PHYSICS TUTORIAL NOTES

Astrophysics

Option 9.7

These notes are meant as a guide only and are designed to focus your thoughts on the dot points mentioned in the syllabus. They give a very brief overview of the topic and should be used in conjunction with your class notes, your textbooks and your research from other areas such as the library and the Internet.

Notes compiled by:

CARESA EDUCATION SERVICES

ASTROPHYSICS

“1. Our understanding of celestial objects depends on observations made from Earth or from space near the Earth.” (syllabus)

Students learn to: “Discuss Galileo’s use of the telescope to identify features of the Moon.” (syllabus)

In the early 17^th century, Galileo learned that a Dutchman, Hans Lippershey, had developed a telescope, so Galileo proceeded to build one for himself. He turned his telescope towards the Moon and observed that its surface was not smooth but covered with mountains and craters. He calculated the height of some lunar mountains. His observations challenged accepted beliefs that the Moon was not perfectly smooth and spherical as all extraterrestrial objects were claimed to be.

Some sites that cover Galileo’s work in more detail are

www-astronomy.mps.ohio-state.edu/ ~pogge/Ast161/Unit3/galileo.html

www.seds.org/messier/xtra/Bios/galileo.html

www-gap.dcs.st-and.ac.uk/ ~history/Mathematicians/Galileo.html

galileoandeinstein.physics.virginia.edu/ lectures/galtel.htm

csep10.phys.utk.edu/astr161/lect/history/galileo.html

es.rice.edu/ES/humsoc/Galileo/Things/telescope.html

www.crs4.it/Ars/arshtml/galileo3.html

www.galileo-galilei.org/galileo-telescope.html

Students learn to: “Discuss why some wavebands can be more easily detected from space. ” (syllabus)

Gamma rays and X-rays are absorbed by the upper atmosphere. Long wavelength infrared and short wavelength microwaves are also absorbed by the upper atmosphere.

Water vapour and gases in the atmosphere reflect some I.R., while the ozone layer of the upper atmosphere absorbs some U.V. radiation. Astronomers rely mostly on visible light and radio waves for ground-based observations. Other regions of the spectrum are best analysed from space-based observations.

Students learn to: “Define the terms “resolution” and “sensitivity” of telescopes. ” (syllabus)

The resolution of a telescope is its ability to form separate images of two objects that are close together.

The sensitivity of a telescope refers to the amount of light it can gather.

Students learn to: “Discuss the problems associated with ground-based astronomy in terms of resolution and absorption of radiation and atmospheric distortion. ” (syllabus)

The movement of the atmosphere and the variation of its density at different heights and temperatures decreases the resolution of ground-based telescopes. The selective absorption of E.M. radiation by the atmosphere restricts observations to the visible and radio bands.

Students learn to: “Outline methods by which the resolution and/or sensitivity of ground-based systems can be improved, including

- adaptive optics

- interferometry

- active optics” (syllabus)

Adaptive optics involves measuring atmospheric distortions and compensating for them. With a computer and flexible mirror the telescope is able to detect atmospheric distortions and allow for them. Its main difference to the active optics is the response speed. It can make around 1000 wavefront corrections per second and this results in a pronounced improvement in the resolving power of the telescope.

Interferometry involves the study of interference patterns formed by two coherent electromagnetic radiation sources. Two nearby telescopes can simultaneously view the same star and interference patterns because of path difference of the light rays can be observed. By observing the changing patterns as the Earth rotates, such information as the star’s size can be determined.

Active optics is a system of computer-controlled actuators that correct imperfections in the shape of the mirror and compensates for distortions due to temperature variations. A slow feedback system between the mirror and eyepiece (or camera) that detects how the incoming light has been altered and then makes the necessary adjustments to the mirror.

A couple of good sites describing Adaptive and Active optics are:

http://www.ing.iac.es/~crb/wht/aointro.html

http://www.physics.usyd.edu.au/physopt/ao/ss.html

Students: “Identify data sources, plan, choose equipment or resources for, and perform an investigation to demonstrate why it is desirable for telescopes to have a large diameter objective lens or mirror in terms of both sensitivity and resolution. ” (syllabus)

Sensitivity:

Telescopes can give a brighter image by collecting more light. Shine light through a hole onto a light meter. Replace the hole with a larger one. As the diameter of the hole increases, so too should the intensity of light.

Resolution:

High resolving power results in an image that is more clear because points close together can be separated into separate images.

View dots that are close together, e.g. newsprint, through a lens of small diameter then one of larger diameter.

“2. Careful measurement of a celestial object’s position in the sky (astrometry) may be used to determine its distance. ” (syllabus)

Students learn to: “Define the terms parallax, parsec, light-year.” (syllabus)

Parallax is the change in position of an object against its background due to the change in position of the observer.

A parsec (Pc) is the distance corresponding to an annual parallax of 1 second of arc.

A good approximation is to take the radius of the Earth’s orbit as an arc and use the formula;

arc = rq where r is the radius and q is the angle subtended in radians.

1.5 x 10¹¹ = r x 1/3600 x 2p/360 r = 1 Pc = (1.5x10¹¹ x 3600 x 360)/2p

= 3.09x10¹⁶ m.

A light year (L.Y.) is the distance light travels in a year.

L.Y. = 3 x 10⁸ x 365.25 x 24 x 60 x 60 = 9.47 x 10¹⁵ m.

1 parsec = 3.26 light years.

Students learn to: “Explain how trigonometric parallax can be used to determine the distance to stars.” (syllabus)

The star whose distance we want to measure is carefully lined up with a background star. Six months later the angle between the background star and the near star is measured. Since the background star is considered to be at infinity, the angle is the same as measuring the angle to the star from opposite sides of the Earth’s orbit. Knowing the diameter of the Earth’s orbit and the angle subtended to the star from opposite sides of this orbit make it easy to determine the star’s distance by trigonometry.

Students learn to: “Discuss the limitations of trigonometric parallax measurements.” (syllabus)

Trigonometric parallax is limited by the ability to measure the small angles and also by atmospheric movements.

Students: “Solve problems and analyse information to calculate the distance to a star given its trigonometric parallax using:

d = 1/p.” (syllabus)

Use the formula d = 1/p to complete the following table:

Name of star	Parallax in seconds	Distance in parsecs	Distance in light years
Rigel (b Orion)	0.0040
Hadar (b Centauri)		120
Aldebaran (a Taurus)	0.0476
Canopus (a Carina)			196
Pollux (b Gemini)		11

Name of star	Parallax in seconds	Distance in parsecs	Distance in light years
Rigel (b Orion)	0.0040	250	815
Hadar (b Centauri)	0.0083	120	391
Aldebaran (a Taurus)	0.0476	21	68.5
Canopus (a Carina)	0.0166	60	196
Pollux (b Gemini)	0.0909	11	36

Students: “Gather and process information to determine the relative limits to trigonometric parallax distance determinations using recent ground-based and space-based telescopes.” (syllabus)

Measurements made by satellites such as Hipparcos are made above the atmosphere and do not suffer from atmospheric distortion. Whereas ground-based measurements are limited to about 30 – 40 Pc it is possible to measure distances up to about 1000 Pc by making measurements from space.

Hipparcos: http://astro.estec.esa.nl/Hipparcos/

“3. Spectroscopy is a vital tool for astronomers and provides a wealth of information.” (syllabus)

Students learn to: “Account for the production of emission and absorption spectra and compare these with a continuous blackbody spectrum.” (syllabus)

A spectrum is a wave that has been separated into its component wavelengths. A continuous spectrum has all wavelengths present e.g. a rainbow. While all wavelengths are present in a continuous spectrum they are not of equal intensity. A blackbody emits a continuous spectrum with the most intense wavelength depending on the temperature of the body. A line spectrum is black except for a few lines of characteristic wavelengths.

An emission spectrum consists of the wavelengths emitted by a low-pressure gas that is in an excited state because energy has been put into it. It arises from electrons returning from a high-energy state to a low energy state. Low-pressure gases in an excited state produce an emission line spectrum. Heated solids and high-pressure gases produce a continuous emission spectrum.

An absorption spectrum is formed when a continuous spectrum shines through a gas that is cooler than the continuous spectrum source. The gas absorbs energy at its characteristic wavelengths and re-radiates it in all directions. It transmits other wavelengths of the continuous spectrum so that the result is a continuous spectrum with black stripes.

A good site on this is

http://www.astro.uiuc.edu/~kaler/sow/spectra.html

Students learn to: “Describe the technology needed to measure astronomical spectra.” (syllabus)

A telescope fitted with a spectroscope is needed to study astronomical spectra. A telescope fitted with a photomultiplier can also be used. The photomultiplier can accurately measure the intensity of different wavelengths, thus indicating spectral characteristics.

Students learn to: “Identify the general types of spectra produced by stars, emission nebulae, galaxies and quasars.” (syllabus)

Stars:

Stars consist of an intensely hot core and a surrounding layer of cooler gases called the atmosphere. As light from the core passes through the cooler gases surrounding it, the cooler gases absorb energy at their characteristic wavelengths, and re-radiate it in all directions. The result is the continuous spectrum produced by the core, with a series of dark lines where the light has been absorbed and re-radiated in other directions. By analyzing the wavelengths of the dark lines it is possible to determine the composition of gases in the atmosphere. Such a spectrum is called an absorption spectrum.

In 1814 Joseph von Fraunhofer discovered that the Sun’s spectrum was not continuous but was crossed by a large number of dark lines. He identified 576 of them and it was found that these lines correspond to the bright lines of the emission spectra produced by certain elements. These lines, called Fraunhofer lines, enable the composition of the Sun’s outer layer to be determined. The composition of more distant stars can be analysed in a similar way.

Emission Nebulae:

These result from a cloud of gas absorbing energy from a nearby star and re-radiating it. The result is the emission spectra of the gases present in the cloud. Most of these lines are from hydrogen and ionized oxygen.

Reflection Nebulae:

These result from clouds of gas reflecting light from a nearby star. The scattering of light is proportional to the fourth power of the frequency. Since blue light has about twice the frequency of red light it is scattered 2⁴ i.e. 16 times as much. Unless the nebula is directly between the star and Earth we observe the scattered light and consequently reflection nebulae appear blue. (This is similar to why the sky appears blue due to scattering of sunlight by the Earth’s atmosphere.) Reflection nebulae show faint absorption spectra.

Galaxies:

Galaxies contain stars of many spectral types as well as nebulae and other celestial objects that produce spectra. Consequently galaxies produce spectra that have a combination of all of these objects. In addition, most galaxies are moving away from us and the spectral lines show a red shift that depends on the speed of recession. More distant galaxies are receding faster and show a greater red shift.

Quasars:

Quasars were first called “quasi-stellar radio sources” because they look like stars and emit an enormous concentration of radio waves. Analysis of their spectrum shows that they have very large red shifts and consequently must have very fast speeds of recession and be a long way away. Because they can be seen at these enormous distances (billions of light years) they must be emitting energy at a tremendous rate. Whereas stars produce strong absorption lines, quasars emit strong emission lines.

Students learn to: “Describe the key features of stella spectra and describe how these are used to classify stars.” (syllabus)

Although most stars are similar in composition their spectra can vary greatly. It has been found that the absorption spectrum depends to a large extent on temperature (determined by Wien’s law or colour index). Helium, for instance, is shown in the hottest stars. This is because to form the dark line of the helium spectrum, the helium atom must absorb a large quantum of energy from the hot body behind it. As the energy produced depends on the temperature of the body, it is only the hot stars that emit quanta of sufficient size to cause electrons in the helium atom to jump up to a higher energy level, so that they can cause the dark line in the spectrum as they return to their original energy level.

In 1863 Angelo Secchi classified stars according to their spectral characteristics. The stars were originally placed in alphabetical sequence. However, some were originally placed in the wrong temperature order and some of the original classifications have been dropped. What remains are the spectral classes represented by the letters O,B,A,F,G,K,M. (The astronomer Henry Russell suggested remembering these by the mnemonic “Oh Be A Fine Girl (or Guy) Kiss Me”). Each of the classes is subdivided into ten divisions: 0,1,2, ……..8,9. In each case the sequence is listed in order of decreasing temperature. Hence O0 is the hottest and M9 is the coolest.

The characteristics of stars of each spectral class are listed in the table below.

Spectral Class	Temperature (K)	Colour	Principal Spectral Lines	Example
O	Over 25 000	Blue	Ionised helium	10 Lacertra
B	11 000-25 000	Blue	Neutral helium & hydrogen	a Crucis
A	7500-11 000	Blue-white	Strong hydrogen lines	Sirius
F	6000-7500	White	Ionised metals	Canopus
G	5000-6000	Yellow-white	Ionised calcium + ionized neutral metals	The Sun a Centauri
K	3500-5000	Orange	Neutral metals	Aldebaran
M	Under 3500	Red	Neutral metals + molecular titanium oxide	Betelgeuse

Students learn to: “Describe how spectra can provide information on surface temperature, rotational and translational velocity, density and chemical composition of stars.” (syllabus)

Surface Temperature: The fundamental spectrum of a star is that of a black body curve. By determining the wavelength of maximum intensity it is a simple task to calculate the surface temperature by the application of Wien’s Law: l_max = W/T.

Another method of determining the temperature is to determine the apparent magnitude of the star at two different wavelengths. The most common of these is the blue (photographic) wavelength and the yellow (visual) wavelength. The difference in magnitudes at different wavelengths, known as the colour index, is related to the stars black body spectrum and can be used to determine the star’s surface temperature.

Rotational and Translational Velocity: The velocity of a star can be measured by its red or blue shift due to the Doppler effect. As a star is moving towards us the wavelengths in its observed spectrum become shorter. As they are shifted towards the blue end of the visible spectrum it is called a blue shift. Similarly as a star is moving away the wavelengths of its observed spectrum become longer as it exhibits a red shift. Since the amount of the red or blue shift is determined by the velocity of the star relative to the observer it is possible to determine the translational (straight line) velocity of stars that are moving towards or away from us. By examining the variation in red and blue shifts it is possible to identify some binary stars (spectroscopic binaries) and to calculate their period of rotation and rotational velocity.

Density: When atoms are well apart (low density) they absorb quanta of specific energy levels only. The result is sharp, narrow spectral lines. As the density of the gas increases and the atoms become closer together, electronic interactions tend to modify the atomic energy levels so that some become slightly larger while others become slightly smaller. The atoms absorb quanta of slightly higher and lower energy levels with the consequent broadening of the spectral lines. The greater the density of the star then the greater is the amount of spectral broadening.

Chemical Composition: Each element has its own characteristic spectrum. The pressure at the centre of a star is high so it produces a continuous spectrum. As the light from this spectrum passes through the cooler outer atmosphere, the atoms of the outer atmosphere absorb energy at their characteristic wavelengths and re-radiate it in all directions. The result is a continuous spectrum with black lines on it (an absorption spectrum) indicating the chemical composition of the outer atmosphere.

Students: “Perform a first-hand investigation to examine a variety of spectra produced by discharge tubes, reflected sunlight, or incandescent filaments.” (syllabus)

To do this investigation in the laboratory you will require a spectroscope (the hand held type is easiest to use), an incandescent lamp and some discharge tubes. To get the high voltage needed for the discharge tubes you will probably need an induction coil and leads as well. Use coloured pencils to record the spectra that you observe.

View reflected sunlight (e.g. a window) through a spectroscope and record its spectrum. N.B. Do not look at the Sun directly, even through a spectroscope. It can cause permanent blindness.

View an incandescent lamp through a spectroscope and record its spectrum.

Set up a number of discharge tubes by connecting them to a high voltage supply. Observe them through a spectroscope and record their spectra.

Some interesting sites to observe spectra are listed below:

http://230nsc1.phy-astr.gsu.edu/hbase/hyde.html

http://cat.middlebury.edu/~chem/chemistry/class/general/ch103/hatom/index.html

http://csep10.phys.utk.edu/astr162/lect/light/absorption.html

http://www.physchem.co.za/Atomic/Hydrogen%20Spectrum.htm

Students: “Analyse information to predict the surface temperature of a star from its intensity/wavelength graph.” (syllabus)

A graph of radiant energy against wavelength for three different stars is plotted below. Calculate the surface temperature for each of the three stars.

Solutions.

Star 1.

l_max = 1.7 x 10^-7m (from graph) T = W/l_max = 2.89 x 10^-3/ 1.7 x 10^-7 = 17000K

Star 2.

l_max = 2.9 x 10^-7m (from graph) T = W/l_max = 2.89 x 10^-3/ 2.9 x 10^-7 = 10000K

Star 3.

l_max = 5.9 x 10^-7m (from graph) T = W/l_max = 2.89 x 10^-3/ 5.9 x 10^-7 = 4900K

“4. Photometric measurements can be used for determining distance and comparing objects.” (syllabus)

Students learn to: “Define absolute and apparent magnitude.” (syllabus)

Magnitude of stars is a way of classifying them on a logarithmic scale according to brightness. Apparent magnitude refers to the magnitude of stars from where they are observed (Earth) whereas absolute magnitude refers to the magnitude that a star would have if viewed from a standard distance of 10 parsecs.

Hipparchus (2^nd century B.C.) developed a method of classifying stars according to their brightness. He defined the brightest stars as having magnitude 1 and those just visible to the naked eye as having magnitude 6. In the 19^th century, John Herschell determined that magnitude 1 stars were about 100 times as bright as magnitude 6 stars. This led to Norman Pogson in 1856 putting forward the DEFINITION of 5 units of magnitude (6 –1) being equal to 100 units of brightness. Hence each unit of magnitude implies a difference of brightness of ⁵\/(100) i.e. 2.512. Hence a star of magnitude 4 is 2.512 times as bright as one of magnitude 5 and 2.512² (i.e. 6.310) times as bright as one of magnitude 6.

Students learn to: “Explain how the concept of magnitude can be used to determine the distance to a celestial object.” (syllabus)

The difference between the apparent magnitude (m) and the absolute magnitude (M) is known as the distance modulus. Distance modulus = m – M

This can be used to find the distance by applying the formula: m – M = 5 log (d/10)

Students learn to: “Outline spectroscopic parallax.” (syllabus)

The term “Parallax” is a bit misleading in this context because it is the properties of the star on the H.R. diagram rather than its displacement against background stars that is used to determine its distance.

The apparent magnitude of the star is determined by photometry and spectroscopy is used to determine its spectral properties and class. The spectral class is located on the horizontal axis of the H.R. diagram. By drawing a vertical line to the appropriate star group and then drawing a horizontal line to the vertical axis, the absolute magnitude can be determined. Since we now know both the apparent and absolute magnitudes we can use the distance modulus to determine the distance to the star.

Exercise: An astronomer analysed the spectrum of a star and found that was spectral class G0. She measured its brightness and found that it had an apparent magnitude of 3.5. Show how she could use spectroscopic parallax to determine the distance to the star and hence determine the distance.

Answer: G0 is the hottest type of G star so she would draw a vertical line from the hot end of the G class stars on the horizontal axis of the H-R diagram.

Continued…

Where the vertical line intersects with the main sequence stars she should draw a horizontal line so that it cuts the vertical axis. This will show the absolute magnitude of the star as about 3.0.

The distance modulus m – M is determined as 3.5 – 3.0 = 0.5

m – M = 5 log d/10 log d/10 = 0.5/5 = 0.1 d/10 = 1.3 d = 13 Pc

Students learn to: “Explain how two-colour values (i.e. colour index, B-V) are obtained and why they are useful.” (syllabus)

The blackbody graphs of intensity v^s wavelength show that hot objects emit different amounts of energy at different wavelengths. The difference between the intensities at two different wavelengths can be used to determine the temperature of the body.

By measuring magnitudes at two different wavelengths it is possible to determine the temperature of a star. Photographic emulsions are most sensitive to blue light so the photographic magnitude is written as m_B or more commonly just as “B”. The human eye is most sensitive to yellow light so the visual magnitude is written as m_V or more commonly just as “V”. The difference between the two magnitudes B - V is called the colour index. The temperature of a star can be determined by determining its colour index. ”. A star with a surface temperature of 10 000 K emits the same amount of blue light as yellow light. Hence B = V. For stars hotter than 10 000K, B < V while for stars cooler than 10 000K, B > V. The lower the colour index, the hotter the star.

Exercise. The table below lists the surface temperature and colour index of several stars. On the axes provided, plot a graph of temperature against colour index.

Star	Spectral Class	Colour Index B-V	Temperature K
Proxima Centauri	M5	1.97	2800
Aldebaran	K5	1.53	4350
Arcturus	K2	1.23	4760
Pollux	K0	1.00	4900
Sun	G2	0.79	5800
Procyon	F5	0.41	6700
Canopus	F0	0.16	7400
Sirius	A1	0.00	10000
Rigel	B8	-0.12	14000
b Centauri	B1	-0.23	25000
a Crucis	B0	-0.26	28000
Naos	O5	-0.30	34000

Solution: Graph of Temperature against Colour Index.

Students learn to: “Describe the advantages of photoelectric technologies over photographic methods for photometry.” (syllabus)

Because stars emit radiation of many wavelengths and the intensity of the radiation is different at each wavelength, the magnitude of the star depends on the sensitivity of the receiver at a particular wavelength. For instance, the human eye is most sensitive to the yellow-green portion of the spectrum while photographic emulsions are most sensitive in the blue-violet portions of the spectrum. Hence a blue star will appear brighter on a photograph than to the eye while a red star will appear dimmer in the photograph. Photographic emulsions also vary and different films (e.g. different brands) are likely to be most sensitive to different wavelengths.

Photocells measure the total light of all wavelengths. By using suitable filters it is possible to measure the light of a particular wavelength.

Students: “Solve problems and analyse information using M = m – 5log (d/10) and I_A/I_B = 100^{(mB – mA)/5 (}*see 2 lines down) to calculate the absolute or apparent magnitude of stars using data and a reference star.” (syllabus)

*The index of 100 is (m_B- m_A)/5 My version of Microsoft Word does not have the Microsoft Equation program so I can’t write subscripts in indices.

Q.1. Sirius is 2.65 Pc away and has an apparent visual magnitude of –1.46. What is its absolute visual magnitude?

A.1. M = m – 5log (d/10) = -1.46 – 5 log (2.65/10) = -1.46 –5log 0.265

= -1.46 - -2.88 = 1.42

Q.2. Beta Centauri has an apparent visual magnitude of 0.60 and an absolute visual magnitude of –5.0. How far away is it?

A.2. M = m – 5log (d/10) -5 = 0.6 – 5 log (d/10) 5 log (d/10) = 5.0 + 0.6 = 5.6 log (d/10) = 5.6/5 = 1.12 d/10 = 13.2 d = 132 Pc.

Q.3. Sirius A has an apparent visual magnitude of –1.46 while Sirius B (a white dwarf) has an apparent visual magnitude of 8.68. What is the ratio of the intensities of light received from Sirius A and Sirius B on Earth?

A.3. I_A/I_B = 100^{(8.68 - - 1.46)/5} = 100^10.14/5 = 100^2.028 = 11376:1

Q.4. Alpha Crucis (the brightest star of the Southern Cross) has an apparent visual magnitude of 0.9 whereas Beta Crucis (the second brightest star of the Southern Cross) has an apparent visual magnitude of 1.26. What is the ratio of the intensities of light received from Alpha and Beta Crucis on Earth?

A.4. I_A/I_B = 100^{(1.26 – 0. 9)/5} = 100^{0. 36/5} = 100^0.072 = 1.4:1

Students: “Perform an investigation to demonstrate the use of filters for photometric measurements.” (syllabus)

Purpose: To determine the surface temperature of the Sun.

Equipment: Light meter, filters.

If these are not available it may be possible to borrow a camera with a built-in light meter and some photographic filters to measure the intensity of light at various wavelengths.

Method: Place the light meter on a flat surface when the Sun is clearly visible (no clouds). Measure the intensity of the sunlight.

Place a filter over the light meter and again measure the intensity of light. The drop in intensity from the unfiltered light to the filtered light represents the intensity of light at the frequency of the filter. Repeat the procedure for as many filters as possible.

Wavelength of filter	Unfiltered intensity	Filtered intensity	Intensity at that wavelength

Plot a graph of Intensity (vertical axis) against wavelength.

Determine the wavelength at which the peak intensity occurs.

Apply Wien’s Law; l_max = W/T where

l_max = most intense wavelength

W = Wien’s constant = 2.89 x 10^-3 mK

T = temperature (kelvin)

to determine the temperature of the Sun.

A good range of filters that transmit a narrow band of light are the Schott broadband filters.

Code	Maximum Transmitted Wavelength (nm)	Thickness (mm)	Range of Transmitted Wavelengths (nm)
WG 305	305	3	290 - 320
GG 385	385	2	360 - 400
GG 395	395	2	365 - 415
GG 455	455	2	440 - 465
GG 495	495	2	480 - 510
OG 590	590	3	580 - 600
RG 630	630	3	620 - 640
RG 715	715	4	690 - 730

If these are available they can be used with a light meter to determine the transmitted intensity at each wavelength and a black body curve of intensity v^s wavelength can be constructed. The temperature of the Sun can be determined as above by applying Wien’s law to the most intense wavelength.

Students: “Identify data sources, gather, process and present information to assess the impact of improvements in measurement technologies on our understanding of celestial objects.” (syllabus)

The sensitivity and resolution of a telescope is governed by the size of the mirror. Large mirrors suffer from “sagging” where the weight of the mirror distorts its shape and limits its optical effectiveness. By incorporating computers into the construction of telescopes and by the use of active optics it is possible to construct large mirrors that are quite thin but will retain their shape due to constant corrections applied by computer controlled actuators situated behind the mirror.

The thin mirror reduces weight so that lighter support structures can be used and the reduced weight of the mirror and supports makes the telescope easier to drive.

Computers also allow for adaptive optics to correct for atmospheric turbulence. With a computer, flexible mirror and tip-tilt mirror the telescope is able to detect atmospheric distortions and allow for them. Its main difference to the active optics is the response speed. It can make around 1000 wavefront corrections per second and this results in a pronounced improvement in the resolving power of the telescope. Speckle Interferometry takes many short-exposure images from a telescope that will freeze the atmospheric blur. A computer then processes the many exposures and removes the blur.

The development of CCDs (cooled charge-coupled devices) has improved analysis of data compared to photographic detectors since they allow data to be fed directly into computers in digital format. This allows data to be stored easily and rapidly analysed.

Interferometry has enabled the resolution of radio telescopes to be improved. A large number of radio dishes are constructed in a pattern and then their signals are combined to form interference patterns and give the effect of a single radio telescope with a much larger diameter. Computers analyse the interference patterns and reveal the structure of the radio source. Interferometry techniques also allow very small angles to be measured and this is useful in determining astronomical distances.

Space technology has enabled telescopes to be developed that orbit the Earth outside its atmosphere and so eliminate the problems of atmospheric distortion and the absorption of most frequencies of electromagnetic waves. While the Hubble space telescope is the best known of these there are also Hipparcos, The Chandra X-ray Observatory, The Space Infrared Telescope Facility (Spitzer Space Telescope) and the Compton Gamma Ray Observatory. While these telescopes have the advantage of eliminating atmospheric distortion and recording radiation that would normally be absorbed by the atmosphere, they are expensive to launch and normally only last a few years until orbital decay brings them crashing back to Earth.

Some websites that may be of assistance are:

Hipparcos: http://astro.estec.esa.nl/Hipparcos/

CGRO: http://cossc.gsfc.nasa.gov/

Chandra: http://chandra.harvard.edu/

Spitzer Space Telescope: http://ssc.spitzer.caltech.edu/geninfo/

Measurement of small angles: http://cfa-www.harvard.edu/cfa/ep/brochure/vlbi.html

NASA’S High Energy Astrophysics Science Archive Research Center: http://heasarc.gsfc.nasa.gov/docs/corp/goto_text.html

A couple of good sites describing Adaptive and Active optics are:

http://www.ing.iac.es/~crb/wht/aointro.html

http://www.physics.usyd.edu.au/physopt/ao/ss.html

“5. The study of binary and variable stars reveals vital information about stars.” (syllabus)

Students learn to: “Describe binary stars in terms of the means of their detection: visual, eclipsing, spectroscopic and astrometric.” (syllabus)

Binary stars are two stars that are close together and revolve around their common centre of gravity. They are grouped into types according to their method of detection.

A visual binary star consists of two stars that can be resolved visually by means of a telescope.

An eclipsing binary consists of two stars that are orbiting “end on” to Earth so that the first moves in front of the second, decreasing brightness, and then the second moves in front of the first, also decreasing brightness.

The two stars in a spectroscopic binary can be identified by their red and blue shift. One star will exhibit a red shift as it moves away while the other exhibits a blue shift as it moves towards us. Half a period later the red and blue shift of the two stars is reversed.

Astrometric binaries are identified by the perturbations (wobbles) in the orbit of a star as it is affected by the gravity of an unseen companion.

Students learn to: “Explain the importance of binary stars in determining stellar masses.” (syllabus)

The total stellar mass of a two star system can be determined from Kepler’s law;

r³/T² = GM/4p² where M is the total mass of the two star system, r is the distance between the stars, T is the period of revolution of the stars and G is the universal gravitation constant. If the barycentre (the common centre of gravity) can also be identified, the masses of the individual stars can be calculated by the formula: m₁r₁ = m₂r₂, where m₁ and m₂ are the masses of the stars and r₁ and r₂ are their distance from the barycentre.

Students learn to: “Classify variable stars as either intrinsic or extrinsic and periodic or non-periodic.” (syllabus)

A variable star is one that changes in brightness.

An intrinsic variable star changes in brightness due to changes within the star itself e.g. a pulsating variable star. An extrinsic variable star changes in brightness because of the way it is observed e.g. an eclipsing binary star.

A periodic variable star changes in brightness at regular intervals e.g. cepheid variables, whereas a non-periodic variable star changes brightness at intermittent intervals e.g. novae.

Students learn to: “Explain the importance of the period-luminosity relationship for determining the distance of cepheids.” (syllabus)

In 1784 John Goodricke found that a star in the constellation Cepheus varied in magnitude on a periodic basis. This type of star has been called a Cepheid variable after the constellation of its first discovered member. Further studied of periodically variable stars showed that there were three main types.

Cepheid variable stars are a type of pulsating variable with a period of 1-50 days. They are luminous yellow supergiants with type I cepheids being younger than type II. Type II are also known as W virginis stars.The luminosity of each type is related to its period of pulsation. Type I cepheids are brighter than type II cepheids with the same period. Type I cepheids are metal rich stars while type II are metal poor.Type I cepheids are found in or near the galactic plane whereas type II are found in the galactic halo.

Another type of periodic variable is the RR Lyrae star found in the galactic nucleus and galactic halo. These periodic variables have a period from ¼ to ¾ day and an absolute magnitude between 0 and 1.

In 1910 Henrietta Leavitt found a number of Cepheid variable stars in the Small Magellanic Cloud (SMC) and found a direct relationship between their period and mean apparent magnitude. Since all of the stars in the SMC are essentially the same distance from Earth the relationship would also apply between their period and absolute magnitude. Unfortunately Ms Leavitt was unable to determine the distance to the SMC or the distance to any known Cepheid variable. However, since then, cepheids have been found in open clusters of known distance and the relationship between period and absolute magnitude has been established.

The distance to a Cepheid variable can be established by observing its apparent magnitude and period. Its absolute magnitude can be determined can be determined from a period-luminosity graph and the distance determined by applying the formula: m – M = 5 log d/10.

Period-Luminosity Relationship for Cepheid Variables

Exercise: An astronomer observed a Cepheid variable star and found its mean apparent magnitude to be 3.6 and its period to be 4.0 days. How far away is it?

Solution: From the graph, the star’s absolute magnitude is -3.2

m – M = 5 log d/10 3.6 - -3.2 = 5 log d/10 log d/10 = 6.8/5 = 1.36

d/10 = 22.9 d = 229Pc

Some websites dealing with Cepheid variables are:

http://chris.kingsu.ab.ca/~brian/astro/course/lectures/winter/var.htm

http://outreach.atnf.csiro.au/education/senior/astrophysics/variable_cepheids.html

http://www.astro.psu.edu/users/stark/ASTRO11/lab13a/PL.html

http://www.astronomynotes.com/ismnotes/s5.htm

Students: “Perform an investigation to model the light curves of eclipsing binaries using computer simulation.” (syllabus)

The graph below demonstrates how brightness varies with time for an eclipsing binary. The graph begins with the stars side-by-side as viewed from the Earth. The dimmer star then moves in front of the brighter star and produces a large drop in brightness. The stars are again side-by-side and the brightness returns to its previous level. The brighter star then moves in front of the dimmer star and produces a small drop in brightness. The brightmess then returns to its previous level as the stars are again side-by-side and the cycle repeats itself.

Graph of brightness against time for an Eclipsing Binary

To perform an investigation using computer simulation, visit one of the following websites:

A good site to try this simulation is the Terry Herter site: http://instruct1.cit.cornell.edu/courses/astro101/java/simulations.htm

Or you can try the “Imagine the Universe” site:

http://imagine.gsfc.nasa.gov/YBA/HTCas-size/binary-model.html

Or try the AstroTutor site:

http://www.brookscole.com/astronomy_d/special_features/astrotutor/eclipsingbinary/animation.html

Students: “Solve problems and analyse information by applying:

m₁ + m₂ = 4p²r³ /GT².” (syllabus)

Q. Alpha Centauri A has a mass of 1.1 S.M. and Alpha Centauri B has a mass of 0.9 S.M. They have a period of rotation of 80 years. The mass of the Sun is 2 x 10³⁰ kg and it is 1.5 x 10¹¹ m from Earth.

(i) What is the distance in A.U. between Alpha Centauri A & B?

(ii) How far in A.U. is Alpha Centauri A from the barycentre (common centre of gravity)?

A. (i). m₁ + m₂ = 4p²r³ /GT² r³ = (m₁ + m₂) GT² / 4p²

r³ = (1.1 + 0.9) x 2 x 10³⁰x 6.67 x 10^-11 x (80 x 365 x 24 x 3600)² / 4p²

= 4.35 x 10³⁷

r = 3.52 x 10¹² m = 23.4 A.U.

A. (ii) Let Alpha Centauri A be a distance d from the barycentre.

Then Alpha Centauri B is a distance 23.4 – d from the barycentre.

m_Ar_A = m_Br_B 1.1 d = 0.9(23.4 – d) 1.1 d = 21– 0.9d 2d = 21 d = 10.5 A.U.

“6. Stars evolve and eventually ‘die’ ” (syllabus)

Students learn to: “Describe the processes involved in stellar formation.” (syllabus)

Matter to form stars comes from the interstellar gas, mostly hydrogen, in the galactic plane. While the gas occurs everywhere in the galactic plane it is most concentrated in clouds called nebulae; remnants of supernovae explosions. As well as hydrogen there are small amounts of other gases as well as minute amounts of dust.

Small condensations form within gaseous nebulae, possibly as the result of shockwaves from a supernova or possibly from the gravitational disturbance of another star passing nearby. These condensations have higher gravity than the regions around them so they attract more gas to them. The initial condensation is quite rapid and the cloud of gas will condense from several times the size of the solar system to about the size of Mercury’s orbit in about 100 years. As the cloud collapses, the potential energy of the gas is converted into kinetic energy and the temperature rises. A stage will be reached where the gravitational collapse of the star is balanced by the release of radiant energy. During the next stage that lasts for tens of millions of years the temperature and pressure at the centre of the cloud will gradually increase. When the central temperature reaches around 10 million Kelvin, the energy of the nuclei is sufficient to overcome the electrostatic repulsion of the protons so the hydrogen nuclei combine with each other to form helium nuclei in the process of nuclear fusion. At this stage the star becomes a main sequence star.

Students learn to: “Outline the key stages in a star’s life in terms of the physical processes involved.” (syllabus)

Protostar: This is described above and involves the gravitational collapse of the gas cloud and the subsequent heating as potential energy is converted to kinetic energy.

Main Sequence: This is the hydrogen fusion stage and involves the conversion of four hydrogen nuclei to a helium nucleus and the corresponding conversion of matter into energy. This is described below.

Red Giant: When most of the hydrogen in the core has been used up, nuclear reactions begin in the region immediately surrounding the core. This shell burning causes the outer envelope of the star to expand due to the increased radiation pressure. Meanwhile the core is contracting and getting hotter as the increased pressure increases the rate of nuclear reaction. When the core temperature reaches around 10⁸K the helium flash occurs and helium is converted into carbon. For stars up to three solar masses this is the last nuclear burning stage but for more massive stars, as the helium fusion diminishes, the core collapses and heats up causing helium shell burning. As the core reaches about 600 million Kelvin, carbon undergoes fusion to form heavier elements such as oxygen. In this way, massive stars form a series of concentric shells in which the outermost shell consists of hydrogen being converted to helium, the second helium to carbon and so on until at the core, iron is created out of the lighter elements.

Instability Strip:As these massive stars undergo their series of nuclear reactions, they move back and forth horizontally on the H.R. diagram, passing through a region called the instability strip or Hertzsprung gap each time. This is the region of pulsating variables, i.e. stars that vibrate at their natural frequency in much the same way as gas in an organ pipe. A layer below the surface of the star consists of partly ionised helium. As the star contracts, ionisation increases and the layer more effectively traps the outgoing radiation. Eventually sufficient pressure is built up to push the outer layers of the star outwards so that they cool down and allow the stored radiation to escape. The star then contracts to begin another cycle.

During a star’s giant or supergiant stage it loses matter as some of its atmosphere is ejected. What happens to the star at its next stage depends on the mass of the remaining core.

White Dwarfs:

Stars up to a size of 4 solar masses on the main sequence will leave a core of about 1.4 solar masses. This is the upper limit for the star to become a white dwarf. As the nuclear reactions die out the star will shrink due to gravity and become much smaller, about the size of the Earth. The reason for this is that the matter becomes degenerate. Because of the large mass and absence of radiation pressure, gravity will cause the white dwarf to collapse. The orbits of the electrons are compressed until they begin to encroach on each other’s space. Since no two electrons can have exactly the same quantum numbers (Pauli exclusion principle) then one of the electrons will gain momentum and so exert an outward pressure to balance the inward pressure of gravity. The degenerate nature of matter in white dwarfs makes them extremely dense with a typical white dwarf being a million times as dense as water. Although white dwarfs are initially very hot they are also quite dim. They radiate their energy into space, but because of their small surface area, this is a very slow process. Over a period of billions of years white dwarfs cool to become black dwarfs.

Supernova: Stars of 3 solar masses and above have a core above 1.4 solar masses. This is known as the Chandrasekhar limit, after Subrahmanya Chandrasekher, who showed that the degenerate pressure of the electrons in a white dwarf couldn’t support masses above this limit. Stars with cores above this limit will undergo a massive explosion called a supernova at the end of their red giant or supergiant stage.

Neutron Star: Stars between 3 and 10 solar masses (core 1.4 – 3 solar masses) will undergo a supernova where a cloud of gas called a nebula is blown off and a dense core remains where the pressure is sufficient to transform protons and electrons into neutrons. Neutron stars have masses between 1.4 and 3 solar masses but are only about 20 km in diameter. To conserve angular momentum they spin very fast with periods usually less than a second. Because of their strong magnetic field and fast rotation they emit regular bursts of radio waves, and because of this they are often called pulsars.

Black Hole: For stars above 10 solar masses the gravitation is so great that the neutron star collapses in on itself to form a singularity i.e. a region so dense that light cannot escape.

Students learn to: “Describe the types of nuclear reactions involved in Main-sequence and post Main-sequence stars.” (syllabus)

Nuclear reactions take place in the core of the star rather than at the surface. Hydrogen is converted to helium by nuclear fusion and liberates energy in the form of gamma radiation. The photon of gamma radiation will soon collide with another atom, which will absorb the photon and re-emit it, not necessarily as gamma radiation but possibly as two or more photons of lower energy radiation. Thus over a period of millions of years the energy gradually makes its way to the surface where it is emitted as lower energy photons such as light or radio waves.

The nuclear reaction proceeds by one of two processes: the proton-proton reaction or the carbon cycle.

The proton-proton reaction takes place where the core temperature lies between 5 and 16 million Kelvin and can be represented by the following series of equations.

₁H¹ + ₁H¹ " ₁H² + e⁺ + n

₁H² + ₁H¹ " ₂He³ + g

₂He³ + ₂He³ " ₂He⁴ + 2 ₁H¹

At temperatures above 16 million Kelvin the carbon cycle occurs; i.e. a series of reactions in which carbon acts as a nuclear catalyst.

₆C¹² + ₁H¹ " ₇N¹³ + g

₇N¹³ " ₆C¹³ + e⁺ + n

₆C¹³ + ₁H¹ " ₇N¹⁴ + g

₇N¹⁴ + ₁H¹ " ₈O¹⁵ + g

₈O¹⁵ " ₇N¹⁵ + e⁺ + n

₇N¹⁵ + ₁H¹ " ₆C¹² + ₂He⁴

The net effect of each type of reaction is that four hydrogen atoms combine to form one helium atom.

When most of the hydrogen in the core has been consumed the nuclear reaction decreases. Gravity takes over causing the star to collapse further and heat up. As the core heats up, radiation passes through the region surrounding the core, which was previously not hot enough to sustain nuclear reactions i.e. the stellar envelope. The radiation pressure causes the envelope to expand and therefore cool.

The energy from the collapsing core heats up the hydrogen at the inner boundary of the envelope, causing it to undergo nuclear fusion. The shell burning of hydrogen increases the total luminosity of the star, which brightens. On the H-R diagram, the star leaves the main sequence. It moves upwards and to the right as it brightens and cools. The expanding envelope in the meantime causes the star to expand to many hundred times its original size, becoming a red giant or supergiant.

As the core collapses the temperature rises. At about 100 million Kelvin the temperature is sufficiently high for helium to undergo fusion to form carbon. The reaction, called the triple alpha reaction, involves three helium nuclei forming a carbon nucleus.

₂He⁴ + ₂He⁴ g ₄Be⁸ + n

₂He⁴ + ₄Be⁸ g ₆C¹² + g

The point at which this begins is called the helium flash.

Students learn to: “Discuss the synthesis of elements in stars by fusion.” (syllabus)

The first synthesis is the fusion of hydrogen into helium in main sequence stars. This is followed by the fusion of helium into carbon in the red giant stage. For stars the size of the Sun, this is where fusion stops but larger stars use carbon as a fuel to form oxygen, neon, sodium and magnesium and subsequently use these fuels to form elements with atomic numbers up to that of iron. At the end of each stage of core burning, shell burning of that particular element continues. The star now consists of an outer shell where hydrogen is being converted to helium surrounding a helium shell being converted to carbon and so on, in a series of concentric shells where each shell is composed of a heavier element than the ones surrounding it. The shells continue until, at the core, iron is being created out of lighter elements. Iron is the most stable nuclear element with the binding energy per nucleon of iron being greater than for any other element. The formation of elements with atomic numbers above iron involves endothermic nuclear reactions, where energy has to be put into the reaction. This occurs in supernovas where the high temperatures involved provide the energy to produce heavy elements from the fusion of lighter ones.

Students learn to: “Explain how the age of a globular cluster can be determined from its zero-age main sequence plot for a H-R diagram.” (syllabus)

The key to this method of age determination is the fact that large stars spend less time on the main sequence than small stars. If we assume that all of the stars in a globular cluster were formed at around the same time then the large stars will have completed their lifetime and will have evolved into either neutron stars or black holes. These do not appear on the H.R. diagram.

The most luminous stars on the main sequence of a globular cluster are at the end of their main sequence lifetime. Since we can relate the luminosity of a star to its mass and to how long it will live as a main sequence star, the most luminous stars remaining tell us the age of the globular cluster.

Main Sequence Lifetimes

Spectral Class	Temperature (K)	Luminosity (Compared to Sun)	Mass (Solar Masses)	Time on Main Sequence (Million years)
O	40 000	400 000	40	1.0
B	15 000	3 000	10	30
A	10 000	60	3	500
F	7 000	5	1.5	3 000
G	5 800	1	1	9 000
K	5 000	0.5	0.75	15 000
M	3 000	0.1	0.5	50 000

Exercise: A globular cluster is represented by the following H-R diagram. Estimate its age.

Hertzsprung-Russell Diagram of a typical Globular Cluster.

The largest stars on the main sequence of the above H-R diagram are F class stars with a surface temperature of around 7 000K. These stars have a main sequence lifetime of around 3 billion years so the age of the globular cluster is around 3 billion years.

Students learn to: “Explain the concept of star death in relation to:

-planetary nebula

-supernovae

-white dwarfs

-neutron stars/pulsars

-black holes.” (syllabus)

Planetary Nebulae:

As a red giant collapses to form a white dwarf it produces UV radiation that will ionise the outer layer of gases and cause it to expand and glow. The expanding shell of ionised gas is called a planetary nebula although it has nothing to do with planets.

Supernovae:

Stars with cores above the Chandrasekhar limit of 1.4 solar masses undergo a supernova at the end of their red giant or supergiant stage. Above this limit the degenerate electron pressure is unable to counteract the pull of gravity.

Type I supernovae occur when one of a binary pair is a white dwarf and it receives matter from its companion to put it over the Chandrasekhar limit. Since these all occur with stars of the same mass they are all of the same luminosity or absolute magnitude. When these are observed in distant galaxies it is possible to measure their apparent magnitude and so calculate the distance to the galaxy.

Type II supernovae occur when large stars (above 4 solar masses, i.e. with a core above 1.4 solar masses) reach the end of their giant or supergiant stage. As the cores of successive heavier elements are exhausted a series of shells is built up with hydrogen in the outer shell and iron in the innermost shell. Iron is the most nuclear stable element with the binding energy per nucleon being greater than any other element. The nuclear reactions in the core stop and the electron degenerate pressure is insufficient to stop the gravitational collapse. As the core collapses the temperature rises to several billion K. At this temperature photodisintegration of the iron nuclei occurs and as the gravitational collapse continues, electrons combine with protons to form neutrons. As matter continues to rush in it bounces off the neutron core and generates a huge amount of heat in the process. As it moves back towards the surface of the star it accelerates due to the density of the upper layers of the star decreasing. It becomes a shock wave and will blow the stars outer layers away from the core.

White Dwarfs:

Neutron Star:

Stars between 3 and 10 solar masses (core 1.4 – 3 solar masses) will undergo a supernova where a cloud of gas called a nebula is blown off and a dense core remains where the pressure is sufficient to transform protons and electrons into neutrons. Neutron stars have masses between 1.4 and 3 solar masses but are only about 20 km in diameter. To conserve angular momentum they spin very fast with periods usually less than a second. Because of their strong magnetic field and fast rotation they emit regular bursts of radio waves, and because of this they are often called pulsars.

Black Holes:

Stars above 10 solar masses leave a core of more than 3 solar masses. The gravity of such matter is so strong that it is able to overcome the degenerate pressure of the neutrons. The nucleus then collapses in on itself to form a singularity. Such a singularity is so dense that the escape velocity at its surface is greater than the speed of light so that even light cannot escape.

Students: “Present information by plotting Hertzsprung-Russell diagrams for: nearby or brightest stars, stars in a young open cluster, stars in a globular cluster.” (syllabus)

Exercise:

The Pleiades (M 45) is an open cluster that is about 410 light years from Earth. It is sometimes called “The Seven Sisters” even though there are many more than seven stars in the cluster. The names of the Seven Sisters as well as their apparent magnitude and spectral class are shown in the table below.

Name of Star	Apparent Magnitude	Spectral Class
Alcyone	2.90	B7
Atlas	3.62	B8
Electra	3.70	B6
Maia	3.87	B8
Merope	4.18	B6
Taygete	4.30	B6
Pleione	5.09	B8

(i) Calculate the absolute magnitude of each of the stars.

(ii) Plot the positions of the stars on the H-R diagram below.

(iii) Explain how this diagram differs from one for a globular cluster.

Exercise:

The cluster M 41 (NGC 2287) is in Canis Major, just south of Sirius. The apparent magnitudes and colour index of some of the stars is shown in the following table.

Star	Apparent Magnitude m_v	Colour Index m_B-m_V
1	9.0	-0.11
2	9.4	-0.02
3	10.3	-0.04
4	10.5	-0.01
5	10.7	0.00
6	11.1	0.06
7	11.4	0.11
8	12.5	0.40
9	12.5	0.36
10	12.9	0.46
11	13.5	0.60
12	14.0	0.71

(i) Determine the temperature of each of the stars listed.

(ii) Plot the positions of the stars on a H-R diagram.

(iii) Is the cluster an open cluster or globular cluster? Justify your decision.

(iv) Determine the absolute magnitude of each of the stars listed.

(v) Determine the distance modulus (m-M) for each of the stars.

(vi) Determine the distance to M 41.

Data for H-R plot: http://members.frys.com/~kgc/HR%20diagram%20hw.pdf

http://www-laog.obs.ujf-grenoble.fr/activites/starevol/FILES/plot.html

good data for H-R plot

http://www.mscd.edu/~physics/astro/activities/activity8.htm

The following site gives details of the Stromlo Summer School 1999 project of plotting a H-R diagram of the globular cluster NGC288:

http://www.mso.anu.edu.au/~rsmith/summer_school.html#clusters

Students: “Analyse information from a H-R diagram and use available evidence to determine the characteristics of a star and its evolutionary stage.” (syllabus)

CHECK

Some H-R diagram plotting exercises are available on:

http://www.astro.ubc.ca/~scharein/a311/sim/hr/HRdiagram.html

http://www.smv.org/jims/l6a.htm

Students: “Present information by plotting on a H-R diagram the pathways of stars of 1, 5 and 10 solar masses during their life cycle.” (syllabus)

The evolutionary timescales for stars of 1, 5 and 10 solar masses are shown in the table below. Their evolutionary paths are shown in the H-R diagrams.

Timescales for Stellar Evolution
Solar Masses	Pre main sequence (years)	Main sequence (years)	Giant Phase (years)
1	1 x 10⁸	9 x 10⁹	1 x 10⁹
5	5 x 10⁶	6 x 10⁷	1 x 10⁷
10	6 x 10⁵	1 x 10⁷	1 x 10⁶

Evolutionary track of star of 1 solar mass

Evolutionary track of star of 5 solar masses

Evolutionary track of star of 10 solar masses

Note the following differences between the diagrams:

(i) More massive stars join the main sequence at a higher point i.e. greater luminosity

(ii) Stars below the Chandrasekhar limit of 3 solar masses (1.4 solar masses core) form a white dwarf after going through the red giant stage. This is true of 1 solar mass stars such as the Sun.

(iii) Stars above the Chandrasekhar limit form red giants or supergiants that are unstable. They tend to move back and forth along the instability strip where they become Cepheid variable stars. Smaller stars tend to form RR Lyrae variables on the instability strip.

(iv) Stars above the Chandrasekhar limit end their giant or supergiant phase as a supernova i.e. a massive stellar explosion. A 5 solar mass star would form a neutron star and gaseous nebula while a 10 solar mass star would form a black hole and gaseuos nebula.

q = 1.22l /d where

q is the angular separation of the two objects in radians,

l is the wavelength of the light or radio waves in metres and

d is the diameter of the telescope objective or mirror in metres.

q = 2.1 x 10⁵l /d where

q is the angular separation of the two objects in seconds of arc,

l is the wavelength of the light or radio waves in metres and

d is the diameter of the telescope objective or mirror in metres.

3. A radian is the angle subtended by an arc of length 1 radius.

q = 1.22l /d = 1.22 x 500 x10^-9 / 114 x 10^-3 = 5.3 x 10^-6 rad.

q = 2.1 x 10⁵l /d = 2.1 x 10⁵ x 500 x10^-9 / 114 x 10^-3 = 0.92 arc seconds.

800/500 = 1.6 times
Infrared radiation cannot be seen so photographic methods of detection would have to be used. Most infrared radiation is reflected by water vapour in the atmosphere so only weak signals would be received.
The refractive index of the Earth’s atmosphere varies, often over an area smaller than the aperture of the eye or telescope. This causes the image of the star to break up into a number of distorted images.
The absence of atmosphere results in a sharper image as there is no atmospheric distortion. ( No variation in the refractive index, no movement of atmosphere, no water vapour or dust)

Active & Adaptive optics: http://www.eso.org/projects/aot/introduction.html

The VLT Active optics system: http://www.eso.org/projects/vlt/unit-tel/actopt.html

Adaptive optics (&active): http://www.ing.iac.es/~crb/wht/aointro.html

Astronomical Images: http://www.aao.gov.au/images/general/full_list.html

Interferometry: http://www.europhysicsnews.com/full/12/article13/article13.html

UNSW Links: http://www.phys.unsw.edu.au/hsc/astrolinks.html

Hubble Telescope Site: http://www.stsci.edu/resources/

Nasa Site: http://www.nasa.gov/

Sydney Uni Links: http://science.uniserve.edu.au/school/curric/stage6/phys/astrphys.html#gen

Woodward links: http://science.uniserve.edu.au/school/curric/stage6/phys/stw2004/woodward.pdf

Big list of astronomy websites and info: Jeff Bondono’s Astronomy Page

http://members.ozemail.com.au/~swadhwa/amastrolinks.htm

Stellar Evolution – Charles Darwin Uni - http://www.ntu.edu.au/faculties/science/astronomy/SPH210/notes/sec9note.html

1. This page was copied from Nick Strobel's Astronomy Notes. Go to his site at www.astronomynotes.com for the updated and corrected version.

Main Sequence Lifetimes

Spectral Class	Temperature (K)	Luminosity (Compared to Sun)	Mass (Solar Masses)	M/L	M/LT	Time on Main Sequence (Million years)
O	40 000	400 000	40	1x10^-4	1x10-4	1.0
B	15 000	3 000	10	3.3 x10^-3	1.1x10^-4	30
A	10 000	60	3	0.05	1.0x10^-4	500
F	7 000	5	1.5	0.3	1x10^-4	3 000
G	5 800	1	1	1.0	1.1x10^-4	9 000
K	5 000	0.5	0.75	1.5	1x10^-4	15 000
M	3 000	0.1	0.5	5	1x10^-4	50 000

7. l_max = 2.89 x 10^-3 / 8000 = 3.61 x 10^-7m

A diffraction grating consists of a large number of parallel lines scratched onto a glass, plastic or reflective surface.
Diagram and explanation to be inserted.
Emission nebulae produce mainly the hydrogen emission spectrum with traces of emission spectra produced by other elements. They result from gas clouds of mostly hydrogen being excited by mostly U.V. radiation from a nearby hot star.
Where galaxies are moving away the wavelengths of spectral lines are displaced to become longer. The shift is due to the Doppler effect where wavelengths of receding sources become longer. (Strictly speaking the red shift is not due to the Doppler effect since the Doppler effect refers to a source that is moving. Actually the red shift is due to the space between the galaxies expanding rather than the receding motion of the galaxies through space but the effect is the same)
Fraunhofer lines are a large number of dark lines on the spectrum of the Sun and other stars.
The black lines are absorption spectral lines caused by the absorption of the spectral lines of elements present in the photosphere of stars.
The Sun’s visible spectrum consists of a continuous spectrum of the colours of the rainbow interrupted by thousands of fine dark lines.
The Sun’s invisible spectrum includes gamma rays, X-rays, U.V. rays, I.R. rays, microwaves and radio waves.

source : http://lheawww.gsfc.nasa.gov/users/allen/spectral_classification.html

Conversion of color index to temp

http://www.earth.uni.edu/~morgan/stars/b_v.html

http://iws.ccccd.edu/jwilson/color_index_vs_temp.html

colour index explained http://docs.kde.org/en/3.1/kdeedu/kstars/ai-colorandtemp.html

Telescope design: optical & radio

Hipparcos: http://astro.estec.esa.nl/Hipparcos/

CGRO: http://cossc.gsfc.nasa.gov/

Chandra: http://chandra.harvard.edu/

NASA’S High Energy Astrophysics Science Archive Research Center: http://heasarc.gsfc.nasa.gov/docs/corp/goto_text.html

Spitzer Space Telescope: http://ssc.spitzer.caltech.edu/geninfo/

Measurement of small angles: http://cfa-www.harvard.edu/cfa/ep/brochure/vlbi.html

Filters: http://csown.dhs.org/photo/crc_filter_info.html

Filters & CCDs Astronomy: http://www.aao.gov.au/local/www/sl/filters.html

Construction of Filters: http://www.mic-d.com/curriculum/lightandcolor/lightfilters.html

http://www.telescopes-astronomy.com.au/astronomy129.htm

UK Schmidt Telescope Filters: http://www.aao.gov.au/ukst/tele.html

Royal Astronomical Society of New Zealand

THE BRIGHTEST STARS

	Name	Star	Mag.	Abs. Mag.	Lumin. Sun = 1	Spec Type	Distance Lt Yr	Distance parsec	21 Hr Transit	R.A. hr	Dec.
0.	Sun		-26.8	+4.8	1.0	G0	(1AU)
1.	Sirius	a CMa	-1.46	+1.4	23	A0	8.6 ly	2.6 pc	Feb 16	6.7	-16°
2.	Canopus	a Car	-0.72	-3.1	1400	F0	120 ly	37 pc	Feb 11	6.4	-53°
3.	Rigel Kent	a1 + a2 Cen	-0.27	+4.5	1.3	G0	4.3 ly	1.3 pc	Jun 16	14.6	-61°
4.	Arcturus	a Boo	-0.04	-0.3	91	K0	36 ly	11.0 pc	Jun 10	14.2	+19°
5.	Vega	a Lyr	+0.03	+0.5	52	A0	26 ly	8.0 pc	Aug 15	18.6	+39°
6.	Capella	a Aur	+0.08	+0.1	76	G0	32 ly	9.8 pc	Jan 24	5.2	+46°
7.	Rigel	b Ori	+0.12	-6.4	30200	B8	680 ly	210 pc	Jan 24	5.2	-8°
8.	Procyon	a CMi	+0.38	+2.7	6.9	F5	11.4 ly	3.5 pc	Mar 2	7.6	+5°
9.	Achernar	a Eri	+0.46	-2.6	910	B5	140 ly	43 pc	Nov 30	1.6	-57°
10.	Betelgeuse	a Ori	+0.57 var	-5.1	9400	M0	427 ly	130 pc	Feb 3	5.9	+7°
11.	Hadar	b Cen	+0.61	-3.1	1450	B1	180 ly	55 pc	Jun 7	14.0	-60°
12.	Acrux	a1 + a2 Cru	+0.75	-4.2	3960	B1	321 ly	98 pc	May 14	12.4	-63°
13.	Altair	a Aql	+0.93	+2.2	10.9	A5	16.8 ly	5.1 pc	Sep 3	19.8	+9°
14.	Aldebaran	a Tau	+0.99	-0.63	149	K5	65 ly	20.0 pc	Jan 15	04.6	+17°
15.	Spica	a Vir	+1.06	-3.55	2180	B2	262 ly	80 pc	May 28	13.4	-11°
16.	Antares	a Sco	+1.06	-5.28	10700	M0	600 ly	185 pc	Jul 14	16.5	-26°
17.	Pollux	b Gem	+1.22	+1.09	30.5	K0	33.7 ly	10.3 pc	Mar 3	07.8	+28°
18.	Fomalhaut	a PsA	+1.23	+1.74	16.7	A3	25.1 ly	7.7 pc	Oct 21	23.0	-30°
19.	Deneb	a Cyg	+1.33	-8.7	250000	A2	3200 ly	990 pc	Sep 16	20.7	+45°
20.	Mimosa	b Cru	+1.31	-3.92	3070	B1	353 ly	108 pc	May 19	12.8	-60°
21.	Regulus	a Leo	+1.41	-0.52	134.2	B8	77.5 ly	23.8 pc	Apr 8	10.1	+12°
22.	Adhara	e CMa	+1.52	-4.10	3650	B1	431 ly	132 pc	Feb 20	07.0	-29°
23.	Castor	a Gem, double	+1.58	+0.59	48.5	A0	51.6 ly	15.8 pc	Feb 29	07.6	+32°

A good site for blackbody curves and color index

http://csep10.phys.utk.edu/astr162/lect/stars/cindex.html

Below:

http://www2.dsu.nodak.edu/users/edkluk/Astro110/Cleaex2/CLEAEX2_w.html

One of these filters, the visible filter (V), has maximum transmission at 550 nm (see the vertical line in Fig. 1) and gradually cuts out transmission of photons with shorter and longer wavelengths. The other filter, the blue filter (B), has maximum transmission at 440 nm (see the vertical line in Fig. 1). The respective stellar magnitudes m_V and m_B as measured with use of these filters are defined as follows

where I_V and I_B are measured intensities, and I_V0 and I_B0 are standard intensities for a star with both magnitudes m_V and m_B equal to zero.

O-type Stars: 10 Lacertra, Lacaille 8760, 40 Eridani B.

Mintaka O9 (Orion), Han O9 (Orphiucus), Alnitak O9 (Orion), Hatysa o9, Heka O8, Menkib O7 (Persius), Meissa O6 (Heka), Naos O5,

The difference between these two magnitudes B-V = m_V - m_B is called the color index.

Below:

http://www.frostydrew.org/observatory/courses/physics/booklet.htm

I know of no formula for converting color indices to temperature but the following chart is approximately correct.

http://www.astro.psu.edu/users/stark/ASTRO11/lab13a/

The Period-Luminosity Relation

http://chris.kingsu.ab.ca/~brian/astro/course/lectures/winter/var.htm

As mentioned above, many stars show a strong correlation between the period of their brightness variations and their intrinsic brightness or luminosity. This makes good sense. By analogy consider a pendulum. The longer the pendulum the longer its period. A bigger star has a longer period than a smaller star and bigger would suggest brighter. As we will see in future lectures this provides us with a very useful method for determining interstellar and intergalactic distances. To illustrate this consider the following figure:

We see three of the most important variable classes listed here. The Cepheids actually split into two classes:

Classical Cepheids are very bright and are considered "young" or Population I stars
W Virginis or Cepheid II stars are older members of the Cepheid clan. They are somewhat fainter than Classical Cepheids but their mode of oscillation, periods and overall stellar type are quite similar.

If, for example, we know the period of this graph it is easy on the basis of the period alone to estimate the absolute magnitude of the star. From this we can determine the distance modulus (what additional piece of information do we need?) and from that determine the distance.

Interesting site on astrophysics & cosmology:http://www.vasa.abo.fi/users/tillman/s.htm

Calculating Distances Using Cepheids

Both types of Cepheids and RR Lyrae stars all exhibit distinct period-luminosity relationships as shown below.

http://outreach.atnf.csiro.au/education/senior/astrophysics/variable_cepheids.html

Period-luminosity relationship for Cepheids and RR Lyrae stars.

Bob Emery notes:

Note that initially Leavitt and others thought that all Cepheids could be described by the same period-luminosity relationship. This, however, was not the case. We now know that there are two different types of Cepheid variable, each with its own period-luminosity relationship. Type I or Classical Cepheids are the brighter, more massive (5-15 solar masses), metal-rich, younger, second generation stars, found exclusively in the disc population of galaxies, where they are often members of open clusters (5, 10 & 12). Type II or W Virginis Cepheids are the dimmer, less massive (0.4 to 0.6 solar masses), metal-poor, older, red, first generation stars (5, 10 & 12). Astronomers examine a star’s metal content from its spectrum in order to classify it as either a Type I or Type II Cepheid (3). The period-luminosity relationship for both types of Cepheid variable is shown in the following graph.

This graph is an adaptation of the graph shown on p.49 of Ref.9. Note that the relationship for Type I Cepheids has only been drawn up to a period of 50 days. In reality, Type I Cepheids can have periods up to 135 days. Remember too that the straight lines in the graph are simply showing general trends that are seen in the data. The actual period-luminosity graphs recorded by astronomers consist of bands of stars plotted on and around these lines. See Ref.1 Fig.14.4 on p.545 for example.

Now continuing with our example

Name of Star	Apparent Magnitude m	Spectral Class	Absolute Magnitude M	Distance Modulus m - M	Distance Pc
Alcyone	2.90	B7	-0.5	3.4	47.9
Atlas	3.62	B8	-1.0	4.62	83.9
Electra	3.70	B6	0	3.7	55.0
Maia	3.87	B8	-1.0	4.87	94.2
Merope	4.18	B6	0	4.18	68.5
Taygete	4.30	B6	0	4.30	72.4
Pleione	5.09	B8	-1.0	6.09	165.2

The distance to the Pleiades according to reference sources is about 125 Pc.

Colour Index	Log T	Temperature
-0.6	7.40636	25 million
-0.5	5.9479	887020
-0.4	5.053145	113017
-0.3	4.5264	33605
-0.2	4.22928	16954
-0.1	4.067346	11667
0	3.979145	9531
0.1	3.92712	8455
0.2	3.8903	7767
0.25		7483
0.3	3.8584	7218
0.4	3.8279	6728
0.5	3.7983	6285
0.6	3.7705	5895
0.7	3.74535	5564
0.75		5418
0.8	3.73521	5286
0.9	3.7035	5053
1.0	3.6857	4849
1.1	3.6688	4664
1.2	3.6522	4490
1.25		4405
1.3	3.6358	4323
1.4	3.6191	4160
1.5	3.6009	3990
1.6	3.5775	3780
1.7	3.5395	3463
1.75		3235
1.8	3.4687	2942
1.9	3.3325	2150
2.0	3.0775	1195

2.5	-4.5225	0.00003 ?
3.0	-55.5473	2.836

Formulae that have been suggested: T = 7200/(B-V +0.64) ihug cyoung

T_eff = 1000 + 5000/(B –V + 0.5)

T = 9385/(V-I + 0.95) Iowa Robotic Telescope Facilities

Main Sequence Lifetimes

1. This page was copied from Nick Strobel's Astronomy Notes. Go to his site at www.astronomynotes.com for the updated and corrected version.

Main Sequence Lifetimes

Royal Astronomical Society of New Zealand

THE BRIGHTEST STARS

The Period-Luminosity Relation

http://chris.kingsu.ab.ca/~brian/astro/course/lectures/winter/var.htm

Calculating Distances Using Cepheids

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