HSC Physics Module 9.7 Summary

HSC Physics Module 9.7 Summary 1. Our understanding of celestial objects depends upon observations made from Earth or from space near the Earth Discuss Galileo’s use of the telescope to identify features of the Moon Galileo did not invent the telescope, but was able to build a telescope that produced a clear enough image to observe the features of the moon. He used higher quality glass than used before, and produced his own lenses in order to build an improved telescope that had a magnification greater than 3x. His interest in celestial bodies caused him to point the telescope at the moon, and be the first person to record observations made of the moon from the telescope. He observed and recorded that the moon’s surface was uneven, rough, and full of cavities and prominences. Galileo was able to calculate the height of mountains on the moon from the measurement of their shadows. This challenged the view held by the Catholic Church, who believed that all celestial bodies were “perfect”.

Discuss why some wavebands can be more easily detected from space Nearly all wavebands of the electromagnetic spectrum are present in space, and most are directed towards Earth. Not all wavebands reach Earth’s surface however, as most wavebands are filtered out by the atmosphere. The only wavebands that predominantly reach Earth’s surfaces are visible light, microwaves and radio waves. EMR Waveband

Wavelength (m)

Gamma rays X-rays UV light Visible light Infra-red Radio waves

DO NOT LOOK DIRECTLY AT SUNLIGHT RESULTS/CONCLUSION As aforementioned, number of photons was plotted against diameter squared, yielding a linear relationship => sensitivity of light is proportional to diameter SQUARED RELIABILITY The method was repeated several times, and an average taken Results were compared to other groups, who measured similar results The data points were close to the line of best fit => PRECISION VALIDITY/ACCURACY The experiment and results reflected the aim Only one variable was changed (diameter of circle), all the others were controlled The final result was checked against reliable textbooks, online websites, and scientific journals, which gave the same result. The method was a model, and so was LIMITED due to discrepancies between model and reality (photons are massless, does not show wave properties of light etc.)

2. Careful measurement of a celestial object’s position in the sky (astrometry) may be used to determine its distance Define the terms parallax, parsec, light-year Parallax is the apparent change in position of an object relative to a distant background due to a change in the position of the observer. In relation to celestial bodies, parallax occurs as the Earth orbits the Sun. The changing position of Earth causes celestial bodies to appear to be changing position, as our perspective of celestial bodies relative to their background is changing. Parallax can be demonstrated by holding your finger out in front of your eyes, and covering one eye, then the other. Your finger appears to move relative to the background as your perspective changes, yet your finger has remained still.

A parsec is a measure of distance, commonly used when calculating celestial distances. More specifically, one parsec is equal to the distance from the Earth to a point that has an annual parallax of one arcsecond.

One parsec is equal to 3.26 light-years. Annual parallax will be discussed below. The parsec is used commonly in astrometry, which is the branch of astronomy concerned with the position of celestial bodies. A light-year is the distance that light travels in one Earth year. It is approximately equal to 9.5x1015m.

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

Angles of deviation used in trigonometric parallax calculations are normally calculated at 6 month intervals, where the diameter of the Earth’s elliptical orbit around the Sun is a maximum.

As can be seen in the above triangle the distance of the star from Earth can be calculated using trigonometry.

The large distances from Earth to celestial bodies means the angle of deviation is very small. Proxima Centauri has the smallest angle of deviation, which is 0.772arcsecs. At angles this small, the following approximation can be used.

Also, the radius of Earth’s orbit is 1 AU (astronomical unit). Therefore, the above formula becomes

where d=distance from Earth [parsecs, pc] p=parallax *arcsecs, “+

Explain how trigonometric parallax can be used to determine the distance to stars Trigonometric parallax is a method of determining distances to celestial objects by using parallax. If the change in position of the observer and the angle of deviation due to parallax is known, the distance to the celestial body can be calculated using trigonometry.

Through the tan ratio

Rearranging

Discuss the limitations of trigonometric parallax measurements Trigonometric parallax measurements rely on accurate measurements of the angle of deviation of celestial bodies. Angles of less than 0.01arcsec are impossible to use, as they have an error of 10% due to limits in resolution, such as due to atmospheric blurring. The refraction of light in the atmosphere changes the angle measured, and thus reduces the accuracy of small angles measured. In addition, parallax readings are limited by the precision of the measuring equipment, as the angles measured in trigonometric parallax are very small. This means that trigonometric parallax is only useful for calculating distances up to around 100 parsecs, which is a small distance in astronomical terms.

Gather and process information to determine the relative limits to trigonometric parallax distance determinations using recent ground-based and space-based telescopes 

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Recall that trigonometric parallax determinations are limited due to o Limits in resolution (e.g. atmospheric blurring), which increase the error of reading o The refraction of light in Earth’s atmosphere Ground-based telescope => V.L.T. (Very Large Telescope) o Built by the European Southern Organisation o Minimum angle of parallax: 0.01-arcsec o Allow for distances up to 100pc to be measured o Only a few hundred stars are this close o Limited by atmospheric blurring (despite interferometry and adaptive optics), and refraction of light in Earth’s atmosphere Space-based telescope => Hipparcos (HIgh Precision PARallax COllecting Satellite) o Launched by the European Space Agency in 1989 o Minimum angle of parallax: 0.001-arcsec

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This allows for distances up to 1000pc to be measured => Hipparcos has measured the parallax of around 100 000 stars. o Hipparcos’s parallax measurements are limited by the precision of the reading equipment Future telescope => GAIA o Due for launch in 2012 by the European Space Agency as a follow-up to Hipparcos o Minimum angle of parallax: 10 microarcseconds o Allows for measured distances up to 100 000pc o Parallaxes of >200 million stars can be measured o Limited by the precision of the reading equipment As can be seen, space-based telescopes are able to achieve a much more precise measurement of parallax angles than ground-based telescopes, as space telescopes are not limited by atmospheric effects. Sources of information o http://outreach.atnf.csiro.au/education/senior/astrophysics/parallaxlimits.html o ESO (European Space Organisation) and NASA website

3. Spectroscopy is a vital tool for astronomers and provides a wealth of information Account for the production of emission and absorption spectra and compare these with a continuous blackbody spectrum  

Recall that electromagnetic radiation consists of a wide spectrum of wavelengths Three types of spectra are emission spectra, absorption spectra, and continuous spectra Emission spectra o Produced by energy supplied to a low-density gas, (e.g. a low-pressure sodium lamp) o An atom absorbs the exact required energy, the an electron will become excited and jump from its ground state to a higher energy state (excited state) o When the electron returns to its ground state, it emits photons of discrete frequencies, given by

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If the electron had been excited to an even higher excited state, then it can return to its ground state in one single jump, or by a set of smaller jumps Each particular jump down between energy levels represents different quantities of energy, and so a spectra of discrete frequencies of photons are given off => this is the emission spectra The emission spectra consists of only discrete wavelengths, rather than a continuous spectra (see below) Each element has a unique emission spectra, thus its emission spectra is a ‘fingerprint’ for the element

Absorption spectra o Produced by a relatively cool, non-luminous gas in front of continuous spectra source (e.g. the relatively cool gas overlying the hotter, denser gas of a star) o As mentioned above, for an electron to jump to an excited state, it absorbs a discrete quantity of energy o The gas absorbs the photons from the continuous source, but only at the wavelengths matching the differences in energy levels o The atoms then re-emit the light as the electrons jump back down, but in all different directions => only a fraction of the re-emitted radiation is in the direction of the incidence light o The net effect is that the incident light is deficient in the absorbed wavelengths

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The absorbed wavelengths appear as dark lines on an otherwise continuous spectrum The dark lines on the absorption spectrum for an element correspond to the bands on its emission spectrum

Continuous blackbody radiation o o o o

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Produced by a hot solid, liquid, or high-density gas (e.g. a tungsten filament) Recall that a blackbody is a hypothetical object capable of absorbing all the electromagnetic radiation falling on it A black body re-emits EMR in a continuous spectrum related to its absolute temperature, described by a black body curve (or a Planck curve) As the temperature of the body increases, the peak wavelength becomes shorter, and the intensity of emitted radiation increases

Most high-density hot bodies approximate a black body A continuous black body spectrum appears simply as a continuous spectrum

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Below is a diagram showing the three different types of spectra as applied to a star

Analyse information to predict the surface temperature of a star from its intensity/wavelength graph 

Consider the intensity/wavelength graph for a black body below:

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The peak wavelength emitted depends on the temperature of the black body Stars approximate black bodies, so if the peak wavelength emitted from a star is known, then its temperature can be estimated off an intensity/wavelength graph (provided that the shapes at differing temperatures are known)

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Whilst not required by the syllabus, Wien’s displacement law provides a quantitative means of calculating a star’s temperature if the peak wavelength emitted is known:

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Where: o λmax = maximum wavelength emitted [m] o T = Temperature of black body [K] o W = Wien’s constant = 2.898x10-3 mK This equation may be given however, which would simply require reading the peak wavelength of the graph, and rearranging the formula

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Describe the technology needed to measure astronomical spectra  

Astronomical spectra is measured with a spectroscope A simple spectroscope consists of light source, slits, a prism, and a photographic plate, and a spectrum can be observed by the following steps 1. Light from a telescope is passed through a slit to form a flat, vertical beam 2. The beam passes through a glass prism, which disperses the light into its spectrum 3. The dispersed light falls onto a photographic plate, which records the spectrum

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A simple spectroscope is useful for observing spectra, but there are new technologies available for more sophisticated analysis o Diffraction gratings can be used instead of a prisms to increase the spectral resolution of images obtained o Collimators are used instead of slits to narrow the light beam o Improved lenses and mirrors have been developed (including collimating mirrors) o Photo electric detectors, such as CCDs (Charged coupled devices) are used to detect light, as they convert 80%-90% of incident photons into the recorded image. This is an improvement of photographic plates which only convert 1% of photons.  The S-Cam is a technology currently in development. The S-Cam is a new CCD that can record the position and colour of individual photons of light, and quickly compile the information into a database by a computer. o Sophisticated computer analysis have significantly aided in the analysis of astronomical spectra

Identify the general types of spectra produced by stars, emission nebulae, galaxies and quasars Object Star

Description A large, selfluminous, celestial body of plasma

Emission nebula

Regions of lowpressure gas clouds (mostly hydrogen and helium) that glow due to intense UV light from nearby stars

Galaxy

Collection of billions of stars, gas, and dust. Spectrum dominated by mix of stars

Spectrum Example Continuous spectrum created by the inner layers of a star, which acts as a black body. Absorption spectrum created by the atmosphere of a star. Emission spectrum => dominated by strong emission lines characteristic of the gas composition

Continuous spectrum from the stars in a galaxy ***May be absorption or emission depending on the abundance of nebulae in a galaxy => young galaxies tend to show emission spectra, whilst older galaxies tend to show absorption

Quasar

Very energetic and distant galactic nuclei, which dominate the total energy output

Emission spectrum superimposed on continuous spectrum due to fast moving gas clouds

Describe the key features of stellar spectra and describe how these are used to classify stars    

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Stellar spectroscopy is the analysis of the spectra of stars in order to learn more about their composition, surface temperature, and other features. Stellar spectrum consists of absorption spectrum characteristic of the atmospheric elements superimposed on an approximate black body continuous spectrum for a given temperature Most stars consist of a very similar set of elements and compounds, yet stars can exhibit very different spectral lines Different atoms and molecules produce spectral lines of very different strengths at different temperatures o For example, at lower temperatures molecules can exist near the surface, whilst at higher temperatures atoms become ionised => these two situations produce very different spectral lines Stars have been classified into spectral classes based on their observed spectrum. o The main spectral classes are O B A F G K M in order of decreasing surface temperature (Oh Be A Fine Girl/Guy Kiss Me) o There are also other spectral classes (see below) that have been discovered in recent times o Each spectral class is divided into 10 sub-classes, with 1-10, where 1 is the hottest and 10 is the coolest (e.g. the Sun is a G2 star)

Spectral class

Colour

W

Temperature (k) >50,000

Blue

Strength of hydrogen lines Weak

O

31,000-50,000

Blue

Weak

B

10,000-31,000

Blue-white

Medium

A

7500 – 10000

White

Strong

F

6000 – 7500

White-yellow

Medium

G

5300 – 6000

Yellow

Weak

Other spectral features He, C, N emission lines Ionised He+ lines, strong UV continuum Neutral helium lines Ionised metal lines Weak ionised Ca+ lines Ionised Ca+ lines, metal

% of main sequence stars Extremely rare 0.00003

0.1 0.6 3 8

K

3800 – 5300

Orange

Very weak

M

2100 – 3800

Red

Very weak

L

1200 – 2100

Red

Negligible

T

intensity greatly reduced) RESULTS/CONCLUSION RADIATION SOURCE OBSERVED SPECTRUM Reflected sunlight Continuous spectrum Incandescent filament Continuous spectrum Hydrogen lamp Distinctive violet, blue, green, and red bands Neon lamp Many distinctive blue, green, yellow, orange, and red bands Sodium lamp Yellow doublet

Reflected sunlight and the incandescent filament both produced continuous spectrum The discharge tubes produced emission spectra VALIDITY/ACCURACY The results obtained corresponded to the expected results The observations were compared to reliable sources, such as textbooks, websites, and scientific journals, which corroborated the collected data Natural sunlight was limited when observing the incandescent filament’s and discharge tube’s spectra, thus controlling the variables Qualitative data was collected => reliability of data not important (though repeated observation and comparison to others helps!!!)

4. Photometric measurements can be used for determining distance and comparing objects Define absolute and apparent magnitude The apparent magnitude (m) of an object is a measure of how bright an object appears when viewed from Earth. As it is a measure of brightness, it is influenced by distance, and celestial matter that may alter the brightness of the star. It is measured on a logarithmic scale, where a body of apparent magnitude 1 is 100 times brighter than that of an apparent magnitude of 6. Apparent magnitude ranges from -27, that of the Sun, to around +30, the faintest object detected by the Hubble telescope. The absolute magnitude (M) of an object is the brightness a star would have if it was observed from 10 parsecs away. Absolute magnitude is a measure of luminosity. It is also a logarithmic scale: for each five magnitudes lower, a star is 100 times more luminous. This measurement allows astronomers to compare features of stars more accurately, as absolute magnitude is not influenced by distance.

Explain how the concept of magnitude can be used to determine the distance to a celestial object The two primary factors that influence the apparent magnitude of a celestial body are luminosity and distance. As absolute magnitude is a measure of a body’s luminosity, a relationship exists between apparent magnitude, absolute magnitude, and distance. Take for example the magnitudes of the stars Sirius and Betelgeuse. Sirius has an apparent magnitude of -1.4, and Betelgeuse has one of +0.45. The absolute magnitude of Sirius, however, is +1.4, whilst the absolute magnitude of Betelgeuse is -5.1. As can be seen in this comparison, the apparent magnitude of Sirius is lower than that of Betelgeuse due to the distances to each star (Sirius is much closer to Earth than Betelgeuse), and thus a relationship exists. This relationship is:

The expression m-M is also known as the distance modulus. Rearranging

where M = absolute magnitude [no units] m = apparent magnitude [no units] d = distance to Earth [parsecs, pc]

Solve problems and analyse information using:

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to calculate the absolute or apparent magnitude of stars using data and a reference star As described above, the relationship between the absolute magnitude and the apparent magnitude of a body can be calculated if the distance to Earth from the star is known. The formula is the following:

where M = absolute magnitude [no units] m = apparent magnitude [no units] d = distance to Earth [parsecs, pc] NOTE: log = log10 The ratio of the brightness of two stars can also be calculated by considering that magnitude is measured on a logarithmic scale. For every five magnitudes lower, a body is 100 times brighter. This can be expressed mathematically as the following:

Rearranging

where IA/IB = the ratio of brightness of two body A and B mA = the magnitude of body A mB = the magnitude of body B

Outline spectroscopic parallax  

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Spectroscopic parallax is a method of determining the distance to a star using the H-R diagram and the distance modulus formula The steps involved in spectroscopic parallax are: 1. Measure the apparent magnitude (m) of a star using photometry 2. Determine the spectral class and luminosity class of a star using spectroscopy 3. On the H-R diagram, draw a vertical line from the relevant spectral class to the middle of the star group corresponding to the luminosity class. 4. Draw a horizontal line from the obtained point to the vertical axis, and read the absolute magnitude off the axis 5. Using the distance modulus formula, calculate the approximate distance to the star

The distance measured by spectroscopic parallax has a large percentage error, due to the estimates in determining the absolute magnitude Spectroscopic parallax is useful however for calculating approximate distances, as often there is no other method available to calculate a more precise distance

Explain how two-colour values (eg colour index, B-V) are obtained and why they are useful    

The observation of a star’s colour depends on the sensitivity of the detection method to different wavelengths of light The human eye is most sensitive to the yellow-green (550nm) part of the visual band Photographic film is most sensitive to the blue (~440nm) part of the visual band The overall colour of a star as viewed by the naked eye is both a combination of the star’s spectrum and the spectral sensitivity of the eye o For example, the peak intensity of blue giants lie in the UV/violet part of the spectrum, yet appear blue-white to the human eye

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The brightness or apparent magnitude of stars appears to change when viewed through different colour filters, as shown below.

o A star field viewed through a red (left) and blue (right) filter The apparent magnitude of a star as viewed by the naked eye is called the visual magnitude Star colours can be determined by using a standard set of coloured filters in front of a photometer, and measuring the brightness of each o The three standard coloured filters are ultraviolet, blue, and visual (yellow-green) filters, or the UVB set Filter Central wavelength (nm) Ultraviolet (U) 350 Blue (B) => photographic 440 Visual (V) 550 The difference in brightness seen through different filters is a measure of the colour of a star o This is called the colour index of a star, and is defined by o B = mB = apparent magnitude as viewed through a blue filter o V = mV = visual magnitude The higher the colour index the more red the star is The lower the colour index, the more blue the star is Colour index typically ranges from -0.6 (O spectral class) to +2.0 (M spectral class) AO stars have a colour index of zero Two-colour values such as colour index are useful because: o Colour index can determine the true colour of a star, independent of the sensitive of the detection method to different colours o Colour index can be used to determine the spectral class of a star, which can then be used to determine its distance from Earth NOTE: Absolute magnitudes are also dependent on colour sensitivity, so ensure that when using the distance modulus formula, both apparent and absolute magnitude are of the same coloured filter

Describe the advantages of photoelectric technologies over photographic methods for photometry 

Photographic film records images using light-sensitive film emulsion through the reaction of silver salts to light

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Photoelectric technologies use the photoelectric effect to produce a voltage Photoelectric technologies include: o Photomultiplier tube, which consists of a vacuum tube that multiplies the original signal by millions of times o Photodiode, which consists of a solid state device that acts as a light detector o Charged-coupled device (CCD), which consists of millions of photovoltaic cells that record incident light, and convert it to a digital signal to produce a digital image => also found in digital cameras Advantages of using photoelectric devices over photographic methods include o Sensitivity => a typical CCD records ~70% of incident photons, whilst photographic film records 2%-3% o Response to range of wavelengths => CCDs and other photoelectric technologies can detect infrared radiation (e.g. in night-vision cameras) and UV, whilst photographic film is restricted to the visible light band. o Image manipulation and enhancement => photoelectric devices can record digital images, so computer technologies can enhance, enlarge, add false colour, or subtract selected wavelengths to aid in analysis o Wider detection => CCDs can record many objects at once, whilst photographs can only record a single image o Faster processing => CCDs provide images much faster than photographic film o Increased astronomical sensitivity => CCDs can record images from fainter objects o Greater detection manipulation => CCDs can either record a broad spectrum, or a narrow band of EMR for specific analysis

Identify data sources, gather, process and present information to assess the impact of improvements in measurement technologies on our understanding of celestial bodies 

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Many recent developments in astronomical measurement technologies have had a significant impact on our understanding of celestial objects, and allowed for new directions in astronomical thinking Such technologies include charged-coupled devices (CCDs), space telescopes, and Wilkinson Microwave Anisotropy Probe. CHARGED-COUPLED DEVICES (CCDs) CCDs consist of a light-sensing array developed since the 1970s that records incident photons by means of the photoelectric effect => the same technology (but much simpler) is used in digital cameras CCDs convert photons into electrical signals, which are then used to form a pixilated image

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CCDs are an improvement on previous photographic technologies because… o They can measure a wider range of EMR wavelengths, allowing for a more thorough analysis of stars o They have increased the light-gathering power of telescopes by almost two orders of magnitude o The collected image is immediately computerized, allowing for instant digital storage, enhancement, and analysis CCDs have had a highly significant impact on our understanding of celestial objects, as they have allowed for significantly more accurate analysis of obtained celestial images SPACE TELESCOPES Space telescopes are telescopes launched into Earth’s orbit, taking images outside of Earth’s atmosphere Some space telescopes include HIPPARCOS (launched 1989 by ESA), the Hubble telescope (launched 1990 by NASA), and GAIA (due to be launched 2013 by ESA) Space telescopes are an improvement on previous measurement technologies because… o Radiation detected by space telescopes is not subject to atmospheric distortion, allowing for the most of the EMR spectrum to be detected and analysed => this provides greater understanding of stellar radiation o Radiation detected is not subject to atmospheric distortion, meaning images have a higher resolution without the need for adaptive optics, astrometric measurements are more accurate => this is particularly useful with parallax measurements (astronomers predict that GAIA could measure parallax of >200 million stars) and resolving globular star clusters, which has enhanced our understanding of stellar evolution o Images obtained are much less subject to background radiation, allowing for greater sensitivity Thus space telescopes have had a significant impact on our understanding of celestial objects, as they have allowed for a greater range of the EMR spectrum to be measured, and provided significantly more accurate data of celestial bodies. o e.g. The Hubble telescope allowed the Hubble constant to be calculated within 10%, allowing for a greater understanding of the universe’s expansion WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) WMAP is a NASA Explorer mission launched in 2001 to make fundamental measurements on cosmology (the study of the universe as a whole)

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WMAP was launched on a spacecraft to measure differences in temperature of the Big Bang’s remnant radiant heat WMAP was a significant improvement in measurement technology, as it provided the following data: o Reported the first detection of pre-stellar helium o Placed 50% tighter limits on the standard model of cosmology o Measured, with very high significance, temperature shifts induced by hot gas in galaxy clusters o Improved visual measurements of the polarization patterns around hot and cold spots WMAP led to the production of the new Standard Model of Cosmology Thus WMAP has had a highly significant impact on our understanding of celestial objects through the collection of new and more accurate measurements, and the development of new cosmological models. More information can be found at: http://map.gsfc.nasa.gov/

Perform an investigation to demonstrate the use of filters for photometric measurements 

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METHOD A light box was set up in a darkened room with a red cellophane filter placed in front of the light source. A light meter was directed towards the light source, and readings were taken with no additional filtering, a yellow filter, then a blue filter placed in front of the light meter, and results were recorded. The process was repeated for blue-filtered light. DO NOT LOOK DIRECTLY AT THE LIGHT SOURCE RESULTS Light Filter Intensity (x2000 lux) Red No filter 580 Yellow (Visual) 530 Blue (B) 33 Blue No filter 100 Yellow (V) 45 Blue (B) 60 By considering magnitudes, B-V for the red star produces a positive result (remember magnitudes decrease as brightness increases), which reflects expected results B-V for the blue star produces a slightly negative result, which was expected. As can be seen, a red star has a positive colour index, whilst a blue star has a negative colour index RELIABILITY Multiple results were taken, and an average was obtained The range of results for each data point was minimal, thus indicating precise results Our results were compared to other groups, all of which produced similar results VALIDITY The method tested the aim by demonstrating the use of colour filters in photometric measurements, specifically colour index

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Other variable which weren’t tested were minimised, such as external light The use of technology (i.e. the light meter) produced accurate results The results matched the expected results, and corroborated with reliable information sources such as textbooks, reputable websites, and scientific journals.

5. The study of binary and variable stars reveals vital information about stars

Describe binary stars in terms of the means of their detection: visual, eclipsing, spectroscopic and astrometric 

A binary star system consists of two stars orbiting around their common centre of mass

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The system’s centre of mass lies at the point where the following relationship holds true:

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where from the diagram above o m1 and m2 = the masses of the respective stars [kg] o r1 and r2 = the radii of each star from the centre of mass [m] The brighter star in a binary pair is designated with the letter A, and the dimmer is designated with B Binary stars are classified by their means of detection The classes of binary stars dealt with in this course are visual, eclipsing, spectroscopic, and astrometric VISUAL Can be resolved into two stars by a telescope => they can be detected visually Visual binaries orbit very slowly, and can take many years to be confirmed as a binary pair The period and radius of each orbit can be measured by visual observation and analysis, allowing the mass of the overall system to be calculated (see below) ECLIPSING Eclipsing binaries whose orbital plane is oriented so that it is almost parallel to Earth’s lineof-sight The stars regularly eclipse each other, causing periodic minima in the brightness of the system, as seen on a light curve o The primary minima correspond to the greatest decreases in brightness, which depends on the tilt of the orbit, the relative size of the stars, their surface temperatures, and their atmospheric structures

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Eclipsing binary system are more easily detected if period of each star is short, hence most eclipsing binaries are close systems The diameter of each star can be determined by the duration of each eclipse, and the period of the stars can be determined by the period of either the primary or secondary eclipses A flat-bottomed eclipse indicates a total eclipse, whilst a curved-bottomed eclipse indicates a partial eclipse SPECTROSCOPIC Spectroscopic binaries are detected by the alternating Doppler shifting of their spectral lines As the stars orbit, one star will typically have a component of velocity away from Earth, and the other towards This causes small red and blue Doppler shifts of the system, causing a double-lined absorption spectrum As the stars move in their orbit, they may have no relative motion to Earth’s line-of-sight, thus have no Doppler shift The doubling of spectral lines occurs periodically, indicating the presence of a binary system

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Spectroscopic binaries are best detected if the component of velocity measured by Doppler shift is large (maximised when plane of orbit is parallel to Earth’s line-of-sight), and the period of each star’s orbit is short (i.e. a close system) The period of the alternating Doppler shift reveals the period of each star’s orbit, and the degree of Doppler shift reveals the velocity of each star ASTROMETRIC Astrometric binaries are detected by an apparent wobble in a star’s proper motion One of the stars is too faint to be observed The centre of mass follows a straight path Measurements of the visible star’s wobble reveal the period of orbit and size, allowing for an estimation of the mass of the system

Explain the importance of binary stars in determining stellar masses  

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There is no method of measuring the mass of an isolated star Measuring the gravitational effect of a star on another object provides a method of determining a star’s mass o For this reason binary stars are VERY important, as they provide the only direct method of measuring a star’s mass The analysis of the motion binary stars enable astronomers to calculate the mass of stars due to the presence of gravity between the two stars The above relationship is used to determine stellar masses

Determining the mass of stars allows astronomers to further our understanding of celestial objects, such as through the mass-luminosity relationship o If the luminosity of main sequence stars are plotted against their mass, the following relationship becomes apparent

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This relationship shows that as the mass of a star increases, it’s luminosity increases at a much faster rate Luminosity is a measure of the rate of the consumption of a star’s fuel, thus higher mass stars consume fuel at a much faster rate => high mass stars thus a have a much shorter lifetime Additionally, the luminosity-mass relationship shows that as luminosity increases up the main sequence on an H-R diagram, so does the mass => this provides a new interpretation of the H-R diagram SUCH ANALYSIS DEMONSTRATES THE HIGH IMPORTANCE OF BINARY STARS

Solve problems and analyse information by applying:

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DEFINITIONS m1+m2 = total mass of the binary system (m1 mass of star 1, m2 mass of star 2) [kg] r = separation distance of the stars [m]

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T = orbital period of the binary system (s) G = 6.67x10-11m3kg-1s-2 = Universal gravitation constant

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The above formula can be derived from equating gravitational force to centripetal force around the centre of mass, and substituting in the orbital speed The full derivation can be seen on p.307 of Jacaranda Physics

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REMEMBER Check the units and dimensions at every line of working Convert all units to S.I. units when using a formula with a constant such as G r is the distance between the centres of mass of each star => you must add the radius of the star if the distance from the surface is given.

Classify variable stars as either intrinsic or extrinsic and periodic or non-periodic    

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Variable stars are ones that appear to vary in brightness with time Most stars vary in brightness over time, e.g. the Sun’s solar flares cause brightness variations of ~0.1% Other stars significantly vary with brightness, and are tracked on a light-curve Below is a diagram showing the classification of variable stars

EXTRINSIC VARIABLES The variation in brightness is due to a process external to the body of the star itself Extrinsic variables include… o Eclipsing binaries => the variation in brightness is due to one star of the binary star system eclipsing the other o Rotating variables => Large cool/hot spots cause the brightness to noticeably change as the star rotates INTRINSIC VARIABLES The brightness variation is due to internal changes of the star => the luminosity (power output) of the star varies Many intrinsic variables occupy specific locations on an H-R diagram (see below0 Intrinsic variables can be further classified as non-periodic and periodic

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NON-PERIODIC Variation in brightness does not follow a regular intervals => the variation is non-periodic Also called cataclysmic or eruptive stars Such stars include supernovae, novae, symbiotic stars, flares stars, R Coronae Borealis, and T Tauri Type Brightness variation Physical description Supernovae Increase to M outbursts from red giant fall onto white dwarf Flare stars Sudden increase >2 Red dwarfs experiencing intense magnitudes, then fade outbursts of energy from a small area within hours on surface R Coronae Borealis Sudden decrease of about 4 Yellow supergiant accumulates carbonmagnitudes, slowly rich dust clouds that obscure surface fluctuating back to normal T Tauri Vary irregularly Young protostar still contracting from gas cloud in which they lie

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Show periodic brightness variations Period can range from hours to hundreds of days, and is mostly sinusoidal Brightness variation occurs generally as the stars pulsate in size, surface temperature, and colour 1. Pulsation occurs due to disequilibrium between gravitational force and radiation pressure, the two primary forces that determine a star’s size

 Type Cepheid Mira RV Tauri RR Lyrae 

Periodic variables include Cepheids, Mira, RV Tauri, and RR Lyrae variables Period (days) Brightness change Description (magnitudes) 1-50 0.1-2.0 Very luminous yellow supergiant. Type I (young) and Type II (older) 80-1000 2.5-10 Red giants and supergiants 20-150 No typical value Yellow supergiants small, old, red, first-generation stars 3. Astronomers can determine the Cepheid type from spectral analysis The following graph demonstrates the period-luminosity relationship

As the graph shows, the luminosity (or absolute magnitude) of a Cepheid can be determined from the period of brightness variation Consequently, the distance to a Cepheid can be calculated using the distance modulus formula METHOD FOR CALUCLATING DISTANCE TO A CEPHEID 1. Establish the type of Cepheid through spectral analysis 2. Determine the period from its light curve 3. From the period-luminosity relationship, use the period to determine the star’s average absolute magnitude (M) 4. From direct observation, measure the star’s average apparent magnitude (m) 5. Use the distance modulus formula to calculate the distance to the star

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Thus the luminosity-period relationship has proved significant in determining distances to Cepheids o This has allowed for distances to be calculated within our galaxy, and to neighbouring galaxies as well

Perform an investigation to model the light curves of eclipsing binaries using computer simulation

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METHOD A java application on the following website was used to simulate light curves of eclipsing binaries: http://www.astro.cornell.edu/academics/courses/astro101/herter/java/eclipse/eclipse.htm. The spectral classes of each main sequence star were altered (F and F, F and B, F and M) whilst the separation (12 solar radii) and angle of view (5°) was kept constant. The resulting light curve was recorded and compared. The spectral class (B and F) and angle (5°) was then kept constant, and the separation was altered. RESULTS Spectral classes B and F, 5° angle to plane, 12 solar masses separation

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Spectral classes B and F, 5° angle to plane, 25 solar masses separation

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Spectral classes F and F, 5° angle to plane, 12 solar masses separation

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Spectral classes M and F, 5° angle to plane, 12 solar masses separation

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The first simulation demonstrates total eclipses, indicated by the flat-bottomed troughs As the more luminous star is eclipsed by the less luminous star, a primary trough occurs. As the less luminous star is eclipsed by the more luminous star, a secondary trough occurs. In this simulation, luminosity is related to spectral class as main sequence stars are modelled. The troughs have a shorter duration as separation increases, as the orbital velocity of each star is faster. RELIABILITY/VALIDITY The computer simulation provided a highly accurate model of our current understanding of eclipsing binaries. The results are based on precise mathematical analysis, hence are very reliable. Variables were controlled in the simulation, thus a valid method was followed.

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6. Stars evolve and eventually die Describe the processes involved in stellar formation 

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A star forms from a region of large quantities of interstellar medium called a nebula, which consists of interstellar dust and gas (mostly hydrogen molecules, but also helium and other ionised gases) o Interstellar medium forms after the death of larger stars, hence matter is essentially recycled o The dust in gas clouds obscures or blocks light coming from nebula, thus it is difficult to observe the process in stellar formation => X-ray, infra-red and radio wave radiation penetrate dust, so these telescopes provide the information on stellar formation The gas cloud is triggered into gravitational collapse, such as the explosion of a nearby star (e.g. supernova), the first burst of radiation from a nearby star, or collisions between gas clouds. The gas cloud starts contracting due to the gravitational between particles to form a denser core The density of the core gradually increases over time, which increases the gravitational forces between the core and gas molecules, thus the star contracts even faster o The rate at which a star contracts depends on its size => a star of 0.2 solar masses would take billions of years to contract to a protostar, whilst a star of 30 solar masses would take only 30 000 years.

As the core contracts, the gravitational potential energy of the gas particles are converted to kinetic energy, so the gas cloud heats up

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The heat creates an outwards pressure that opposes the gravitational collapse (called the hydrostatic equilibrium), only slightly at first, but it gradually builds When the star is hot enough, the outwards pressure stabilises the size of the core whilst the surrounding gas continues to fall inwards => at this stage the star is called a protostar

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With no source of energy, the gas cloud in a protostar continues to collapse, which increases the temperature of the core Once the core’s temperature reaches approximately 10 million Kelvin, the fusion of hydrogen is triggered, which provides a long-lasting energy source that stabilises the star => the star is now a zero-age main sequence star

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Not all gas clouds form stars however…

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Gas clouds of less the 0.08 solar masses cannot heat sufficiently to trigger the fusion of hydrogen o Gas clouds of greater than 30 solar masses are too unstable during collapse due to overheating, and blow apart to form smaller stars The above processes have been limited the formation of a single star in a gas cloud, but most often more than one star forms from a gas cloud, creating a binary star system or a cluster of stars o The contracting gas clouds could form multiple cores, which would eventually form multiple stars o Multiple stars can also be formed from the fragmentation of a star, as a star spins faster as it contracts (conservation of angular momentum) => this can also lead to a system of planets around a star Summary…

Outline the key stages in a star’s life in terms of the physical processes involved 

The key stages in a star’s life are summarised in the flow-chart below:

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NEBULA TO MAIN SEQUENCE A star initially forms from a cloud of dust and gas (interstellar medium) called a nebula. The nebula gradually collapses under its own gravity to form several cores of matter The gravitational potential energy is converted to kinetic energy, thus the core heats up, and provides radiant pressure to oppose the gravitational forces inwards

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The core is called a protostar when the core stabilises (i.e. forces are balanced), and the surrounding cloud becomes luminous. The star continues to shrink and heat up => when it reaches a temperature of around 1x107K and sufficient pressure, hydrogen fusion commences in the core, and the star becomes a Main Sequence star o There is a balance between the gravitational force inwards and radiant pressure outwards, thus the star stops contracting o The star remains on the Main Sequence of the H-R diagram for around 90% of its lifetime o The position a star enters the Main Sequence depends on its mass => a higher mass star will start on the Main Sequence at a higher point See above for a more detailed description of star birth, see below for a description of thermonuclear reactions in Main-Sequence stars POST-MAIN SEQUENCE Recall the mass-luminosity relationship, which demonstrates that a star of higher mass consumes its fuel at a higher rate (thus giving higher surface temperatures) o A star of 0.3 solar masses stays on the Main Sequence for around 30 billion years, whilst an O class stays on the Main Sequence for only 30 000 years

When the helium content in the core reaches around 12%, the fusion of hydrogen stops The future of the star depends on its mass: o A small mass star (less than 0.5 solar masses) will not be able to fuse heavier elements, so the star collapses to form a white dwarf, which are the hot remnants of a star o A core of a star of greater than 0.5 solar masses is able to reach high enough temperatures to commence helium fusion to carbon, with hydrogen fusion in the core => this star is called a Red Giant o When a star runs out of fuel in the core (i.e. it cannot fuse heavier elements), the star collapses under its own gravity, and can form a white dwarf, neutron star, or black hole depending on its mass RED GIANT As mentioned above, when a Main Sequence star runs out of hydrogen fuel it starts contracting due to the lack of energy to oppose the gravitational force If a star has a high enough mass, the temperature and pressure in the shell surrounding the core will have the required temperature and pressure to allow hydrogen fusion => this is called shell burning.

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This expands and cools the star, causing it to move off the Main Sequence towards the top-right of the H-R diagram As the star contracts, gravitational potential energy is converted to kinetic energy, so the core heats up, and allows the fusion of helium if a star has a high enough mass, and fuse hydrogen in the shell o This may happen in a helium flash (helium fusion starts suddenly in the core) for stars less than 2.6 solar masses, or smoothly for higher mass stars o After helium fusion starts, the star contracts again, thus the star’s surface temperature increases, and it moves towards the left of the H-R diagram

A star remains a red giant until the fusion of heavier elements stops (i.e. the star runs out of fuel) See below for a more detailed description of star death

Describe the types of nuclear reactions involved in Main-Sequence and post-Main Sequence stars  

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As mentioned above, main-sequence stars remain stable due to the energy radiated by thermonuclear reactions The two main nuclear reactions in Main-Sequence stars are the proton-proton chain and the carbon-nitrogen-oxygen (CNO) cycle => MEMORISE THESE REACTIONS PROTON-PROTON CHAIN The proton-proton chain occurs in all stars once they reach the main sequence, but only eventually dominates in cooler Main-Sequence stars like the Sun The proton-proton chain consists of the following three reactions:

Where… o ν = neutrino (small, massless, chargeless particle) o e+ = positron (positive electron) o γ = gamma photon o Hydrogen-2 = deuterium The first two reactions must proceed twice before the last reaction takes place As six hydrogen nuclei go into the reaction but two come out, the overall reaction is… The mass of four hydrogen nuclei more than the mass of a helium nucleus => the lost mass is converted to energy according to E=mc2, which provides the energy for the star

CARBON-NITROGEN-OXYGEN (CNO) CYCLE The CNO cycle is another thermonuclear reaction in stars, but only dominates in stars more massive than the star, where the core temperature exceeds 1.6x107K=> both reactions can still proceed simultaneously however

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The CNO cycle consists of the following reactions:

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Note that the carbon-12 acts as a catalyst The net reaction is still that four hydrogen nuclei combine to produce a helium nucleus, and release the energy similarly to the proton-proton chain due to the overall mass deficit.

POST-MAIN SEQUENCE

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Helium fusion occurs in the core of a star through the triple alpha reaction (recall that a helium nucleus is called an alpha particle)

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Carbon then can fuse with a helium nucleus to produce oxygen

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Elements up to iron can be fused in the core to provide energy for the star => beyond

Discuss the synthesis of elements in stars by fusion    

Hydrogen and helium were the only elements present in the primordial universe => all other elements have been synthesised in stars All Main Sequence stars fuse hydrogen nuclei to produce helium nuclei through aforementioned thermonuclear reactions in the core The mass of a star determines the elements that can be further fused in a post-Main Sequence star through exothermic nuclear reactions Elements up to iron (atomic number 26) can be fused in the shell of a post-Main Sequence star => elements beyond iron are fused in endothermic reactions, thus are not fused in the core of a star o A supergiant can develop an onion-like structure of many layers of shell burning of different elements, though only for a short period of time (heavier elements fuse more quickly) => the fusion of silicon to iron typically lasts only for one day Fusion fuel H He C Ne O Si

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Core products He C, O Ne, Na, Mg, O O, Mg Si, S, P Ni to Fe

Core temperature (K) 4x106 120x106 600x106

Mass (solar masses 0.1 0.4 4

1.2x109 1.5x109 2.7x109

Approx. 8 Approx. 8 Approx. 8

Elements beyond iron are produced in two ways: o The slow capture of neutrons in a helium-burning shell of a red giant can produce elements up to lead o The fast capture of neutrons in a supernova explosion, which provides enough energy to produce elements up to uranium

Explain the concept of star death in relation to:     

planetary nebula supernovae white dwarfs neutron stars/pulsars black holes

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A star ‘dies’ after it stop fusing element to produce the energy required for stable existence The processes involved in star death depend on the mass of the star

PLANETARY NEBULA Occurs for stars of less than 5 solar masses A star of this size can fuse helium in the shell, but does not fuse oxygen in the core The unsupported hells become unstable, and produce bursts of energy known as thermal pulses and high ‘superwinds’ These pulsations blow eventually around a quarter of the star’s material away from the star’s core, which eventually forms an expanding shell-shaped nebula => this is called a planetary nebula o The name planetary nebula is historical, as early astronomers believed these nebula to be planets

WHITE DWARFS Occurs for stars of less than 5 solar masses White dwarfs are the remnant core of a star after material has been blown off to form a planetary nebula

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No fusion reactions proceed, so the core collapses due to gravitational forces to a size around the size of Earth, forming a very dense, glowing core called a white dwarf The balancing force comes from electron degeneracy pressure, which results from a quantum effect where closely-spaced electrons are prevented from being on the same energy level A white dwarf has a surface temperature of around 10 000K but a relatively low luminosity due to its small size, so it exists at the bottom-left of an H-R diagram A white dwarf eventually radiates its remnant energy, and becomes a brown dwarf White dwarfs have a maximum mass of 1.4 solar masses (can be higher for rotating white dwarfs) => beyond this mass the gravitational forces are too strong for electron degeneracy pressure to balance the force SUPERNOVAE This occurs for stars of greater than 5 solar masses Larger stars are also subject to the pulsations that blow material away, and form a rapidly contracting core The high mass of the star however means that electron degeneracy pressure is not enough to balance the gravitational forces, so the star continues to contract until degenerate neutron pressure halts the contraction. The surrounding layers are bounced back, causing a supernova explosion

A significant quantity (~1046J) of gravitational energy is emitted in a few seconds, leaving behind a very dense core Iron and other heavy nuclei are ripped apart, releasing a large number of neutrons => these neutrons can be captured to produce heavier nuclei NEUTRON STARS/PULSARS Occurs for stars between 8 solar masses and 25 solar masses If the residual matter from a supernova forms a core of between 1.4 and 3 solar masses, a neutron star will be formed The mass of the star means that gravitational forces overcome the electron degeneracy pressure, crushing the protons and electrons together to form a sea of neutrons Below 3 solar masses, the neutron degeneracy pressure balances the gravitational forces, so the collapse halts to form a neutron star of around 10km in diameter

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A neutron star formed is a very dense, hot star that is rapidly rotating (dozens of times per second) due to conservation of angular momentum as the star shrinks to a significantly small fraction of its initial diameter Neutron stars also have a very strong magnetic field (108T) that emits an beam of electromagnetic radiation As the magnetic axis rarely aligns with the rotation axis, the electromagnetic beam sweeps across space as the neutron star rapidly rotates If the Earth is aligned with the beam, the neutron star can be detected (most commonly with radio telescopes) from the very regular pulsations of radiation detected => this is why neutron stars are also called pulsars

BLACK HOLES Occurs for stars greater than 25 solar masses When the remnant core of a supernova is greater than 3 solar masses, the gravitational forces are strong enough to overcome the neutron degeneracy pressure No known force is able to counter the significant gravitational forces, so the matter is crushed to a single point of infinite density called a the singularity The gravitational forces are so strong that not even light can escape the singularity from a certain radius called the event horizon, hence the celestial object is called a black hole Black holes cannot be directly detected due to the lack of EMR emitted, but can be detected from its effect on surrounding objects o For example, material accelerated into a black hole emit X-rays that can be detected

Explain how the age of a globular cluster can be determined from its zero-age main sequence plot for a H-R diagram 

A globular cluster contains hundreds of thousands of old stars in a sphere of around 100 light-years in diameter that have evolved from a giant molecular cloud o This is in contrast to an open cluster, which contains a few hundred young stars in a group of around 10 light-years in diameter

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The relative ages of open clusters and globular clusters is known because open clusters contain O and B spectral class stars, whilst globular clusters do not => higher mass stars have a shorter lifetime, so a cluster containing high mass stars is a relatively young cluster The diagram below shows an H-R diagram of an open cluster and a globular cluster against a Zero-Age Main Sequence (ZAMS) line o Luminosities can easily be determined, as the stars in a cluster are about the same distance to Earth

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The globular cluster has a lower ‘turn-off’ point from the ZAMS line and a more developed giant branch than the open cluster o In other words, the highest remaining point of the Main Sequence group is lower for the globular cluster Lower mass stars have a longer lifetime than higher mass stars due to the mass-luminosity relationship, so if more low mass stars have become red giants, then the cluster must be older Thus as the star ages, it appears to peel off the main sequence, as higher mass stars progressively become red giants In other words, the lower the turn-off point from the ZMAS line in an H-R plot of a globular cluster, the older the cluster The age of a globular cluster can be estimated by considering the lifetimes of the stars that have left the ZAMS line, and the lifetimes of those still on the line o In the above diagrams, the open cluster on the left is estimated to be about 600 million years old, whilst the cluster on the left is about 13 billion years old

Present information by plotting Hertzsprung-Russell diagrams for: nearby or brightest stars, stars in a young open cluster, stars in a globular cluster

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Consider the H-R diagrams below of (left to right) nearby and brightest stars, stars in a young open cluster (such as the Pleiades), and stars in a globular cluster

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The plot of the nearby or brightest stars shows a random sampling, so all the prominent star groups are present Star clusters were formed at the same time however, so they are not a random sampling since they are all of the same age The plot of the young open cluster lies almost entirely within the ZAMS line The plot of the globular cluster however consists of the bottom half of the ZAMS line, and a number of stars occupying the red giant region, indicating that the missing Main Sequence stars have become red giants and shifted to the right

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Present information by plotting on a H-R diagram the pathways of stars of 1, 5 and 10 solar masses during their life cycle 

Stars of 1, 5, and 10 solar masses enter the H-R diagram on a track shown below:

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The H-R diagram below shows the life cycles of stars of 1 and 5 solar masses:

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For a star of 1 solar mass… o The star enters the Main Sequence at the position indicated o Once it has fused all the hydrogen in the core, it proceeds to burn hydrogen in the shell, causing expansion and cooling of the star’s surface o The core contracts due to gravity, causing the temperature and pressure to increase => once it reaches the right conditions, helium begins to fuse to carbon in a ‘helium flash’ o The star contracts until helium fusion stops in the core and shell burning again, causing it to contract o No further fusion occurs so luminosity decreases, and the star contracts due to gravity, causing the surface temperature to rise => the star becomes a white dwarf o NOTE: The evolutionary track from the red giant to white dwarf should be lower (i.e. the star does not sweep upwards as shown) For a star of 5 solar masses… o The star follows a similar track to the 1 solar mass star, but enters the Main Sequence at a higher point due to its higher mass o The star experiences a supernova however and becomes a neutron star, so does not exist on the H-R diagram after the supernova Below is an H-R diagram of a star of 10 solar masses…

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For a star of 10 solar masses… o The star follows a similar evolutionary pathway to the 1- and 5- solar mass stars, but at a higher luminosity into the supergiants region o The star is able to fuse elements heavier than helium in the core, so moves left and right in the giant region more often o The star then goes to a supernova and a black hole, and thus exits the H-R diagram in the supergiants region

Analyse information from an H-R diagram and use available evidence to determine the characteristics of a star and its evolutionary age  

Many characteristics of a star can be determined directly from an H-R diagram, such as colour, surface temperature, chemical composition, and luminosity. The position of a star on the H-R diagram can also help determine what type of star it is:

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Thus from an H-R diagram, we can determine the characteristics and evolutionary age of a star according to its position For a protostar (lower than Giants, to the right of Main Sequence)… o It is at the beginning of its life o No fusion proceeds in the core o Very cool but luminous, therefore large in size For a Main Sequence star… o It is in the middle of its life, and remains in the Main Sequence for the majority of its lifetime o Hydrogen fusion in the core to produce helium either by the proton-proton chain or the CNO cycle o A main sequence star towards the top left of the H-R diagram is young, o Stars higher on the Main Sequence burn their fuel at a much higher rate, and are larger in size than those lower on the Main Sequence => stars towards the bottom of the Main Sequence are considered dwarfs For a Giant… o It is towards the end of its life, as it has consumed most of its fuel o It is relatively cool but luminous, thus is large o It is fusing hydrogen to helium in its shell, and may be fusing helium to carbon through the triple alpha reaction in its core For a Supergiant… o It is towards the end of its life, as it has consumed most of its fuel o It is relatively cool but very luminous, thus it is very large o It is fusing heavier elements in its core, and fusing various other elements in its shell in an onion-like structure For a white dwarf… o It is at the end of its life, as it is no longer fusing elements to produce energy o It is a relatively hot star but not very luminous, thus it is very small (i.e. is a dwarf) o It is the remains of a former star, and is probably surrounded by a planetary nebula