Celestial Sphere

Mike Hewson
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In what sense is 'astronomy'

In what sense is 'astronomy' a science?

Short Answer : There is evidently only one universe. We are inside it. You can't rewind and play it again. Any consistent description must take account of these features. To avoid touching any philosophical issues then stop reading here !! :-) :-)

Long Answer : My reading seems to indicate that over known history it has varied widely amongst cultures and times as to our relationship with the sky overhead. Meaning that there have been many viewpoints taken to the idea of whether we are part of, or separate from, the things we see in the sky. I don't mean clouds and birds flying around but the Moon, the Sun, and the Stars ( including Planets ). We have always sensed them to be well away from the surface of the Earth - of whatever overall shape - that we live upon. Certainly one aspect that has reinforced them as real and in some way related to us is persistence : my ( distant ) ancestors have given descriptions which aren't radically different to what I may see myself. I know this all seems 'self evident' but it was not always the way. I'd credit the Eastern and Arabic astronomers first, followed by the Europeans, for whom the penny dropped. Newton for instance made a huge leap by stating that the force which binds us to the Earth's surface ( including that apocryphal apple ) is one and the same that binds the Moon to the Earth, and by extension onwards and outwards. Galileo is oft quoted as ( risking a fair bit of trouble with the church by ) discovering the centrally bound behaviour of the Jovian moons. Copernicus wasn't the first to create a Sun centric system, and indeed had more epicycles in his model than Ptolemy had in his! Dear Kepler replaced these unreliable epicycles ( circles with centres moving on the circumferences of other circles ) with much smoother and eventually simpler and more accurate ellipses. Newton again was masterful by explaining tides here on the Earth - yes, you could go to the ocean shore and see for yourself - in due relationship to Moon phases and the position of the Sun as well. So we are part of the Universe, not separate or outside of it and we are well influenced too. Try a spot of jet lag, say, in order to convince yourself that your brain can indeed be influenced by the Sun !! :-)

Now despite some ( quantum mechanical ) pleas to plurality we have yet to fully describe the universe in anything other than the singular. That is the physical laws as seem to be obeyed are quite inclusive and consistent with us only having the one connected entity. So unless you go for solipsism* you're out of choices for which Universe you want to be into today.

Now the more significant bugbear is that we can't do repeatable experiments on a cosmic scale. This is not just lack of influence, or energy, or choice per se. It's more the nature of time - whatever that is - to proceed regardless. To be circular in thinking you could pop that into some definition of time. However you sort that out ( the 'onwardness' of time ) you still have the pragmatic point that information we receive is always dated and generally doesn't recur. Not exactly anyway. Contrast this with that of a 'reproducible experiment'. This is the ideal that with respect to parameters of interest one can replay certain physical scenarios and have consistent explanations that reconcile from one example to another. Of course with each run of some experiment there will be, say, atoms in the Andromeda Nebula that had influenced the first run in a way differing from said influence on the second run.

And here lies a risk of progressive recursion in our thinking, probably insanity perhaps, and certainly no gain in understanding. So we have to accept a threshold of detail that we will not go below, or like some really long book risk never finishing the reading of. That's rather like asking how many numbers are between zero and one? That depends. How many would you like? What sort? Integers? Rationals ? Irrationals ? Reals ? Complex ones ( yes, I can go along the real axis in the Argand plane between [0, 0] and [1, 0] )?

So what's the upshot then ? Picture William Herschel in his backyard sitting on his telescope frame, at night with clear skies, looking down the optics and watching the universe rotate on by. He grasped - and was probably the first to announce this so clearly - that what we see 'now' is but a tiny sliver of time out of a very long interval indeed. For all our modern talk of the evolution of, say, binary star systems we have never followed even one example from go to whoa. We do much the same as Herschel did which is to survey and for a given 'type' of object. Of course such classification schemes may vary, indeed selecting the 'right' scheme is more or less a key step in understanding.

A good example from here at E@H is pulsars. Get data from as many pulsars as you can lay you hands on, naturally having first defined what a 'pulsar' is, and then look at what regularities or patterns may emerge. Alot of pulsar-ish thinking pivots on understanding the relationship b/w frequency and frequency derivative. You'll find that typically ( yes, there are always exceptional cases which require special understandings ) the faster rotating pulsars are usually the ones that are 'spinning down' at the greater rates. So we have identified a general characteristic that is more or less independent of other data about pulsars. [ Actually I have jumped the gun a bit here by assuming that pulsars are spinning objects at all, but go with flow here, eh? ] What to make of that? Now we come to the area of modelling. And all models are a work in progress, meaning that new data or a better look at old data will force a revision of said model**. So you're model now has to account for that relationship.

And so we iterate. Back and forth. Back and forth. Between model and observation. A new pulsar? OK. How does this one fit in? It looks to be 'like' other pulsars that we have classified as 'disrupted recycled'. So what's one of those when it's at home? Well it's ......

And so on. Herschel put it well when comparing objects in the sky to plants in a garden, but we only see specific stages of the ( presumed ) evolution within classes of objects. An example of a 'protostar', a different object being on the 'main sequence', yet another looking like a 'dying' star etc. We have never seen one specific object evolve. We assume an evolution of objects by cognitively connecting current examples in order to then place them upon some pathway of change. Next up :

Hereabouts

Cheers, Mike.

* The position that only one's own mind is sure to exist, the rest being a clever ploy. A real cutdown version of 'I think therefore I am'. So you wake up each morning with only the memory of yesterday(s) in your head, but then again that memory may be a fake too. So yesterday(s) never happened etc. See The Matrix for an extreme take on this.

** Oddly, in my opinion, science is often criticised for being both dogmatic on the one hand and also for being too flexible on the other. This is a key misunderstanding of many that look into science practice and invest their own preconceptions. The problem here isn't that science is 'broken' but that others can't respect domain knowledge. The example I like to give is a chap called Craig Lowndes, a racing car driver DownUnda. He'd be just real keen to tear around the track in a thumping V8 in front of everyone else. Now, who drives Craig Lowndes' car in a V8 SuperCar round? Well, it's Craig Lowndes. That's because he's a racing car driver. So none of his mechanics, say, will be competing even though they may have excellent knowledge of his car, they won't have the knowledge/experience to NOT pile into the fence at the first corner encountered. If you want you're own personal definition of science practices then go out and get on with it, put your stamp on it, publish and defend the theses. Poop or get off the pot! :-)

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

mikey
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RE: RE: Some people say

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Some people say that we are less then 400 thousand years from 'seeing' the big bang happen, I don't know if it's true, but it is interesting to me.

I think that refers to the cosmic microwave background which is an imprint of the radiation produced with the Big Bang but representing a time some ~300K years after that. Around that time most photons generally 'decoupled' or largely ceased interacting with matter as the temperature had subsided to the point where electrons and protons could form low mass atoms like hydrogen and helium ie. not ionised. That 'gas' of non-interacting photons has likewise 'cooled' with the further expansion of the universe since and in fact is one of the main points of evidence that there was such a thing as a Big Bang at all.

Cheers, Mike.

That is probably what they meant that I heard, they just did not use those terms. My hope is that one day, in MY lifetime, they can figure out where all that 'stuff' came from to go BOOM! One of the Laws says 'matter cannot be destroyed' but it can be changed from type to type solid, liquid, gas, plasma, whatever. But to get ALL that we can 'see' now all that stuff had to be here before, so what form was it in? Dark Matter doesn't seem a likely scenario but the research is on going and who knows what they might find out today, tomorrow or even next week or in the next decade. But if the Laws are correct, and there is currently no reason not to believe they aren't, then all the 'stuff' was here and then went BOOM, and we have what we have today. LOT of questions, but not too many answers just yet!

When I was younger I 'decided' that Universes were formed by black holes, with the matter in the 'new' Universe being formed at the bottom of the black hole by all the stuff the black hole 'devoured'. Hawking now says that was stupid of me and that all the 'info/matter/whatever' is stored at the entrance around the opening and nothingness is what is inside. Since Hawking, and a whole lot of other people, are a whole lot smarter then I can ever wish to be, then I am like most people, still wondering!

One thing that does call into some question is the idea of 'premonitions', are they just fake or are they just real lucky guesses? Or is there really a way for some people, or even all of us, to 'see' the future before it happens? If they are real then how do they coincide with the future being the future and it not happening yet? How can anyone 'see' what hasn't happened yet if it hasn't happened yet? LOTS of questions but not many real answers just yet, I like looking for Pulsars, they are easier and more tangible!!

Mike Hewson
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RE: ... that all the

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... that all the 'info/matter/whatever' is stored at the entrance around the opening and nothingness is what is inside.


The basic idea here is that quantum mechanics comes into play and allows for the black hole to radiate particles from the horizon, hence a mass/energy loss. Eventually the hole will 'evaporate' at a rate which is inversely proportional to mass. So tiny holes go pop with a quick flash of radiation, but larger ones ever so slowly. Applying thermodynamic rules one can view the hole as lying within the heat bath of the cosmic microwave background ( CMB ) radiation. It's a cool bath actually. In any case the hole has an effective temperature going like the inverse of it's mass ie. big is cool, small is hot. Weird huh ? Currently the numbers are such that stellar mass plus black holes are cooler than the CMB, so it'll be a while before the CMB glow fades such that they will be warmer than their surroundings. Then they'll lose energy and begin to dissipate by an excess of radiated loss over absorbed gain.

Now the information/entropy content of the hole is not going to be visible to the rest of the universe until it begins significantly emitting. Hawking's interpretation is thus that the info is stored until a way later time : some humungous figure like 10 to the sixtieth power years ( we're only around 10 to the power of nine years currently ). Could be more. Long time.

You could heuristically explain this as saying that since, from the perspective of a long distance from the hole, the horizon has 'frozen' features - time dilation is so severe that clocks ( thus frequencies too ) go to a zero rate. So when the horizon is shrunk with hole evaporation, so nearby spacetime returns to flat, horizon clocks come back up to speed and the information 'wakes up' and returns to the remainder of the universe. So in a weird sense a black hole operates like a time machine : you get to go to a very distant future time with only a short elapsed proper time. Well I wouldn't personally survive intact but that information which is encoded in my body structure would! :-) :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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RE: .... that information

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.... that information which is encoded in my body structure would! :-) :-)


Well, that information thus merged with all other captured stuff .....

Hereabouts

The metre. Our current basic scientific unit of length in the MKS system ( Metre/Kilogram/Seconds ). Some use CGS ( Centimeters/Grams/Seconds ). Originally defined as the length of a pendulum with a given period, then one ten millionth of an Earth quadrant i.e. pole to equator - though never measured exactly as such - then a series of metal bars of particular production and situation. Now defined as that distance traveled by light in a certain time*, though in practice remains a certain count of wavelengths of specific emitted standard light sources. Whatever. Take it as read that we know what a metre is and can when needed line other things up against that to make statements like "the distance from Melbourne to Frankfurt is ... " accurate/sensible/consistent.

So we can hence forth measure any and all things here on Earth - the word 'geometry' means exactly that - and even these days have very accurate measurements by remote calibration. This is otherwise known as the Global Positioning System which relies upon highly specified coordinate entries, base points if you like, combined with time of flight measurement of light signals in the form of radio waves. Think of the speaking clock ( an 'old' service but still operating ) per phone emitting the likes of 'at the third stroke it will be ten forty five am and fifty seconds .... beep .... beep .... beep'. Suppose I heard that very same message but by several distinct paths, and the 'origin' of each path ( satellite in this case ) encoding it's position when the time signal was emitted, then you have ( with suitable General Relativistic corrections for the variation in the Earth's gravitational field hereabouts ) sufficient information to deduce/calculate where you are. For instance I could hear that third beep earlier from one satellite than from another. Of course in practice it isn't quite like that but that's the simplest explanation I can think of without going on about spacetime events and reference frames et al. Naturally there is all manner of issues as regards accuracy for any given measurement instance eg. how many distinct satellite contributed signals were enjoined.

An obvious question here is : how does the satellite 'know' where it is? It cannot inform me with my GPS unit as to my whereabouts if it doesn't itself have a well specified position from moment to moment. You can sense the inevitable answer : it must check with base camp(s). They do and indeed crosscheck against each others' positions too. Again complex in the detail.

The deeper point here is that we are at the first rungs of a ladder. As one steps to another scale one still refers back to - calibrates against - a prior scale. You can do this in either direction of course ie. go up to higher scales ( longer lengths ) or to lower scales ( smaller lengths ). In fact it is not only distance that is subject to that ie. ditto in principle for other quantities. We are human and thus to make sense of our investigations, for our brains, the information has to return to our scale - which notably is of the order of a few metres, a few kilograms and a few seconds - to be absorbed and pondered upon**. So the hard drive which is storing this post is going to be doing that according to some agreed pro-forma/encoding, but we are not personally inspecting the small magnetic domains on the platter that serve to represent such information. We have constructed circuitry to do that, and said circuitry was manufactured using other calibrated instruments to enable that. Et cetera ... in a chain of mutually calibrated procedures and devices.

Einstein was exceptional in that he really took to task the explanation/practice of measurement. This is what is frequently found annoying by newcomers to relativity discussions. Why do they go on about rulers and clocks and observers ? Isn't that obvious ? What's the issue here ? Well you can't actually consistently discuss space without mentioning time, you'll hit it sooner or later, so we ought not go onwards to cosmic scales without tackling :

The Speed Of Light

Cheers, Mike.

*Interesting that we've come full circle to a time based definition of length, which of course regresses to what is such a time interval anyway? Note that for small amplitudes all pendulums of the same length will have the same period, this being first ( well recorded/apocryphal ) noted by Galileo sitting in church watching the candelabras suspended from the ceiling. It won't depend on the mass attached to the end of the pendulum but does depend on the local gravity field ie. acceleration due to gravity at the point of operation of the pendulum. Interestingly the current length definition has a caveat regarding general relativity essentially acknowledging that yes, clocks in different parts of a gravitational field will run at different rates.

**Alas some navel gazers have taken this to support their contention that science is 'socially defined' and cannot make absolute comments about anything. See my earlier comments about Craig Lowndes, but apart from that it is simply the case that scientifically slanted slugs, or questioning quokkas, or intelligent motivated asteroids are all going to fail to reach the speed of light if they try to travel that way. My money will be on the photons arriving first. Science may have various social/language encodings - somewhat like the choice of ASCII for computer character data - but an independent universe ( and the same one in each case ) does exist regardless. You can always regress to measurement. A black hole will swallow up all comers. Of course I appreciate that how people interact and behave in common endeavours - like scientific studies - has lots of cultural spins and group thinks. But let's not confuse that with falling to the ground when you trip.

( edit ) "You can always regress to measurement". This probably is the line in the sand with respect other disciplines of study that claim the 'science' label. Bringing back information to human scale does not lock us to a human centric view. In fact technology has allowed many surrogate viewpoints to arise eg. an image of the Earth from the Moon, or writing little messages with an electron microscope. You can think outside your own head.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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The Speed Of Light How

The Speed Of Light

How would you know if a clock was regular or not ?

I mean you can start with a standard metre ruler, thus when lined up against another ruler you want call 'a metre long' and note that the ends coincide. Do that for as many rulers as you please and you can lay them out end to end to get some number in a row, say X of them. Then you get a total of X metres from the start of the first one to the end of the last one*. Fine. Subdivide the metre by say producing smaller equal length ones, and if say Y of those end to end gets a metre then you could say that each was 1/Y of a metre. Mutatis mutandis. Note that we assume a 'regularity' in distance by doing this.

But time. How to measure that? Einstein said that "time is that which is measured by clocks" and thus neatly ducking** - like everyone else - what time 'really means'. It was a pragmatic decision, and to be useful you need an agreed upon and workable standard. You want something, some method, some gadget, that can be used as a time ruler. His breakthrough was to claim, as a fundamental axiom not to be otherwise explained but simply assumed, that the speed of light is constant ( in a vacuum when it's not bouncing off things, irrespective of direction, regardless of speeds of observer or emitter ). That way you can take your metre ruler, run a photon from one end to the other and claim that the interval in time for that to occur is a metre of time.

What? Hah! Well you've be doing that anyway, but with a different length - 300,000 kilometres - and calling that 'one second'. Up until now you've had 'one second' as having meaning with regard to the Earth's rotation. Assume that the rotation rate is constant, take one 'day' as a single rotation per the reference of directions to distant stars***, then subdivide by 24 to get an 'hour'. Then by 60 to get a 'minute'. Then by 60 again to get 'one second'. If we translate our metre of time to this standard then our metre ruler is transited in about 3 billionths of a second. See ? ;-) :-)

So how do we take our metre of time and turn it into a clock? Easy. Keep bouncing the photon back and forth with mirrors at the metre rod ends. A round trip is two metres of time, two round trips is four metres of time, three round trips is six metres of time .... 1.5 * 10^8 round trips will count out one second ( close enough ).

Now in practice we're not really going to do that. At a minimum to measure a single photon ( e.g. has it hit a mirror ? ) then we must absorb it or otherwise interfere with it and so our single photon clock gets stopped quite early into it's operation. Thus in practice we use large bunches, measure some and let the rest reflect. Use a laser maybe so that all photons are the same ( within QM limits anyway ). You get the idea.

In summary : take a standard length interval ( however defined ), assume light has a constant speed of propagation, use that to define time intervals. You can flip this over if you like : take a standard time interval ( however defined ), assume light has a constant speed of propagation, use that to define distance intervals.

You may feel uneasy about this. Fair enough. It is not particularly intuitive. It simultaneously pokes, prods and challenges your basic ideas of distance and time, and also fails to state what they 'really' are. If you think this is a bit intellectually cheeky then you are right. It is audacious. Thank Mr A. Einstein for that. However it is consistent, it is practical, and most importantly when carried out the predictions are spot on.

So the final answer to the original question is : 'regular' clocks are those constructed using light traversals across fixed distance rulers. Because we define the speed of light as constant then they must be so.

Now one last deep plunge on this : it's not really about the rulers and the clocks. Say if I'm an electron sitting here and you're a proton sitting over there. You wiggle. I am not going to know of that 'immediately'. In fact 'immediate' now means for me : when I feel the effect of your wiggle. Photons are being passed as force mediators. The reception of a photon is the 'feel' of the force. If there is another charged particle between us then that particle feels the force before I do. Always before I do IF it is closer to you, the proton. This can be denoted as a 'transitive' property. In an arrangement of three distinct particles we could be passing photons around like this, who gets what and when, but always you will find it to be the case that a given triangle side is no longer than the other two combined. Known in maths as the Triangle Inequality. It's the physics definition of 'shortest distance'. It's the path of a photon when uninterrupted. The 'straight' line. Even when you throw gravity into the mix. Yup, the universe really does work that way. Next up ( it gets easier ) :

The Solar System

Cheers, Mike.

*Ah, the joy of Cuisenaire rods! Where did they go? Are they still in use? I can and did play all day with them. :-)

**If you think carefully we also duck any definition of what distance 'really means' too. We just don't think that we do! I reckon that's because humans are intensely visual creatures, compared to say a wombat which has outstanding smell but wouldn't recognise itself in a mirror. We thus have an inbuilt comfortable spatial sense that we don't question, because we 'know'. Take the hummingbird for instance. Does it have a time sense, finely honed and far more exact than us ? My guess is that it would have to have something to that effect.

***Yes, pedants will note some complications here : sidereal, solar, averages et al. Important but too arcane for here.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

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The Solar System These

The Solar System

These days we have a pretty good definition for this, but that wasn't always so. For a start it is only in recent centuries that the Sun was given central prominence. Now it's true that as per The Relativities we can locate a viewpoint anywhere in space and time, and still be able to translate findings to some other viewpoint. So you can have a Sun-centric model, an Earth-centric model, a Moon-centric model, a hanging-upside-in-a-tree-in-my-backyard-centric model etc .... each varying in the origin, arrangement and sense of some frame of description. But some viewpoints are simpler, or seem so, than others.

So if you look at the magnitude of influences, then here you find that the Sun is absolutely way more massive than anything else in nearby space. That matters. When you have things orbiting other things and then introduce a much smaller 'test' mass then things become interesting. Lagrange sorted this out and produced a set of points in two body systems - that could be calculated given certain quantities - where 'dynamic equilibria' apply. There's one Sunwards of Earth ( L1 ) where we have SoHo the solar flare satellite, there's one away from Earth ( L2, anti-Sunwards ) where the Planck satellite sits. The original Lisa plans were for the array to be in the vicinity of the lagging or L5 point. You get the idea.

I can pretty safely say that for most positions in the solar system the Sun has the dominant effect on my movements. My point is that when we say 'the Sun is the centre of the solar system' it's not just a casual choice of coordinates or language emphasis. It really does affect things like that. I don't change the Sun's mass when I move somewhere else and look in a different way at things.

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For many social reasons over history this was thought of as Bad News. It displaced humans and their constructs from the focus of things. Giordano Bruno's life ended badly not only for saying that, but also for wandering too close to a Pope. Gallileo merely got house arrest. Copernicus said the same thing but from a safer distance.


We still need an absolute scale though ie. how far actually is it from the Earth to the Sun ? We have the Keplerian revelations about orbit sizes around a common central object. For a given central mass, but much larger than any other in the system, you can express this as a type of power law between the time it takes to complete an orbit versus the extent of the orbit. Radius cubed goes like period squared. So you'll find that Jupiter is about five times further out than the Earth and takes about 12 times longer to orbit ( ie. 5 cubed = 125 ~ 144 = 12 squared ). But that doesn't nail the absolute scale. We know what 12 years is but still lack a good Earth to Sun distance estimate. So what you need is one good distance measurement and the timing of all the other planetary orbits will give the absolute distances for the whole setup.

It wasn't until the 1960's that this was done really well. Take the Goldstone radio telescope facility and use it as a radar! I think they first pinged Venus thus giving the difference b/w Earth's and Venus's orbital sizes. Hit the 'send' button and start your stopwatch, listen for a return echo, and when heard stop the stopwatch. Multiply that time interval by light speed and there you have it. Well, you are going to do that a few times, while accounting for where Venus is in it's orbit at the times of measurement with respect to where the Earth is in it's orbit. In any case you now have a separate equation to fold in with the Keplerian one and both Venus's and Earth's orbital dimensions emerge. Thus the entire system. Naturally you have to go beyond Kepler for a full description. Jupiter is a big planet and is going to be the next dominant effect after the Sun. I think JPL keeps track of all these planetary motions.

Ole Roemer had a neat idea. With a good telescope one can look at the moons of Jupiter. From time to time a given moon would go behind Jupiter - be eclipsed that is - from our viewpoint. That event is a specific moment that we can time and record, meaning the time here on Earth. He found that a specific moon would have some average time interval between these eclipses - it's orbital period around Jupiter evidently* - but it would sometimes arrive early at the eclipse event and at other times arrive late. What's going on?

This turns out to be a rather slow-motion example of the Doppler shift. If you think of a given moon's eclipse event as generating a 'click' on each occasion, then speed things up, you will get a rhythmic change interpreted as like the Dopplering of vehicles passing you by. So if I'm sitting around up at Jupiter Station I'll hear a constant note, but down at Earth the note goes up and down in pitch - but the average will be the same as for Jupiter Station. Sometimes the whole Jupiter system is coming towards us ( rising pitch ), sometimes it's going away ( falling pitch ). Depending on what you think you know best : you can estimate** say, the speed of light ( which is what Roemer and colleagues did ), or the Earth's orbital diameter.

[ You'll revisit this Doppler theme many times in astronomy, it's a really useful tool. ]

So we get the diameter of the Earth's orbit. This is a key rung in the distance ladder because we can now reach out of the solar system and begin to determine distances to things which exhibit :

Parallax, or Why Binocular Vision is Good

Cheers, Mike.

* You can redo all the Kepler stuff for Jupiter and it's moons too. A smaller scale version of solar system behaviours.

** What they were actually intending to do was characterise the orbit of Io so well that it could be used as a 'clock' in the sky. If you could see it at sea you could determine a time relative to Greenwich ie. deduce your longitude. That didn't work out, but this is a neat example of how asking questions for one reason can unexpectedly/inadvertently answer others.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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Parallax, or Why Binocular

Parallax, or Why Binocular Vision is Good

Triangulation is the other name in general use for this type of trick, because we are going to construct a triangle and do some some trigonometry! :-):-)

The purpose of the exercise is to determine the distance to some object out in space, well out of the solar system. But not too far away.

So we reckon we know the diameter of the Earth's orbit or have it's ( essentially ) elliptical shape suitably understood, so that we can determine the distance b/w any two points in the orbit. Select two points in the orbit, and call the line between those two the baseline. It is one side of a triangle that we will now build.

The third point of the triangle is out there in space at the star/whatever that we wish to determine the distance to. So the other two non-baseline sides each extend from the two selected orbital points to the object in question. Call those two sides the long sides. They are quite long in fact, and way longer than the baseline.

Now we're not going to be walking up and back to that star are we? No. We ain't. The whole point of parallax is to avoid that because we can't do that. But what we can do is form two angles, one b/w each long side and the baseline. That we achieve by measurement of where the star appears to be in the sky at those two orbital points as above. If you know two angles in a triangle you can calculate the third one, as all three must add up to 180 degrees. Where is that third angle ?

In your mind go up to the star and look back towards the solar system. You'll have a long triangle side to your left side and another to your right side. Each of those lines is ending at one of the chosen orbital points. The third angle ( or 'parallax angle', or just plain 'parallax' ) sits b/w the two long sides. ( Actually the parallax angle is officially half that third angle as I have just defined ). It's likely to be a small angle - hence the triangle is very pointy there - because the stars are ever so far away. For all practical purposes the two long sides are of equal length, so let's call that common length the distance to the object. This makes our triangle of the isosceles type, and we now have enough to calculate those long side lengths. I'll skip the math.

OK, so that's the geometry for sure but let's think a bit deeper. We actually do this everyday without realising. It's called distance or depth perception and is available to those of us who have two good eyes ( or alternatively if only one eye is good then you can move your head side to side ). The baseline is the distance between the two eyeballs. In effect mathematics is performed as the distance judgment then translates to movements, manipulations etc. Binocular vision.

This works best when the distance to be judged is directly in front of you and not so well in the peripheral vision. Likewise for star positions it's best to have the star 'square on' to the baseline, and that way the long triangle sides will actually be equal too. The accuracy is better for a longer baseline, so if we choose our orbital positions close as can be to six months apart the baseline is the full width of the Earth's orbit.

Alternatively one can measure as accurately as possible the position of a star against the background of more distant objects, as often as you can throughout a year, say. That's the same gig as looking at your finger with alternate eyes : it apparently moves to the left and right in front of the background objects. The star will appear to oscillate from side to side and the maximum excursion is your parallax.

[ To be more precise the sequence of positions will trace out an ellipse upon the sky. The ellipse will collapse to a line if the star is in the ecliptic plane. It will 'fatten' to a circular if the star is perpendicular to that plane. Again, let's skip the trig. ]

Now an important astronomical unit. The parsec. It's a distance. In the MKS system it is about 3 x 10^16 metres, light takes about 3.25 years to travel that far. The Alpha Centauri system is around 1.3 parsecs away. A very long way for us as humans. A thousand of them is a kiloparsec, a million a megaparsec. The diameter of our galaxy is about 50,000 parsecs depending on who you ask.

The word parsec means 'the distance such that the star's parallax angle is one arc second'. We have 360 degrees in a full circle, each of those degrees can be divided into 60 equal angles each called an arc minute, and each of those are divided again 60 ways to arc seconds. So 60 * 60 = 3600 arc seconds make up one single degree of angle ( and 1,296,000 arc seconds get's you a full circle of arc ). So that's really small, right ? No point standing out looking at Alpha Centauri night after to night in the hope of noticing that with the naked eye! :-)

Now there is a distance beyond which we cannot use parallax simply because we can't measure really small angles well enough. I believe that's about 100 parsecs with the basic technique discussed here ( telescope on Earth's surface ). I've just looked up the Hipparcos satellite and they do around five times better. In another thread I've looked at the LSST which is a whole other level on that. Next up :

What's beyond parallax ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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What's beyond parallax ? :

What's beyond parallax ? : Part A

Recall the distance ladder concept. Where have we come from and where are we now ? We started with a metre ruler, however defined, and then gauged the size of the Earth and things upon it ( terrestrial mapping ), then reached out to the planets ( ultimately via radar ) to then scale the solar system. Then knowing the dimensions of the Earth's orbit we could do some trigonometry on star positions ( parallax ) to find some distances to our galactic neighbourhood. But that had limits due to difficulty with measuring really small angles.

Enter Henrietta Leavitt and 'standard candles'. To understand this we first need to think of how light intensity decreases with distance. You are familiar with this in everyday life as the obvious dimming of objects as you get further away from them. And of course the converse, getting brighter as you get in closer.

In the first instance think of a lone light source in a vacuum ( ie. no absorption by intervening material ) : take up some position at a distance ( R ) from that source, and consider a sphere of radius R centred on the source. So we are sitting on the surface of said sphere much like we are now sitting ( more or less ) on the surface of the Earth. Now if the source is truly alone then there are no other sources within the volume of the sphere, and if there is no absorption then all photons generated by the source at the sphere's centre are going to make it to the surface of the sphere at radius R. For discussion think of the source as producing X photons per second of some fixed energy per photon. So by the above if I count all the photons that made it to any point on the sphere then I am going to come up with the same count ( per second ie. X ). This if you like is conservation of energy - during the outward transmission of the photons - because none are destroyed/lost or produced/created.

But let's face it : I am unlikely to achieve that because with the really big spherical surfaces astronomy will give us I won't be able to count them all, because I won't get to sample all across a sphere's surface. Enter the idea of solid angle. Since this is a higher dimensional analogue of angle in radian measure I'll start with explaining that.

Radian measure. Take a circle of some radius R, and as you may know it's circumference = 2*PI*R where PI is that funny number with all the decimal places ( 3.14159 .... ). If I divide the circumference by the radius I then get the pure number 2*PI, which represents a full circle of angle.

Quote:
Why? Because I have just defined it to be so, that's why. :-)


If I only had a half circle then the length of that arc ( part of the circumference ) is PI*R or half that of a full circle, which when divided by the radius R is just PI. For a quarter circle the radian measure is PI/2 = 2*PI/4. If you like radian measure is a measure of the angle enclosed by some arc of a circle, regardless of the size of the circle's radius. So any two semi-circles will have the same angle measure in radians irrespective of their size. This is one of the features of angles that you may not have appreciated : it is scale invariant, meaning the angle in some geometric figure doesn't change simply because you make all lengths larger or smaller.

Quote:
That's a feature of Euclidean space at least, and we wont go to other geometries here .... :-)


Solid angle is the same sort of a deal but up a dimension in definition. Instead of a circle with a circumference I now have a sphere with a surface area. For a given R the sphere's surface area = 4*PI*(R^2), where R^2 = R squared = R * R. Analogously I divide by R^2 now to get 'solid angle'. Thus 4*PI becomes a whole spherical surface of solid angle, a hemisphere 2*PI etc. Again because I define it that way. So all half spheres have the same solid angle, being scale invariant in Euclidean space etc.

Quote:

Note that solid angle and radian measure don't/can't have the same conceptual meaning. By construction they ain't the same type of cow. Radian angle measure is a ratio of two lengths ( arc length to radius ) and solid angle measure a ratio of two areas ( surface area to radius squared ).

Also note that while I can produce a given contiguous ( no gaps ) radian measure in many different ways, each example is able to be overlaid with another by a simple rotation. Contrast with a given amount of contiguous ( no gaps ) solid angle that can be constructed in many different ways, but not necessarily simply rotated one to another ie. for some fixed amount of area of a sphere's surface the 'shape' may vary.


The upshot is that now I have a firm definition of what could be meant by a fraction of a sphere's area. If I could catch all the source photons because I totally surrounded it with detectors then I'd call that '4 PI coverage in solid angle'. If I only had a hemisphere then I have '2 PI coverage', and on down to smaller fractions intercepted by my detectors. Specifically if I hold up a square meter of detector surface here at Earth, and have it facing directly at Alpha Centauri ( AC ), measure the number of photons ( each of some energy ) arriving in one second then I have a thing called 'flux'. If I know how big a sphere would be centred on AC at the distance I am measuring at ( R ) then I can deduce what ( quite small ) fraction of a sphere's surface my detector area is. I could - assuming that the flux was streaming away from AC equally in all directions ( isotropy ) - then deduce the total number of photons per second being produced by AC.

Quote:
Clever punters will note that I have essentially outlined an 'inverse square law of flux' here. If I keep my detector area constant and move twice as far away I will intercept only 1/4 of the whatever fraction of the total spherical area that I intercepted originally. If I move in three times closer I will intercept nine times the flux, if I keep the detector area constant. So inverse square laws are firmly rooted in the geometric concept of surface areas varying with distance, and boy does that say alot about how the Universe works!! See Warped Passages* by Lisa Randall for a great discussion on this very aspect.


Now you need to be careful as flux can be defined differently in different contexts**. Generally it is a 'rate of flow of a property per unit area' so it's a something per unit of area per unit of time. In this thread I have used it as 'photon flux' meaning some number of photons hitting my detector per area of that detector per second of time. Knowing the photon energies I can convert that to energy hitting my detector per area of that detector per time, and so call it an 'energy flux'.

Quote:
Go down to you local cafeteria. Set up a rigid wire frame enclosing a square one metre a side, then erect it suitably. Things can pass through that frame like the goal frame in soccer ( but nothing behind to catch anything ). Now have a food fight. Count say donuts passing through the frame as the seconds go by. You could then talk of a 'donut flux'. There is a subtlety here : you have to track which way the donuts are going and thus state a 'nett donut flux to the right' for instance. For some one second time interval take the number that went through the frame going toward the cafeteria door say, then subtract the number going the other way toward the kitchen. This implies that our detection area has a 'positive' direction associated with it, which may give rise to vectors constructed perpendicular to the area with a certain direction 'sense'. We're not worried about this point with our AC photon flux, but in general you have to at least consider the possibly of nett flows.


There's an extra notion though, which is why I said 'facing directly at Alpha Centauri'. By that I meant 'square on', or in mathematical vector terms a vector perpendicular to that metre of area ( the 'normal' ) is pointing right at AC. Clearly I have to require something like this, because if I align the detector's area 'edge on' to AC ( the normal is thus perpendicular to the line to AC ) then the photons will shoot right on by either side of that surface and none will be recorded. In mathematical terms this is handled by including either the sine or cosine of an angle ( depending on definition ) in order to project some slanted actual detector area onto an equivalent one ( 'en face' to the source ) that truly represents the ability to capture the photons ( or donuts etc ).

In summary : if I have a known source-detector distance, a known detector area, a measured flux at the detector, and the assumption of isotropy THEN I can state the total source flux.

Now you can flip this over : if you know the source flux, the detector area, the measured flux, and assume isotropy THEN I can deduce the distance to source.

Cheers, Mike.

* Pages 42 thru 46

** About 35 years ago I would have paid a decent sum to have all the uses of 'flux' ( plus related terminology ) disambiguated for me. You get this 'transport' concept for free today .... :-)

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
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Addendum Re :

Addendum Re : Parallax

I've just researched and recalled a few other parallax examples, illustrating that this technique is indeed quite ancient. Here's a neat summary of same that I found, the diagram on the last page gives a good general idea of how to track an object against an assumed distant background. The errors weren't in the use of the technique so much as which quantities were better or worse known at the time.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

Mike Hewson
Mike Hewson
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What's beyond parallax ? :

What's beyond parallax ? : Part B

An ancient ( Greek mythic ) king of Aethiopia gave his name to the constellation of Cepheus, in the northern sky. Delta Cephei was known from the 18th century to vary in it's brightness with a regular period of about 5 + 1/2 days. It's about six times the Sun's mass and about 50 times it's diameter. The outer layers are around 2000 times brighter than the Sun and they literally pulsate in and out, giving rise to the variation in luminosity. It has pretty well finished burning hydrogen in the core and is now working it's way into the higher elements.

It's not the only star in the cosmos to behave like this. The overall category/type are called Cepheid variables ( although two subtypes are also defined ). They have the same overall mechanism but do vary individually, especially with regard to the luminosity change and the time to do that. Like all stars there is a battle on the one hand with gravity wanting to compress the object - thus heating the interior and triggering nuclear fusion - against on the other hand said heat causing gas pressure to expand it outwards. You may as well think of Cepheids as not being quite able to decide on a given constant state ( in the sense that the Sun has ) but it oscillates around a mean behaviour where gravity will eventually win when the fuel runs out. The probable truth is a mechanism thought up by Arthur Eddington, a really clever astrophysicist from early last century, that relates to how the transmission of radiation through a gas depends upon it's density, ionisation and other features ( complex ).

In any case we now have a curious, and rather useful, group of objects. If I measure the period of a given Cepheid's fluctuation - and all I need is a telescope and a clock for that - then I can also closely predict the variation in intensity and the intrinsic luminosity. By intrinsic luminosity I don't mean the brightness per se as seen here on Earth, because that will also depend on how far away it is. But suppose I had a group of Cepheids all at the same distance from me, then they would all suffer the decrease in brightness with distance equally ( inverse square law ). So the differing brightness that I measure from one example to another within the group will then reflect the true difference in photons that each star produces. This is also called absolute magnitude, or even bolometric intensity if I account for radiation across all wavelengths and not just the ones visible to humans. It turns out that the brighter Cepheids, in the absolute sense, have the longer periods of fluctuation. For this we can primarily thank Henrietta Leavitt.

She was initially employed at the Harvard College Observatory to measure and record photographic plates for cataloging purposes. Her job title was 'computer'. She studied thousands of stars in the Greater and Lesser Magellanic Clouds, and discovered a subset that varied as per Delta Cephei et al. Under the reasonable assumption that each star in each of the above Clouds were at more or less the same distance ( per cloud ), then she deduced a logarithmic relationship called the 'period-luminosity law'.

Now we've run three bases but have still to get to home base. Nearly there. Although we talk of absolute luminosity we haven't got an absolute distance yet. We haven't connected to the next rung down in our distance ladder. For that we need one or more Cepheids that we know the distance to, but measured by some means independent of brightness, apparent or absolute. So suppose there were Cepheids that were known by parallax, say, AND well characterised in their brightness variability. And we do. Originally this was only for a couple of dozen examples as measured on Earth, but the Hipparcos satellite has rounded up over two hundred.

In summary : if we can identify a given Cepheid ( this will take time ) then we can characterise it's period of fluctuation, then deduce it's absolute luinosity per Leavitt, and hence calculate how far away it would have to be to appear to have the brightness that we actually measure ( inverse square ).

Of course there are a host of details left out of this explanation. One would be extinction, being a loss of light intensity due to absorption by intervening material. The Magellenic Clouds had the advantage of being outside the main body of our galaxy and in a direction not toward the centre ( which obscures ). The two Cepheid subsets have different exact mathematics relating period to luminosity, but within said subsets you can still predict one from the other quite well. I've also ridden roughshod over a few other points of logic .... :-)

Leavitt died of cancer, without being awarded a Nobel Prize many contemporaries thought she deserved. Partly that was because she was dead when the question arose, but also because her supervisor Harlow Shapley ( upon inquiry ) deflected undeserved credit to himself for her work. Next up :

Birds Of A Feather

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

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