Neutrino Mass

I had an interesting chat with a neighbour of mine yesterday regarding the mass of neutrinos. Now the issue of whether neutrinos have mass or not was once contentious but it is now (I think!) generally accepted that they do have mass. The evidence for them having mass is given from the fact that neutrino oscillation has been observed. What is this?
Well, there are three types of charged lepton – electrons, muons and tauons with the last two being unstable and decaying to electrons in a short time when they are created. Each of these has a neutrino pair – so there are three types of neutrino (and each of these particles has an anti-particle also). Now in the sun nuclear reactions occur which spit out neutrinos. Clever astrophysicists can work out how many neutrinos one should expect if you try and detect them (and this is no mean feat!). It turns out that the number detected is less than what is predicted. Hmmm.
The solution lies in some basic quantum mechanics. The neutrino wave-function is a superposition of 3 types of neutrino mass eigenstates. Each of these mass eigenstates is a superposition of the 3 “flavour eigenstates” mentioned above (and we could go in a circle here!). Over time as the neutrino travels from the sun to us, the larger mass eigenstates travel slower than the lighter ones so in a way we get interference between the 3 flavour states. In general then the neutrino is a superposition of the 3 eigenstates. So if the neutrino started as being an electron neutrino (probability of measuring electron neutrino =1) after travelling some distance it would then turn out to be a superposition of the 3 flavours (probability of measuring electron neutrino = 1/3 say). –>. This solves the solar neutrino problem where experiments detected a deficit in the number of neutrinos expected from the sun. These exps were only sensitive to one flavour, so could only detect 1/3 of the neutrinos hitting the Earth.
As this oscillation of flavours is observed we know that neutrinos have mass (massless particles do not change types like this – there is only 1 type of photon for instance!). Now the interesting question is what is there mass? One thing is for sure it is VERY SMALL!! We can constrain this using cosmological obs and theory. The mass of a neutrino is less than 1eV (and that is bloody small!!).
The fact that neutrinos have mass is significant – for instance the standard model of particle physics treats them as massless (uh oh). Also the question of why neutrinos have the masses that they do have is a big one? There is no reason for their masses to be on the scale that they are … or so i thought. This was the topic of my neighbourly chat. Apparently in extra-dimensional QFT models of particle physics the mass scale of the neutrino arises naturally. Ok cool – what is it about? Well apparently the flavour eigenstates are not only superpositions of mass eigenstates but of massless eigenstates. In fact one of the massive eigenstates is apparently of a huge mass (the left handed one is high mass, right handed low mass, or vice-versa, i can’t recall!). The overall mass of the neutrino is not “huge” as the high mass eigenstate’s contribution is diluted by that of the massless eigenstates.
Apparently this state of affairs falls out of a higher dimensional theory and could explain the obs mass scale of neutrinos. This is kinda cool and i will definitely be trying to find out more about this during my random internet wanderings …

Tachyons – Do They Exist?

I just realised i never actually filled in this post! I had been meaning too but i guess it just went deeper and deeper down my to do list.

I should say that well – tachyons don’t exist*. But anyway …

One thing that i wanted to note about tachyons is that if they did exist there would be a way of investigating for their existence – via ‘Cerenkov Radiation.’

What is this? Well we know that the local speed of light can be surpassed. This occurs when a charged particle passes through a medium with refractive index n, say. Now the refractive index of a medium is just the ratio of the speed of light in vacuum to the speed of light in that medium. So if c=300,000 km/s in a vacuum and n=1.5 (for glass say) then light travels at v=200,000 km/s in this medium. But this speed of v as relativity tells us that only the speed c cannot be surpassed. When this occurs the particle will emit Cerenkov radiation.

Now suppose tachyons existed and that they carried charge** then they would contunously be losing energy by Cerenkov radiation as they always travel above c. But as tachyons speed up as they lose energy*** the tachyon goes to infinite velocity. At infinite velocity it annihilates with another tachyon which is OK as infinite velocity means zero energy but finite momentum***

Weird stuff indeed!

* In QFT tachyons are quanta (particles) of fields with negative squared mass. When they arise in quantum field theories (or string theories) they signify that the theory is unstable, i.e. we are at an unstable point in the vacuum so we would roll down to a stable minimum and then the particles will no longer be tachyons.
* if tachyons exist then they could certainly carry charge – sure why not!
** see previous post about odd things like this

Tachyons – What Are They?

Some people are curious about tachyons. So what the hell are they? I will endeavour to explain.
Tachyons are particles with spacelike four-momentum and imaginary mass. What does this mean? Well it is probably best to explain this using a space time diagram.
Here the ordinary spatial dimensions are in the x-y plane and the z axis represents time since an event. The event is at the origin. The -z axis is thus the past and the +z axis the future.
A particle travelling at the speed of light is represented as a line with slope c [the slope is drawn as slope=1. This is because some people, especially particle physicist like setting everything =1. Other people actually define the z axis as ct instead of t as i said]. This “world-line” defines the “light cone” shown in the diagram. Any world-line represents a path of a particle moving at a constant speed. As we’ve said the slope of the line gives the speed in units of the speed of light c. So particles travelling at v>c will have world-lines outside the light cone. The lines inside, along and outside the light cone are called “time-like”, “light-like” and “space-like” in the jargon of relativity.
Tachyons are the space-like particles. Now one important aspect of special relativity is that everything is relative – you see particles move differently based on your reference frame. If you know what a world-line looks like in one reference frame you can see what it looks like in a reference frame by applying a Lorentz transform. These are just equations that relate position and velocity between two reference frames (discovered by Joseph Larmor and Hendrik Lorentz with Lorentz getting the credit as he had a knack of doing as Ludvig Lorenz would have attested to).
An important result is that Lorentz transforms transform time-like to time-like and space-like to space-like, i.e. if you are in the cone you will be seen differently by different observers but you will always be seen to be moving along a line inside the cone. Likewise if you are outside the cone you will be seen differently by different people but you will always be outside the cone. Now this is very important! As z>0 is the future, z=0 is now and z<0 is the past then this means that if you are moving along a space-like world-line you can simply move to a refrence frame in which your world-line points backwards in time!! You can then send messages back to yourself and others through time!
There are many examples of why this is bad (not in the sense of bad for book-makers, lottery operators, etc. but bad in a logical sense! google the grandfather paradox to see what i mean) and this is why tachyons are held by most as not existing. They are generally viewed as hypothetical objects.
The second thing i mentioned is that they have imaginery rest mass. The energy of a particle is :

But if v>c then the denominator is imaginery (it contains the square root of a negative number). So as energy is not imaginery it means that the mass must be to cancel the imaginery denominator.
It is worth noting that this problem can easily be circumvented by defining the mass* to be:

So the energy equation of relativity becomes:

which basically says that while ordinary particles cannot accelerate to the speed of light (it would require infinite energy) tachyons cannot slow down to the speed of light!
If i think of other basic properties of tachyons i will add them here.

* note that this is perfectly fine to do mathematically speaking

Beautiful Physics Result II

Another result which i always found cool was the calculation that shows why orbits get synchronised. The orbitting body in question (the Earth’s moon is the prime example) orbits the system barycentre, i.e. the system’s centre of mass (this is a point below the Earth’s surface in the case of the Earth-Moon system). Over time the rotation speed of the Moon about the Earth and the rotation speed of the moon about its own axis become equal. This means that the Moon rotates about it’s axis (1 lunar day) in the same time that it takes for it to orbit the Earth. This means (think about it!) that we will only ever see one side of the moon. Which is what we do see!! The far side of the moon was not seen by man until the the Soviet space probe Luna 3 tooks some photos of it in 1959.
For those of you interested in the calculation here it is:
Let the ang. frequency of the moon about the earth be and let the ang. frequency of the moon about its own axis be . Then:


But the angular momentum is conserved so:


So we can combine these for the expression:

As the moon orbits around the earth energy will be dissipated by friction so we have:

So then if we have
then
and if we have
then
so either way and the orbits get synchronised!

A Watched Kettle Will Certainly Boil…

The saying: “a watched kettle never boils” might upset a physicist or two. It violates all that is good about quantum mechanics. Indeed I would argue that i could round up some people who would take the view that an unwatched kettle doesn’t boil. Surely it is the acting of observation which makes the kettle boil … have you not heard of Schrodinger’s kettle?!
Well anyway why the rant about kettles? Well it struck me as i was making tea for myself that all physicists in the world will have at some time or another watched and timed a kettle boiling. I thought that was a weird thing that people should know!
Why you ask? Well if you know how much water you have and what temperature it is at you can work out how much energy is needed to boil the water. If you time this you get the power rating of the kettle. You can then check this against the value on your kettle to see if you managed to do the calculation correctly. Very nerdy but somebody is probably doing it right now in some high school/college physics 101 class somewhere and that has to make you smile!

Beautiful Physics Result I

When i was with my girlfriend recently I remebered something which i thought was pretty cool. This is a not advanced physics or anything. Anybody who has studied Archimedes’ Principle probably knows this if they have thought about it (so i guess high school physics/leaving cert./A level and equivalent is probably more than required…). Enough babbling – what is the result?
Lets say you have a glass of water with ice cubes in it. The glass is filled to the brim. As you will know the bouyant force makes the ice-cubes float. The force equals the weight of water displaced so the ice cubes will not be completely below the surface of the liquid – they stick out above the water (you can very quickly check this if you have an ice box!).
So what happens when the ice cubes melt? All the ice turns to water. But the glass is full to the brim so you might think (well i did anyway!) that the ice turning to water will make the glass overflow – wrong. As you may or may not know ice expands when it freezes and contracts when it melts. The amount it contracts by is exactly the amount that stuck out above the level of the water. So when the ice cubes melt the glass remaind full to the brim and none spills. When i worked this out first i thought it was pretty cool!
For those interested in showing this – here it is (it’s nice and short):
If the volume of the ice-cube is V and the volume of the ice cube under water is V’ then our force balance equation (which keeps the cube floating) is just:


So we know that …

Now when the ice cube melts the mass of material remains the same whether it be in ice or water form. In other words:



Lets say that the volume that the melted ice cube water takes up is V”. Then:


But this means V”=V’, the volume of the ice cube which was under water in the first place! So the volume has shrunk by the exact amount which was above the water in the first place.
NB As the density of water and ice is close the amount of the ice cube above the water to start with isn’t much – I guess the effect would be more dramatic for a different liquid which we could choose based on on its density to maximise (1-V’), the volume above the water. You would just have to replace “water” with “liquid X” in the above few lines and you would get the same result.

Beautiful Physics Results

I thought i would add a section about beautiful physics results which i have come across during my years trying to learn the ins and outs of the subject. Here i will post some things that i think are pretty – the kinds of things that when you work them out you say “ahh!”. As these things pop into my brain I will add them to this section …

Equations … excellent!

Ah so i have figured out how to use equations in my posts! Good old wordpress allows you to use LaTeX (pronounced “lay-tech”). Hmmm i think i will practice …
These are some of the most important equations in physics…

Classical Mechanics
Hamilton’s Equations …



Lagrange’s Equations …

Special Relativity
The Minkowski metric …

Time Dilation …

Fitzgerald-Lorentz Contraction …

Energy-Mass-Momentum Relation …

Electromagnetism
Maxwell’s Equations (in vacuum) …

General Relativity
Einstein’s Field Equations …

The Einstein Tensor …

Quantum Mechanics
The Schrodinger Equation …

Quantum Field Theory
The Dirac Equation …

Astrophysics
The Blackbody Spectrum …

The Stefan-Boltzmann Equation …

What is a planet?

When people get wind that i am an astronomer those that don’t confuse me with an astrologer (this has happened a lot – i wouldn’t have thought so many people would be confused on that point but there you go!) ask me certain questions. Of late one question that i have had a lot is regarding Pluto’s status as a planet. So i thought i would explain it here.
Basically, up to last summer Pluto was considered to be one of the nine planets. Then the IAU decided they were going to define “planet” – something which had never previously been done. This was prompted by the discovery of a body called 2003 UB_313 beyond the orbit of Pluto. It was seen to be about the same size as Pluto and hence was being billed as the “10th planet” by many astronomers.
Now for a long time there has been argument over the planetary status of Pluto. As i said the term planet had never been defined and the main reason that Pluto was considered a planet was historical, i.e. it had been declared a planet when it was discovered and this stuck! So basically many people argued that if Pluto was discovered now it would not be declared a planet – simply an Edgeworth-Kuiper Belt Object. With the prospect of more bodies such as 2003 UB_313 being discovered beyond the Edgeworth-Kuiper Belt the IAU decided it was time to define what a “planet” actually was. Either Pluto was a planet and all these new Pluto like bodies would also be given planetary status or Pluto would be downgraded and we would have 8 planets and these newly discovered bodies would be simply referred to as “Trans-Neptunian Objects”.
In fact the decision of the IAU was somewhere in between these 2 possibilities. They defined “planet” and “dwarf planet”. So there were now 8 planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus & Neptune) and 3 new dwarf planets. Pluto and 2003 UB_313 satisfy the criteria for the new dwarf planet category. In addition the criteria elevated the asteroid belt object Ceres to dwarf planet. The IAU also gave a more elegant name to the 2003 UB_313 (thankfully!) and it is now known as Eris.
There is a good image on APOD which shows this new classification very well. Also for those of you interested in what the criteria set down by the IAU are then here they are:
Dwarf Planets must satisfy…
1. Must be in an orbit around the sun
2. Must have sufficient gravity to be in hydrostatic equilibrium (i.e. it must be big enough so that gravity has made it into a spherical shape)
3. It must not be a satellite (i.e. a moon)
A Planet must satisfy these same 3 criteria but with the additional condition that…
4. It must have cleared the matter from the region surrounding its orbit of debris.
We can see clearly why the 3 dwarf planets are so classified now. Ceres has not cleared the area around it’s orbit – asteroid belt! Pluto lies in the Kuiper belt, a region containing many cold rocky objects and comets. Eris is in the Scattered disk region of the solar system, which is just outside the Kuiper belt and has many icy bodies like the Kuiper belt, but is more eccentric w.r.t. the eccliptic.
[ASIDE: Also, it isn't mentioned as far as i know but presumably the planets would have to satisfy a mass condition - i.e that they are all below 13 times the mass of Jupiter. This is the mass at which it becomes possible for nuclear fusion to occur and here we are leaving the realm of planets for that of stars.]

I hope this clears things up a bit!

Evan

Mauna Kea, Hawaii

Mauna Kea is a huge mountain in Hawaii. It’s situation makes it an ideal place to put telescopes! Because it is so high up it is above most of the water vapour and a lot of the atmosphere. This makes for good viewing and as an added bonus there are typically 300 days a year of clear viewing – an astronomer’s dream.
Not being people to wait around astronomers have stuck a load of telescopes up there. It is amazing to see that a huge number of the best telescopes on Earth are all on the same mountain, nay the same road! Hopefully i can visit it some day!