Measuring Stellar Distances

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Venus, Jupiter and the Moon 17/11/2017

Three useful units – which we use depends on how far away the object is. Being able to convert quickly is useful to know.

  • Relatively small distances – Astronomical Unit, AU = average distance between Earth and Sun  (1 AU=150 million km)
  • Light Year. 1 ly is the distance light travels in a year at a speed of 300 million m/s.

Find the distance in km between us and our nearest star Proxima Centauri (4.3ly away) You might like to speculate how long it might take a spacecraft travelling at a maximum possible 25,000km/h to reach it.

  • One parsec (pc) is defined as the distance to a star that shifts by one arcsecond from one side of Earth’s orbit to the other. These angles are incredibly small. They’re too small for degrees to be a practical unit of measurement. There are 3,600 arcseconds (60 minutes x 60 seconds) in one degree. To provide some perspective: one arcsecond is equivalent to the width of an average human hair seen from 20m away.

The nearest star is Proxima Centauri, at 1.3 pc. The Andromeda Galaxy, the closest spiral galaxy to our own, is nearly 800 kiloparsecs away.

If we imagine ourselves taking measurements of an imaginary star at six monthly intervals, it seems to have moved with respect to the Sun, 1 AU away.

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Putting this another way:

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Explaining Trigonometric Parallax.

  • Because of the Earth’s revolution around the Sun, nearby stars appear to move with respect to very distant stars which seem to be standing still.
  • Measure the angle to the star and observe how it changes as the position of the earth changes. In the second diagram if the observation point is at the top of the picture, six months later it will be at the bottom, 2 AU’s away
  • You can use your fingers to show trigonometric parallax. Shut one eye and hold your finger about eighteen inches in front of your face. Observe a distant object and the finger. Keeping still, look with the other eye. The finger represents the near star and appears to have moved with respect to the background. If you ask a friend to hold up his finger and repeat the observation, it would seem to have moved much less.
  • The parallax or apparent shift (from the Greek for ‘alteration’) of a star is the apparent angular size of the ellipse that a nearby star appears to trace against the background stars. Because all parallaxes are small (the stars are very far away), we can use the small angle approximation as shown. If we measure the distance to the star in AU. (astronomical units), then the parallax is given by:Screen Shot 8.png

For example – the six month parallax angle for Alpha Centauri is 1.52 seconds of arc. You might like to calculate how far away this is in light years.

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Luminosity

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A supernova
  • Is the amount of electromagnetic energy a body radiates per unit of time. {J/s (W)}
  • Is intrinsic to a body and is a measurable property which is independent of distance.

Imagine a point source of light of luminosity L that radiates equally in all directions. A hollow sphere centred on the point would have its entire interior surface illuminated. As the radius increases, the surface area will also increase, and the constant luminosity has more surface area to illuminate, leading to a decrease in observed brightness.Screen Shot.png

A is the area of the illuminated surface, a sphere of radius r.

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F is the Flux of the illuminated surface, the energy radiated per second per square metre of surface from a point source such as a star. This means that Luminosity is the total flux in watts

So:Screen Shot 4.png

The sun has a luminosity L = 3.8 x 1026W. If Earth – sun distance is 150m km and the Sun can be considered a point source, we can show that the radiant energy flux at the surface of the Earth is about 1.3kW m-2

Hertsprung and Russell showed that that the luminosity of a star L (assuming the star is a black body – a perfect emitter and absorber – which is a good approximation) is also related to temperature T and radius R of the star by the equation:

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where  ‘sigma’ is the Stefan-Boltzmann Constant  5.67 × 10−8 W·m-2·K-4

L is often quoted in terms of solar luminosities, or how many times as much energy the object radiates than the Sun, so LSun= 1

Questions:

  1. Find the actual luminosity of the Sun, given a surface temperature of 6,000K and a radius of 7 x 108m
  2. Compare with Sirius – a very bright star – temperature 12,000K and radius 2.22 x 109m. How much brighter is Sirius than the Sun – in solar luminosities?
  3. What would be the surface temperature of a star having the same luminosity as the Sun but twice the radius. What would it look like?
  • So a bigger star can be at a lower temperature and yet have the same luminosity, i.e. it looks just as bright
  • A hotter star is more luminous than a cooler one of the same radius.
  • A bigger star is more luminous than a smaller one of the same temperature.

A cool (red) giant star is more luminous than the Sun because, even though it is cooler, it is much larger than the Sun.

Finally, the idea of a Standard Candle is an important one in astrophysics. Certain classes of objects such as supernovae and Cepheid variable stars have properties whereby their luminosities can be determined separately from other measurements. For example, the period of a Cepheid variable star depends on it’s mean absolute magnitude; the more luminous the star, the longer the period.

If we know the luminosity and can measure the energy flux or brightness, then by comparing it with a standard candle then we can figure out how far away it is by comparing it with a standard candle of the same luminosity.

 

 

Resolvance of a Diffraction Grating

Illuminating a diffraction grating with monochromatic light from a He/Ne laser shows a typical pattern, out in the photograph to m=3 on both sides. The spots are equally spaced and we notice that the m=2 spot is hidden under the first single slit diffraction minimum – a “missing order”.

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The geometry is identical to that for a double slit, d being the distance between the centre of one slit and the next. For a bright maximum:Screen Shot

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Unlike two-slit interference, only at very particular angles do the contributions from each slit add constructively. Everywhere else, the contribution from one slit has a partner somewhere else down the grating which cancels its contribution out, hence the very bright spots and a lot of empty space.

You are strongly encouraged to go to the Wolfram Demonstrations Project, download the CDF player and experiment with this demonstration. 1, 2 or many slits -the choice is yours. With 15 slits the pattern is almost indistinguishable from a diffraction grating – screenshot below – the single slit diffraction envelope is clearly shown. Light intensity (y-axis) is proportional to amplitude squared.

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A flame test for sodium displays a very bright yellow emission. This emission is due to the sodium D-lines – two lines very close together.

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The diagram shows the absorption spectrum of the Sun by Fraunhöfer who labelled the lines. The sodium doublet is seen at wavelengths of about 589.0 nm and 589.6nm.

How could these be resolved using a diffraction grating? We recall that a diffraction grating gives sharp, clear orders.

More accurately, the D lines have wavelength1 = 589.592nm and wavelength2 = 588.995nm

We can find the resolvance or the resolving power required for the doublet to be resolved.

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For N lines of the diffraction grating, we can write (without derivation) for the mth order:

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So, in this case, for a required resolvance of about 1000, viewing the second order would need N=500 grating lines to be illuminated – even the coarsest of gratings manages this easily – a grating with 1800 lines per mm is quite common, if rather expensive. The larger N the better the resolution. If third, fourth or greater orders are visible, a coarser hence cheaper grating will do.

Black bodies, Wien, Boltzmann – very briefly

Screen Shot 9.pngA black body is an idealised body which absorbs and emits all radiation incident upon it. For example, this piece of iron which glows orange-red when heated is an approximate black body. The colour tells us the temperature of the metal. If we continue heating the metal, eventually it will glow orange, then yellow then white. By then, the metal would have melted and boiled off. Theoretically, if we kept on heating it, it’d glow blue-white, eventually emitting UV and even X-rays. Similarly, a star is an approximate black body radiator.  For hotter stars, the maximum wavelength emitted shifts to shorter wavelengths, as shown by the graph.

The black body radiation curves for different temperatures peak at a wavelengths inversely proportional to the temperature.

The plot is valid for determining the temperature of any object which is considerably hotter than its surroundings. Wien’s Displacement Law relating wavelength to temperature is shown on the diagram and the displacement constant shown.

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Our Sun has an external T of about 5300K – the intensity peak is at a wavelength of 550nm.

The thermal energy radiated by a hot body per unit area per second  (power/area) is proportional to the fourth power of temperature – Stefan-Boltzmann’s Law

Screen Shot 5.pngSo, for our Sun, at a temperature of 5300K, every square metre of the surface radiates almost 45MW of power. An object which absorbs all of the energy which falls on it is an ideal absorber or blackbody. For such a body e = 1, where e is the emissivity. Most emitters aren’t black bodies, however, so we can amend Stefan-Boltzmann thus:Screen Shot 8.pngwhere e is a number between 0 and 1. e is zero for a shiny mirror (absorption =0) and 1 for a black body.

By contrast, Albedo (in Latin ‘whiteness’) is the fraction or percentage of incident solar infrared energy  reflected from the Earth back into space and is a measure of how reflective the earth’s surface is. Ice, especially with snow on top of it, has a high albedo – up to 90%: most sunlight hitting the surface bounces back into space, whereas the albedo of a summer forest is only about 0.1 or 10%. Recently, a chunk of ice the size of Scotland fell off the Larsen C ice shelf in Antarctica, reducing its size by 12%. Ultimately, it will either melt or break up. Given that sea water has an albedo of only 6%, the reader might like to speculate about the effect this might have on global warming.

Rayleigh’s Criterion.

Here’s an exercise for a physics class. At eye level, make two tiny dots on a whiteboard as close together as possible. Blue, green or red dots work well – try to make them both small and the same size and separation. Then walk backwards looking at the dots with one eye. Eventually the two dots cannot be resolved as separate.

When the light from either one of the dots reaches our pupil, it will be diffracted through a circular aperture and a diffraction pattern is formed on our retina. When light from both dots reaches our eye, the diffraction patterns overlap. BTW, the red one blurs closer than the green (or blue) one. Any idea why?

As a reminder, you will have seen single slit diffraction with a laser, the light passing through a very narrow slit and displayed on a distant screen. The angle in the diagram below is exaggerated for clarity. Notice the central bright maximum is twice as wide as the other secondary maxima on either side of it.

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Each tiny element down the length of the slit ( width a) behaves like a point source which can be thought of as producing a circular ripple, like on a pond. These superpose at the screen. When the path difference between contributions at the top and bottom of the slit is one wavelength, (m=1) each contribution has a partner halfway down the slit which has a path difference of half a wavelength. So, every point source has a partner exactly out of phase. At the screen, all these contributions superpose and we get a dark first minimum. So, we see the familiar pattern of a wide central bright maximum and minima on each side, fainter maxima, minima and so on.

Just while we’re here – as a is decreased, pattern smears out (y increased). Narrower slit means broader diffraction pattern in other words.

If we decrease the wavelength (use blue light), y decreases.

Screen Shot.pngLord Rayleigh ( who told us why the sky is blue and discovered argon) gave us the accepted standard for the measurement of angular resolution. Rayleigh’s criterion is the generally accepted criterion for the minimum resolvable detail – the imaging process is said to be diffraction-limited when the first diffraction minimum of the image of one source point coincides with the maximum of another. This is the definition an examiner might want to see. This image of two circular apertures shows what it means; the middle picture shows two images which are JUST resolved.

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More rigorously, see the middle diagram below. First minimum of one diffraction is exactly underneath central maximum of the other.Screen Shot 1.png

In exams, they sometimes ask you to either draw this or calculate it. Clearly, it’s wavelength-dependent and also dependent on the width of the slit or aperture diameter (a).

Calculation:Screen Shot 8.png

As an example, how far away from two point sources of green paint of wavelength 400nm separated by a distance of 2 mm would you have to stand so they could no longer be resolved as separate?

Solution:

For a circular aperture (our own pupil), we have to invoke the factor 1.22

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If the paint were blue we could walk further away and still resolve them, theoretically.

NB, in reality – this is an upper limit for people with perfect vision – most people can’t do as well as this. Most people would only be able to resolve the dots as separate at about 4m, which makes this a good little exercise for a class.

 

 

Newton’s Laws of Motion

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Fun fact. Newton laughed only once in his life, when somebody asked him what was the point of studying Euclid.

FIRST: ” A body continues in a state of rest or motion at constant speed in a straight line unless acted upon by an unbalanced external force.”

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SECOND: “The applied force is equal to the rate of change of momentum of the body.” . A rather more modern interpretation is here. If cliff-diving appeals to you, watch this video. As long as you don’t scare easily…Screen Shot 1.png

the conveyor belt problem – if we are to keep a conveyor belt moving at a steady speed – for example in a coal mine where mass is being added to it all the time, we require a force to be applied to the conveyor belt.

THIRD: “For every action, there is an equal and opposite reaction”  

Think about a jet propulsion system. Thrust is a mechanical force which is generated through the reaction of accelerating a mass of gas, as explained by Newton 3. A gas or working fluid is accelerated to the rear and the engine and aircraft are accelerated in the opposite direction.

The force on the working fluid is equal and opposite to the force on the engine and aircraft.

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Look here for a very easy walkthrough of all of Newton’s Laws. This is a particularly good treatment so you can work through the videos yourselves.

Doppler Effect

Imagine a Formula 1 car approaching the stands at 60m/s. The frequency of sound made by the engine as heard by a stationary observer in the stands is higher than the actual frequency as heard by the driver. The sound is squashed up – or better, the apparent wavelength is decreased and the apparent frequency increased.

car approaching observer
car receding from observer

 

As the car recedes away from the stands, exactly the reverse happens. the observer waves goodbye to the red line. Think of EEEEYOWWWW as the car approaches then recedes.

For a stationary observer and a moving source, we can write:

These will mostly do – but IB requires us to use these as well:-A quick calculation shows how the first equation works. Let the car be moving towards us in the stands at a speed us of 60m/s and emitting a frequency f of 800Hz.

Speed of sound in air is 340m/s. We can find the frequency f’ as heard by the stationary observer. Common sense tells us whether we add or subtract the velocities – in this case, we subtract and hear a higher frequency as it approaches us. (the EEEE bit)

 

As it recedes, we add, thus: (the YOWWW bit)

Police speed detectors bounce microwave radiation (about 10GHz) off a moving vehicle and detect the reflected waves. Because the car is moving towards the police observer, these waves are shifted in frequency by the Doppler effect and the difference in frequency between the transmitted and reflected waves provides a measure of the vehicle’s speed. Of course it works just as well for recession speeds as well.

Two Doppler shifts because of the reflection from a moving target. c is of course the speed of light

By observing distant galaxies, Edwin Hubble concluded that distance and recession speed were proportional – so galaxies further away are receding faster than closer galaxies. We know this because the atomic fingerprint or spectrum of atomic hydrogen or helium is shifted to the red (long wavelength) end of the visible spectrum. The degree of redshift can be used to find out how far away a galaxy is.

This absorption spectrum shot (idealised) shows what the spectrum of atomic hydrogen might look like from several distant objects like galaxies. The further away, the greater the redshift. Redshifts of up to 0.95c have been observed – the light having taken almost the lifetime of the Universe to reach us.

 

 

Finally, a medical use. Doppler blood flow is a technique whereby ultrasound waves (f about 800Hz) emitted from a piezoelectric transducer (transmitter/receiver) are reflected off red blood cells in an artery or vein as they are moving towards the stationary detector. The more occluded or blocked the artery is (think about a fluid in a pipe) the faster the cells are moving. It can also be used to find blood clots in deep veins – DVT – deep vein thrombosis – can be fatal.

The detector and the moving cells are at an angle hence the cosine term and, like the police car, the factor 2 accounts for the reflection from a moving source.