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Swinburne Online Education Exploring Stars and the Milky Way Module : 4: Module Vital Statistics of Stars Activity 2: Magnitudes and Colours of Stars © Swinburne University of Technology Summary In this Activity, we will investigate the magnitude and colour of stars. In particular, we will discuss: • the meaning of “colour” when applied to stars; • the meaning of magnitude; • the difference between apparent magnitude and absolute magnitude; • luminosity; and • the relationship between a star’s brightness and its size. Stars: what we can measure It would be lovely to be able to measure everything about stars directly... Scales Meter Wow! Thermometer Too too What too far, small, small, an picture interesting won’ttoo reach fuzzy star ......... ? ? ? ? ? … but until humans develop interstellar travel, the measurements we can make from observatories on Earth are limited. What we can measure All we can do is analyse the light that reaches the Earth from the star, and obtain data on: • the energy spectrum of the light emitted; • the energies missing from that spectrum; • the intensity of the light emitted; intensity Tell me about light • the distance of the star from Earth (using Cepheid variables or parallax for example). However, as you will see, this is quite powerful data and we can do a surprising amount of science with it. wavelength Colour of Stars The colour of a star depends on the strongest colour in its spectrum. As you will learn in the next Module, the spectra of stars share some basic features, and there is a common “hill” shape, with a maximum. Flux One with this spectrum will look yellow A star with this spectrum will look blue Ultra-violet And one with this spectrum will look red Visible light Infra-red wavelength Finding temperature from spectra Flux The position of the maximum in the spectrum from a star can indicate its surface temperature. The hotter the star, the more light Ultraviolet X-ray it emits at the blue, short wavelength gamma ray end of the spectrum. Visible light hot star medium star cool star Infra-red wavelength The colour of stars Flux hot star 11 000 - 30 000 K Naos Rigel Sirius Canopus This means that a star that looks blue is likely to be a very hot star, while a reddish star is (relatively speaking) cooler. Our own Sun is medium star a yellow star. 5 000 - 11 000 K Sun Capella Arcturus Aldebaran Antares Betelgeuse cool star 3 500 - 5 000 K wavelength Blue Stragglers The colours of stars in these pictures are artificially enhanced, but still indicate the variation in colour within one star cluster. The picture on the left was taken from the ground. The small rectangle marks the area detailed in the photo on the right, taken by the Hubble Space Telescope. The circles in the Hubble photo indicate “blue stragglers”: young, hot stars found in clusters of much older stars. From spectrum to magnitude The spectra of stars will be studied in more detail in the next Module “Colours and Spectral Types”. Our next topic in this Activity is magnitude. We will have a look at: • how the brightness of stars is perceived; • how it is adjusted to compensate for our limited vision; • how it is adjusted for distance from the source; and • the definition of the various terms used in the process. What is magnitude? In astronomy, the magnitude of a star is a measure of how brightly it shines compared to other stars. The human eye is good at telling the difference between stars that are roughly twice as bright as each other, but not much better than that. Magnitude? Magnitude! Magnitude and History I II Hipparchus in the 2nd century BC classified about a thousand stars into six brightness groups called “magnitudes”. III The first magnitude stars were the brightest. They are about 100 times as bright as the sixth magnitude stars, which were the faintest stars that Hipparchus could see. Each magnitude was about twice as bright as the next magnitude. IV V VI Magnitude and Commonsense Unfortunately, this has left us with a scale which runs counter to common sense in many ways, but its usage has a long history. The way the magnitude scale is defined means that: • the brighter the star, the lower the magnitude • some stars (such as Sirius and the Sun) actually have a magnitude that is a negative number! Sun -26.5 -30 Sirius -1.5 -20 -10 Aldebaran 1 0 Naked eye limit ~6.5 10 Binocular limit ~9 Faintest detected 28.5 20 30 Hubble Space Telescope A very human scale Although the magnitude scale seems strange at first, it is actually quite intuitive. Hipparchus, who developed the magnitude scale, had little instrumentation and relied far more heavily on human senses than we do today. The magnitude scale, along with other measures for things such as loudness and pitch of sound, is based not on a linear scale but rather on the way the human nervous system operates. How bright is that star? I don’t know, but I can tell you how bright it looks ... Two ways of measuring Magnitude is measured on a logarithmic scale. This means that increases in magnitude involve a multiplication, rather than a simple addition. Consider the following: +1 0 km +1 1 km +1 2 km +1 3 km 4 km 5 km Four kilometres is four times further than one kilometre, and twice as far as 2 kilometres. Each ‘one kilometre step’ involves the addition of an extra kilometre, so we say that... Distance is measured on a linear scale. A logarithmic scale Our senses often don’t work in this way however. What seems to a human observer to be a simple increase in loudness or pitch is actually not an addition but a multiplication. In this example, each 3 decibel increase corresponds to a doubling in the intensity of sound. So a sound 9 dB “louder” actually has 8 times the sound energy! 3 dB “louder” Twice the intensity 3 dB “louder” Twice the intensity 3 dB “louder” Twice the intensity Magnitude, like loudness, is measured on a logarithmic scale. Our perception of the brightness of stars works in a similar fashion, and so the magnitude scale is a little odd at first. x 100 I II x 2.512 x 2.512 III x 2.512 IV x 2.512 V VI x 2.512 It turns out that each step in magnitude corresponds to an increase in brightness by a factor of 2.512. Five even steps (from magnitude 1 to magnitude 6) give a factor of 100 in brightness. Magnitude depends on where you are The magnitude or brightness of a star depends on the distance between the star and the observer. The further away you are, the less bright the star will appear to be, according to the inverse square law. The brightness we see from Earth is called the apparent magnitude. That’s a fairly bright star: magnitude 2, I’d say Rubbish! It only looks about magnitude 6 from here! We’d better call it the apparent magnitude, then What’s the inverse square law? Our Limited Vision Infra-red long wavelength low frequency Human perception affects not only how we see brightness but what “colour” we think a star is. Stars radiate at all kinds of wavelengths, from the ultraviolet to the infra-red and radio waves, and beyond. However since magnitude was (and still is) defined by how stars appear to human vision, we may as well go the whole way and customise the definition a bit further to suit ourselves. We can’t see in the infra-red, nor can we see in the ultraviolet, so why bother with them? Ultraviolet short wavelength high frequency Apparent Visual Magnitude, mv “v” is for “visual” If we restrict our definition of magnitude to the “visual” wavelengths ( from roughly 4 000 to 7 000 angstroms, or 400 to 700 nanometers) then we have a pleasant and friendly measure by which we can classify and discuss stars and other objects. The brightness of a star apparent … as seen from Earth visual magnitude … by humans! Stars radiate entire spectrum For Earthers only This definition of Apparent Visual Magnitude may not impress a Martian, but it certainly makes life easier for a life form which evolved on a planet where the atmosphere lets in lots of light in the 400-700 nm range, and so that’s what our eyes can perceive. Venusians might might judge magnitude by these wavelengths Earth people judge magnitude by these wavelengths Martians might judge magnitude by these wavelengths Absolute Visual Magnitude, Mv It’s all very well to say that a star appears dim in the visual wavelengths, but is it really dim? That will depend on its distance from you. Astronomers use another measure, Absolute Visual Magnitude, to compensate for the fact that stars are at different distances from Earth. The Mv of a star is the magnitude that it would have if it were placed at 10 parsecs from Earth. The apparent visual magnitude is 3 Real distance 10 parsec Earth star … but the absolute visual magnitude is 0.8! Some sample values of Mv The apparent (mv) and absolute (Mv) visual magnitudes of some stars are given below, along with their distances (d) from our Solar System in light years. Star Sun Alpha Centauri A Canopus Rigel Deneb mv -26.8 -0.27 -0.72 0.14 1.26 Mv 4.83 4.4 -3.1 -7.1 -7.1 d(lyr) 1.7x10-5 4.3 98 900 1600 Some things to note It will take some time for you to get used to magnitude measurements. For now, you should note that: • A large negative value means a very bright star, and • A large positive value means a very dim star. Consider our own Sun… mv: measured atmEarth Star Sun Alpha Centauri A Canopus Rigel Deneb v -26.8 -0.27 -0.72 0.14 1.26 Mv: measured at M10 pc v 4.83 4.4 -3.1 -7.1 -7.1 The Sun looks extremely bright d Earth’s at the (nil) ... surface but 4.3 would look 98dim at a very 900 distance of 1600 10 pc What does that mean? The faintest object we can comfortably perceive with the naked eye would have an apparent visual magnitude mv of +6.5 or so. An object that was “pretty bright” would have an mv of about 0. The Sun has an mv of –26.8. It is by far the brightest object in the sky. I think I will remember it this way: Something with a plus sign is quite safe to look at, while a minus sign can mean danger! Twinkle, Twinkle, Little Star Consider the star Deneb. Its apparent visual magnitude is 1.26, which means that it is a fairly bright star when seen from Earth. But its distance from Earth is about 490 pc, which means that in fact it shines very bright. When the distance is taken into account, the result is an absolute visual magnitude of –7.1. That makes Deneb one of the brightest objects around. Deneb: 490 pc away mv=1.26 If Deneb was 10 pc away Mv=-7.1 Luminosity Star radiates heaps of energy in all directions Luminosity is the amount of energy a star radiates in one second, and is often quoted relative to the luminosity of the Sun. It would be nice to be able to simply measure the luminosity of a star directly, as this would help us to classify it and describe it (e.g. as incredibly luminous, low-luminosity etc). However all we can measure is the tiny fraction of the radiation that reaches Earth. We can calculate the luminosity from this, but there are a few steps to fill in first. … but only a tiny bit reaches Earth Start with Magnitude mv + 1. Start with the measurement of apparent visual distance magnitude, mv. This measurement can be made using modern photographic equipment (basically, a light meter!). Mv 2. The distance to the star is also measured, using parallax. 3. When both mv and the distance are known, the absolute visual magnitude Mv is calculated. mv Bolometric Magnitude + distance 4. However the light considered in a measurement of mv (and then a calculation of Mv) is in the visual wavelengths only, so a Mv correction must be made to include the + wavelengths outside the visual range. That way we can estimate the total energy non-visible radiation radiated by the star at all wavelengths (not just the visual). The result of the adjustment is called Absolute the absolute bolometric magnitude. bolometric magnitude mv = -0.06 An example: Arcturus + The apparent visual magnitude (as seen distance 11 pc from Earth) of the star Arcturus is -0.06. Taking into account its distance of 11 pc gives an absolute visual magnitude of -0.3. Mv = -0.3 The adjustment to include non-visible + wavelengths doesn’t make much difference non-visible in this case: the absolute bolometric radiation magnitude is still about -0.3. However if a star is very red or very blue the correction can be quite large. Abs. bolometric magnitude = -0.3 And finally, luminosity abm of Acturus = -0.3 5. The absolute bolometric abm of Sun magnitude (abm) of the star is then = +4.7 compared to the absolute bolometric Difference = 5.0 magnitude of the Sun (+4.7). … by definition, this Every difference of 1 in magnitude means a factor of 100 means a factor of 2.512 in the in luminosity luminosity of the star; a difference of luminosity of Sun 5 means a factor of 100. 26 Watts = 4 x 10 The luminosity of the Sun is known, so the luminosity of the star can be calculated. so luminosity of Acturus = 4 x 1028 Watts The Size of Stars Let’s leave brightness for now, and start thinking about stellar size: another important property for classifying stars. It is almost impossible to actually see a star through a telescope and measure its physical diameter. We can do this with objects within the Solar System, but the stars are simply too far away to appear as more than blurry dots. How then, can astronomers confidently state that one star has a diameter a hundred times that of the Sun, while another has a diameter one-half that of the Sun? Well there are some clever observational tricks using pairs of telescopes known as interferometers, but there is often another easier indirect way ... Luminosity Again The luminosity of a star depends mostly on its temperature and its radius. There is a simple physical law which determines how much radiation a black body emits. According to this law, luminosity, L, is related to temperature, T, and radius, R, by: = 3.1415926… = 5.98 x 10-8 m-2 K-4 is the StefanBoltzmann constant. temperature … defines how much energy is given off per square metre. radius … will determine the surface area of the star luminosity … the energy output of the star (per second) How Hot, compared to the Sun? We can measure the temperature of a star by looking at its spectrum. This will be studied more in the next Activity. The hotter the star is, the bluer its light will be. Comparing the spectra of stars lets us compare their temperatures. The spectra on the right show that the star is a lot hotter than our Sun, and would look blue to humans. Apparent Brightness Star’s spectrum Sun’s spectrum Wavelength How Bright, compared to the Sun? We can measure the luminosity of a star (by measuring its apparent visual magnitude and working back through the steps shown earlier in this Activity) until we can compare its luminosity to that of the Sun. Apparent visual magnitude, mv Distance from Earth Absolute visual magnitude, Mv Absolute bolometric magnitude Compare to the Sun Calculate luminosity How Big? So, if we know both the luminosity and the temperature, we can work out relative sizes of stars! Here is an example about the reddish “star” Antares, which is actually a binary system Antares A - reddish, (two stars orbiting surface temp. 3,000oK each other) made up of Antares B - bluish-white, surface temp. 15,000oK - but the combination looks reddish, because the measured light intensity that we pick up on Earth from Antares A 40 x that from Antares B, and as they are close together, that means that Antares A must be approx. 40 times as luminous as Antares B. How Big? Surrounded by a nebula of expelled gas, Antares A is the brightest star in the constellation of Scorpio, and one of the brightest in the night sky. Question. If Antares A is much cooler (3,000oK) than Antares B (15,000oK), how can it be so much more luminous than Antares B? Answer. It depends on their relative sizes. This is because the amount of energy radiated by each square metre of star’s surface does depend strongly on temperature ... In fact, as we will see in the next Module, stars behave very like practical examples of black bodies - theoretical objects with properties that have been determined by classical physicists. (We will leave the tricky question of how someone could describe a star as a black body to the next Module!) This is very useful, because classical physicists in the 1800s worked out lots of laws which apply to black bodies. One in particular, the Stefan-Boltzmann Law, is very important for the study of stars. Stefan-Boltzmann Law The Stefan-Boltzmann Law says that if an object radiates like a black body, then the amount of energy, E, it gives out per second (its luminosity) is related to its temperature T by … for each square metre of its surface. Note the emphasis on each square metre of its surface - if two stars are about the same size, the hotter one will definitely be by far the most luminous, … but a huge cool star can still radiate more than a small hot star, because of all its extra square metres of surface. = 5.98 x 10-8 m-2 K-4 is the Stefan-Boltzmann constant we saw earlier. It turns out that Antares A manages to be so much more luminous (x 40) than Antares B, even though it is a factor of five cooler, because it is 160 times bigger than Antares B, and about 700 times the diameter of our Sun. (in diameter) . Antares B (Antares A is a red supergiant star - more about these in the Module on Stellar Old Age.) Antares A Summary This Activity has shown you how some of the simple measurements that we can make in astronomy can lead to really interesting and exciting facts about stars and other distant objects. In spite of the fact that we are stuck on Earth, our instruments can measure the position of a star as the year passes, the brightness of a star, and the spectrum of light from a star. These allow us to calculate things that we can’t possibly measure directly, such as the distance to the star, the luminosity of the star, the temperature and the size of the star. In the next Module, we will look more closely at stellar spectra. Image Credits The Earth’s Moon: http://nssdc.gsfc.nasa.gov/planetary/banner/moonfact.gif AAO: Antares © David Malin (used with permission) http://antwrp.gsfc.nasa.gov/apod/image/9706/antaresneb_uks_big.jpg Hit the Esc key (escape) to return to the Module 4 Home Page Inverse square law If something is being emitted with equal intensity in all directions from a point source, it will obey the “Inverse Square Law”. Point source of light Closer in, the intensity of light is high as the light is only spread over a small area Further out, the intensity of light is low as the light is spread over a larger area Imagine that a star is emitting light equally in all directions. At planet Alpha, the light is observed as being fairly intense, as it is being shared over a small area: • small radius, therefore • small area, therefore • high light intensity. At planet Beta, the light is being shared over a larger area and so the intensity of the light is far less: • large radius, therefore • large area, therefore • low light intensity. Star Return to Activity Light For thousands of years people tried to come up with a description of light which explained the many surprising things that light can do, including: • reflection; • refraction; • dispersion; and • interference, especially diffraction. Since astronomers learn about other places in the Universe by That star has studying the light we receive got light and dark from them, you will need to patches around it! know a bit about light during No it hasn’t. your study of astronomy. The star isn’t like that; it’s our telescope’s fault An Example of Diffraction This photo shows “spikes” and rings caused by diffraction through the telescope instrumentation: These rings and spikes are diffraction effects What is light? Although reflection can be explained easily enough if light is made up of tiny particles, it is very hard to explain phenomena like diffraction Is it a wave? unless light is a wave. A wave model has its own problems, however. For instance, if light is a wave, and waves require a medium such as air or water to carry them, how does light travel through empty space? Physicists now believe that light is neither a wave nor a particle, but its behaviour sometimes resembles a wave and sometimes a particle. Is it a particle? It is neither, but it’s like both Waves It is convenient to treat light as a wave when discussing colour, and this is a property of prime importance to astronomers (particularly when examining the colours of stars, as we are doing in this Activity). Waves are described in physics by a few standard dimensions. Wavelength = length of one cycle Amplitude A = height of wave above “rest position” A Frequency f = how often the wave passes: longer wavelength means lower frequency Velocity v = speed of wave Frequency and energy Frequency is very important in physics and in astronomy, where we are very often interested in such things as energy and temperature. This is because energy, E, is related to the frequency of light by the formula: = 6.626 x 10-34 Js When writing about light, people often use the Greek symbol (pronounced “noo”) for frequency, and c for the speed of light. In astronomy, you will often see the symbols and c for frequency and speed. Waves in general Light Electromagnetic radiation Electrons accelerate and decelerate releasing energy in the form of EMR Light is just one type out of many types of “electromagnetic radiation” (EMR). EMR is produced when electrons decelerate and lose energy (e.g. in a radio transmitter) or drop from a high energy level in an atom to a lower one (e.g. in the chromosphere of a star), and lose energy. Electrons drop to lower energy levels releasing energy in the form of EMR Observing EMR EMR is absorbed by electrons and is turned into an electrical signal When EMR is absorbed or detected (e.g. by a leaf, an eye, a telescope or photographic film) the reverse happens. The energy of the EMR is absorbed by electrons and converted to electrical energy (e.g. in a solar panel) or it causes an electron to jump to a higher energy level, allowing a chemical reaction to take place (e.g. in the human eye). EMR is absorbed by electrons allowing a reaction to take place Colour of electromagnetic radiation The human eye interprets difference in frequency as “colour”, and calls the range of frequencies that we can see “visible light”. 450 nm ultraviolet frequency 700 nm wavelength infra-red 6 x 1014 Hz There are an infinite number of possible frequencies (and wavelengths) for light, but humans can see only a very small band of them between the ultraviolet and the infra-red. Return to Activity