic need of telescope's diameter should be larger than the length of gratings. And that estimated above is much smaller than usual. (While I use a 600lines/mm grating, its minimum length is 5mm). S o all that I need to consider about is star's brightness it can observed.

ii.Focal length

From Young's experiment of double slits we know that the formula of image width (Y) is given by

Y=mfλ/d

("m" means order, "f" is the distance between slit and screen, and "s" is the separation of the apertures).

Because the secondary maximum (m=1) is weaker than the primarily one (m=0), the more the order is, the weaker the brightness will be.

So I choose the first order, "m"=1.

And I still take the Hα wave for example, "λ"=6.563*10^-7 (m).

Take the ST-4 for example, its each pixel diameter is 13.75*16microns, so the minimum "Y" required is about 1.6*10^-5 m

When using a 600lines/mm grating, its "s"=1.67*10^-6 m, so "f"(focal length) = 1.6*10^-5x1.67*10^-6/1x6.563*10^-7=4.063*10^-5 m=0.0406 mm, it's also much shorter than usual.

So as the diameter, the only one I have to consider about is the brightness. (The longer the focal length is, the weaker the brightness will be.)

(3)Photographic equipment

Resolution of film or CCD

As figured above, the resolution of a CCD camera doesn't matter, not mention to the film. Its particle is always smaller than a pixel of CCD camera.

Exposure time

Reference to previous record:

Celestial name	Magnitude	Instruments	Exposure time
Comet Hale-Bopp	≒4	Meade LX200-20(f/6.3) +武藤CV04	10seconds
M82	8.8	Pentax75SDHF + Meade Pictor416	120seconds

So as taking a single image of my selected fixed star and planets, the exposure time can less than 10 seconds, and that of M31 could be about 120 seconds. (Comparing a star near the lip of plate, both M82 and M 31 has a magnitude about 14~15 in their spiral arm.) But while dealing with spectra, the exposure time need to be gained.

The intensity in the interference pattern at an angle, θ, to the direction of the incoming radiation, I(θ), relative to the central intensity of the pattern, I(0), is given by

I(θ) / I(0)=sin²(πd sinθ/λ) / (πd s inθ/λ)² x sin²(Nπs sinθ/λ) / sin²(πs sinθ/λ)

Where "d" is the width of the aperture and N is the number of apertures. So as Picking up the first order, its brightness will be about 1/10 of total brightness. So the exposure time must increase to 10 times of origin.

Spectroscopic standard stars

Already find on http://vizier.u-strasbg.fr/cgi-bin/VizieR

Instruments that available

For the best instruments in Taiwan, NCU, Institute of Astronomy has a 600mm-diameter telescope, and its focal length is about 6000mm. I've attempted to find out how long should it exposure when using such instrument but failed because that I doesn't know the relation between brightness and items such as diameter, focal length, and the spectrograph settled behind.

(The spectrograph settled behind: the slit area is 1.5 mm* 0.1mm (or 0.05mm), and the focal length is 250mm. )

Reference

The BRIGHT STAR CATALOGUE -- fourth revised edition, by Dorrit Hoffleit.

Optical Astronomical Spectroscopy -- by C R Kitchin.

ASTRONOMICAL OBSERVATIONS an optical perspective -- CAMBRIGE UNIVERSITY PRESS.

A WORKBOOK FOR ASTRONOMY -- by JERRY WAXMAN, CAMBRIGE UNIVERSITY PRESS.

CCD Camera Operating Manual for the Model ST-4X, ST-5 and ST-6 -- Santa Barbara Instrument Group

How to 冷卻CCD --by村上將之,

天體 --by 天文 ,誠文堂新光? ?/LI>

Student: YunJu Tai,. Study in the TFG.
E-mail: yjtai@ms31.hinet.net
Taipei, Taiwan.

Real progress

Date	Item	Result	P.S.
1998/7/2~3	Take pictures on Mt. Ta-Shif	Failed because the Angular of the prism is to big
1998/7/6	Photo spectra of Na by the spectrograph in NCUO	Its resolution can be higher than 2~3angstrom
1998/8/1~2	Take pictures on Mt. Li	Use 50mm camera, with 100ASA film, we can get Jupiter's spectra.	Not preparing well, I can only with poor equipment. >_< And the grating used has a too large disperse angle, so we can't see spectra from camera lens. I can only guess… As a result, I didn't get a right angle for each photograph, the one for Jupiter is the best.