# Why are radio telescopes so large?

Hi,
take a Hubble telescope, it has primary mirror 2.4 meters tall. Now take this: I mean the thing above which is Arecibo radio telescope has diameter of 305 meters. Both things are for the same thing, to observe the Universe.

So the answer lies in the name. Arecibo is a RADIO telescope which means that it works on quite different wavelengths, actually magnitudes bigger wavelengths since the wavelength of for example visible light for Hubble is 550 nanometers which is quite small while radio waves can have wavelength of hundreds of meters.

This is essentially the key. If you want to see clear image in light (that we can see) you need just a small telescope. Both work the same way though from what I understand you need larger area to collect all of those waves and reflect them on the focus which is above. The equation shows it clear:

θ=1.22*λ/D

Where θ shows how close two points can be to each other without you being able to distinquish them. λ is the wavelength of the light and D is the diameter of your telescope. So you will see best when wavelength is small and diameter is huge since this will lower the angle that you are not able to distinquish. Of course that there is huge difference when you insert meters instead of nanometers so you must compensate it with the diameter of the telescope.

If you want to have a clear image in radio waves, well you have to build Arecibo.. really? Isn’t there another option?

Yes there is! You can build a lot of small radio telescope that would alone be very weak but if you take lot of them you can have a Diameter of kilometers. Such a device is called interferometer which means that is “operates by myltiplying the data from each pair of telescopes together to form interference patterns”. There is more of them and this one is ALMA observatory.

So those are huge fields or rows of smaller (even 60 meter) discs that collect data. They have to be extremely accurate what is time concerned (atomic clocks).

Dragallur

1.22 is just an empirical value.

# Can you see a coin from 400 kilometers?

Hi,
no, probably not with naked eye, I think, though if it would be some special coin, maybe? The point is that you can use some cool things to see such a coin, for example Hubble Space Telescope!

Ok, I found some page with physics problems and one of the first was to calculate what is the smallest angle that Hubble Telescope can distinquish. I calculated that it is roughly:

# 0.0127”

This means 0.0127 arcsecond!!! How far away football has to be to have the angular diameter of one arcsecond

What about a human eye how good is it? Well eye can distinquish only things that are 1′ away from each other , which means one arcminutes, one degree has 60 of those so it is pretty good but not so much as Hubble. But anyway, back to the title, how small is a coin?

I measured the second largest Czech crown to have 2.55 centimeters.

Now how far away do you need to be to not be able to see it with naked eye? We will use this formula to get the diameter:

θ=D/r

Where the diameter (D) is 0.0255 meters. The distance (r) is what we are trying to find and the angle eye can distinquish is θ (in radians). 

r=D/θ

If all the calculations went right it should be:

r=87.6625 meters

But beware this is not counting air, humidness and so on so you will most certainly not be able to see coin on this distance.

What about the Hubble telescope? We can use the same equation but for θ we will insert much smaller value!

And yes, you can see the coin from 414,153.744827 meters! Which means that you can see one freaking coin from London 50 kilometers behind Paris! One coin!

This is so cool.

Dragallur

You can try this by drawing two dots on paper and then moving away from it, at one point when you are far enough you should be able to see only one.

This equation is simplified but it should work for small angles.

Again, this works only in space because there is nothing that would block your sight.

# How does atom looks like?

Hi,
this was a question that friend of mine asked me on one contest I was this weekend. I was sure with the answer but after I said it I was not able to come up with the reason for it, at least I was not sure enough to say something clear.

So what exactly needs to happen for you to see it? There must be a photon which is reflected off the surface. How does this look like anyway on the atomic level? Well the light hits some electron in its way. There is lot of free space so this is why things that are not transparent can be if they are thin enough. The electron absorbs the photon, jumps to higher level (excitation), then it emits photon. Now on what you see depends upon its wavelength. So different materials will like to absorb different wavelengths making the object to have color. You can only change in what orbital you will have the electron so I guess that this is the fundamental difference between various colors of objects (though I did not check it).

So when electron emits the photon you simply do not know what the electron looked like. The only thing that you can get is just photon of some wavelength and there simply is not any way to look on some kind of surface of electron. Another factor is that the light has too big wavelength and you can not observe surface with that because the photon kind of just flows around and when you get to wavelength of the size of atom or smaller, the energy of the photon is so huge that the electron is anyway blown away.

There is nucleus too of course and normally photons do not get there because of this electron cloud around and nucleus is tiny. Otherwise from what I found it seems that again the proton and nucleus as whole is way too small and you can not actually map the surface.

But I was talking only about electrons and nucleus. You can actually see atom as whole. Not by microscope because visible light is way to huge. There is what is called Abbé difraction limit so you have to look for atoms in different way, using for example electrons (electron microscopes) and then recreate the image using some cool physics, this is for example picture of silicon carbide: And the one below is picture made by IBM of individual atoms that are shaped like the letters of IBM. So while you can not see the atom, you can observe, not electron though or even the nucleus at least in ordinary way. You can for example measure energy or calculate the shape but you can not see them as physical objects.

Dragallur

Electron microscopes observe how electrons bounce of the surface just like photons. # Optics: 5) Magnifier and microscope

Hi,
today I will finally continue to write about optics. Last time I was talking about dioptre and today I will explain how magnifier and microscope works.

### Angle of view

Angle of view plays important part in magnifier, microscope and so on. The problem why we can not see individual cells in leaf with naked eye is that we are not able to distinquish things that are too close to each other. Human eye is able to distinquish things that are about 1′ (arc minute=1/60° degrees) apart.

You can easily try it when you draw something on paper and then walk away from it. Or you are not able to tell trees apart when you are hundreds meters away from them. (On the picture you can see the angle of view for camera, it can be measure horizontaly, verticaly or diagonally).

To make this angle bigger so we can distinquish everything we can walk towards stuff. But when we have a leaf we can get only limitely close to its surface and our eye can not adjust to something so close. Look at your thumb when you put it three centimeters from your nose. It will be blurry even if you try your best, this is because your eye does not have enough dioptre to make the image clear, plus your eye is going to hurt because of the muscles in eye stretching to make the optical power of your eye bigger. Conventional visual distance is distance for which human eye has to release least effort, this is about 25 centimeters.

To know the distance two objects must be apart to distinquish them we can use tangens:

tg τ=y/d

τ is the angle of view which is for human 1′.
y is the distance of two objects which you are trying to distinquish.
d is the distance from you to the objects.

y=d * tg τ

So the limit of our eye is that it is not able to be powerful enough so we need something which will help us and it has to work the same way as our eye, magnifier!

The light rays are going too much away from each other and your eye is not able to change their direction to create picture.

### Magnifier

There is thing called angular magnification.

γ=τ’/τ = tg τ’/tg τ =  y/f/y/d = d/f

γ is the angular magnification and is the distance to focal point.

Angular magnification says to us how much our magnifier is strong. The formula above works for objects that are right in focal point, otherwise there would be “a” which is the distance to the object. If the object is right in focal point our eye does not need anything to do and as it gets closer the light rays are more and more going apart so that at one point you will need better magnifier and then it just wont be enough so you will have to use microscope.

### Microscope

There are two lenses in microscope. The first one is close to the object and it has the largest dioptre possible, making its focal point small as possible. It is called objective lens.

The second one is not so strong and its role is to make finally adjustment of light rays so they create image in your eye. The picture above which I drew is horrible wrong but I can describe what is going on there. On the right you have the small object (brown). There is light coming from it in all various angle but important is that the lens has enough dioptres to use them all. F’ is the focal point of second lens, this point should be at the distance where all the rays from first lens converge into one point but I was not able to draw it properly. This is the way microscope is designed. Those rays start to go apart again but soon they hit the second lens, converging again and entering the eye in proper angles so that they hit all the spots on red line creating much bigger image.

Microscope is not unlimited source of magnification since when you will try to make bigger something too small you will get into problem with the wavelength of light.

Dragallur

# Optics: 4) Measuring dioptre

Hi,
today I was doing the best thing in optics to date. I was measuring the dioptre of my glasses (yes I wear glasses) and also I measured the dioptre of my magnifier (yes I measured it but then I figured out that I did it wrong so I will skip it).

Ok, before I get to the measuring and how I did it I will explain how lenses work because in last episodes what I did were only mirrors.

The difference between mirrors and lenses is that mirror reflect light while lenses let it through while changing its direction of travel.

There are several types of lenses which can be sorted to two main groups of convex lens and concave lens.

On the huge picture you can see the six types. The first row are convex lenses. First one is called biconvex lens then planoconvex lens and the third is concave-convex lens. You can see than there is always convex which hints for the first row, for convex type.

It is similar with the second type, those are biconcave lens, planoconcave lens and convex-concave lens.

I know this is cool, what can we do with this? This equation which you can see on the left is the equation for lens which is thin. This means that there is no space between the arcs of the lens by this I mean that the arcs touch . Those arcs you can see on the right of the first picture. r1 is radius of the first arc and r2 of the second. f is the focal distance, the distance from focal point to the middle of the lens. The thing here is that lens has two focal distances, that is because it is made of two parts separeted by the vertical axis as you can see on the next picture. Also this whole equation not only equals to 1/f but also to φ(phi). The unit of φ is dioptre so φ=1/f. If f increases dioptre decreases logicly. So if someone has glasses with 4 dioptre his focal distance is 25 centimeters because dioptre is measured in meters!
This equation can be used both for concave and convex lenses of course (but concave lens will have r negative).

n1 and n2 are the refractive index of the glass which is around 1.6 and of the stuff where the lens is in, air, water or something else (n2 is the higher one).

You can find lot of problems on this equation and I did some from one book. It is good to exercise some of them because then you will feel much better on the stuff you are actually doing.

Now last thing before I get to the glasses, lets see how convex lens react to the three main rays which I mentioned in earlier posts (I will do the concave lens next time because I did not get to it yet).
When the candle is in about twice the distance of the focal point you can see that the size is fairly similar and what concave lens does, is that those light rays which are going from each other will be headed back towards the same point where the image will be formed. Of course the problem is that you wont see the picture of something when you put your lens from your glasses on the paper. It is because there is whole other bunch of rays from all different sides that will disturb any image that could be made. When you look on the picture above, you can see that blue and green line were not able to touch anywhere which is the same thing that happened with the mirror when you put something between the mirror and focal point.

This image is enlarged and not true image since the rays are not actually going that way but our eye thinks so.

I was measuring the dioptres of my glasses. For the right eye I have -2 dioptres. You see it is very important that it is minus because that is what is saying that it is concave lens.
I took the glasses and drew line on the paper of their bottom side, which I then expanded and tried as accurately as possible to find out the radius of this circle 9.2 for the inside of concave lens and 12.8 for the outer part.

Do not forget that those glasses are concave convex lens which also means that the inside is -9.2 because it is “negative” of the glass.

When I gave it to the equation I found out that focal distance was 54.52 centimeters and dioptres -1.8342 which is not very close but since the way I was doing this was not meant to be very accurate I could not get anything better. (I took the refractive index of glass to be 1.6).

Dragallur

PS. this was my 100th post!
PPS. I will update about those glasses because I am not totally sure yet how they work so stay tuned.
Picture of equation
Picture of magnifier

# Optics: 3) Spherical mirror

Hi,
hell yeah! Here comes another post about optics. This time I will be writing about spherical mirror, its properties and how light behaves when it hits its surface. On the left you can see the spherical mirror. Black line indicates the edge of the mirror while the other line is the axis of the mirror. C is the center of the mirror. F is the focus and v is the top, or vortex.

Below you can see the light rays colored.

There are three main types of light rays hitting this kind of mirror.

There is the blue one which is called parallel ray. It is parallel to the axis and when it hits the surface of the mirror it is reflected toward focus. Actually when it is far away from the axis it may hit some different point (but that is not important right now).

Green one is called the vortex ray. It is coming to vortex where it is reflected in the same angle. On vortex, light behaves the same as when it is hitting normal flat mirror. The last and red one is focus ray, which comes through focus and is reflected in parallel with axis. The last one is actually reverted parallel ray.

Using these three rays you can get the place of the reflection. Of course in normal light there is tremendous amount of rays which than create an image. On the next picture you can see the image of candle/flame reflected using these three light rays.
This kind of picture is turned upside down and when the candle is so far away it will be smaller. If it gets closer at one point, both sides will be the same and then at on point when the candle will be between focus and vortex, another thing happens. So here you have the candle between focus and vortex. This means that there is no red ray/focus ray because it is not reflected  by the mirror. But your mind things that blue and green ,which should never meet, are going in straight line behind into the mirror which than creates bigger apparent picture.

So this is for the reflection basics now lets see the math behind. This one on the left is rendering equation.
“a” is the distance of the object from the mirror.
“a'” is the distance of the reflection from the mirror.

“f” is the distance of vortex and focus. You can see that if “f” is bigger the fraction is smaller which means that the other side has to be smaller too so “a” and “a'” will get bigger too since the distances are bound to each other (and they are in denominator). If “a” is bigger than the reflection is closer to vortex which means that “a'” is smaller.

The last one is called something like transversal enlargement. From Z which is dimensionless you can know if the reflection is bigger or not because “y'” is the height of the reflection from the axis and “y” is the height of the real object. If Z is 1 then both real object and reflection are the same size. If Z is bigger than one than the reflection is bigger which hints for the last picture of reflection. If Z is smaller than 1 than the reflection is smaller.

Z can be also calculated using the distance from vortex:
Z=-a’/a.
This works similarly. If Z is smaller than 0 then the picture is reverted. And everything else works as well because it is actually the exact same equation.

Also you can get the place where the focus is when you are measuring something which is far away. This is because when it is far away almost all of the light rays are going parallel to the axis. When it is about 10 meters there is problem with your accuracy.

Dragallur

The pictures which were drawn by hand are made by me so I have all the right for them (actually feel free to use them using )!

# Observing Supernova

Hi,
so what I just found was very very interesting and it is about observing of Supernovas which are those exploding stars or white dwarfs.

The problem with observation of these very bright objects is that first of all there is not much of them and second, it is extremely hard to catch the start of the explosion so we do not have much data about it. This problem seems to have really awesome solution that is based on nothing else than Einstein’s gravitational lensing.

Gravitational lensing is effect of very massive thing like black hole or galaxy or even galaxy cluster. It can increase the amount of light coming to us or bend the light. So actually when you are looking to star right next to Sun it may appear on different spot than it actually is! Actually I am pretty sorry but not only that you see everything in past but you see it actually on the wrong place, DAMN! (And if you run it is bluer).

So because of this light bending it can happen that the light even comes from different directions. The picture above shows how we could observe one event (one supernova type Ia) four times in different time intervals just because it was in huge super cluster which bend the light from supernova so much that it came to us in different directions.

This particular observation was done only by accident when one astronomer was looking at pictures from Hubble and he saw it.

Dragallur

PS: today I have reached 400 visitors!

# Optics 2) Snells law

Hi,
here it comes, here it goes! This is gonna be probably my first post where some real physics is involved (actually it is far too easy)! I am going to write about Snells law which is the law describing how light changes the angle of traveling when it enters to different medium.

First of all I have an experiment for you which goes as follows: you take a bowl which for the start will be empty except some small thing that will sink in water later on.
1) place your chin on the edge of table and move the bowl so you can see what is inside.

2) move it on the place where the edge will block your view and then place a water inside.

3) You should see the thing inside again!
This is because the light changes its angle of traveling which makes for example straw in bottle of water look so distorted (or hunters eye cheated when he is seeing the fish on different place then where it actually is (this makes water not look so deep as it actually is)), as that the higher part is not even connected sometimes to the lower one.
On the picture on the left you can see that the lower and upper part does not match which is because of the light which change the angle of traveling when entering glass then water then glass and air.

Fine, so what depends on how much will the angle change?
It is the difference between the speeds of light in both medium.

There is thing which is called refractive index: n. All mediums (which do let some light through) have some refractive index. Vacuum has 1, it does not change the angle of light. Air has about 1,0003 so you will usually see 1 also. But water has 1.33 and refractive index of glass ranges from 1.5 to 1.9 and the highest index known is for germanium=4.

When light enters another medium Snells law comes!  Btw. if you want to know n.. n=c/v (c is speed of light and v is the speed of light in current medium)

On the left you can see the example of light ray traveling from one thing to another. So for example if you did not know the angle in which it will fly in the second medium (water now) you would do this:
alfa=50°
beta=?
n(air)=1
n(water)=1.33

sin 50°/sin beta=1.33/1
sin 50°/1.33=sin beta
0.77/1.33=sin beta
0.58=sin beta
beta=sin^-1(0.58) beta=35.5°
This is all beta about which I am talking here would represent theta two on the picture. So if you have enough information you can either get the angles or the speeds of light in mediums or the refractive indexes.

On the gif you can see pretty neat animation of how the waves of light are slower in the water and the red line clearly represents how the light is refracted.

Dragallur

1st picture source

# Optics: 1) Reflection

Hi,
here it comes, here it goes. I realized that if I want to really know something about astronomy I have to use physics. For now my first goal is to learn something about optics and particularly about binoculars and how they work and what is the math involved! Today I will write about very basics and it is reflection of light. Probably I will add something to it on my YouTube channel with some examples and I bet it will be fun!

When we are talking about optics and stuff around we assume that light or electromagnetic radiation behaves as particle (on the level where I am), the thing is that as rest of the really small particles like quarks and electrons and neutrinos, light behaves both as particle and wave which called wave-particle duality. This is very interesting but let it be for now. As you see on the first picture from wikipedia article reflection, there is laser pointed on mirror which as you can see is reflected to the right.
This is important, the light gets reflected from mirror because photons bounce in the same angle from which the came.
On the right you can see the ray of light starting at P, reflecting from O and flying to Q. Both of the angles are are same from imaginative line that has 90° angle with the mirror. θr=θi  … r stands for reflected and i stands for incident. This also means that the angle of the ray and the mirror equals to the one on the other side. This is for mirror but of course other things are reflecting light too but because their surface is not smooth photons are flying all around and you want get the exact image of the thing that was emitting/reflecting the light first even if it is not absorbing any spectrum like snow. This is also the reason why it is dangerous to not wear anything across your eyes when you are a long time on place where is snow because over time all of this light can blind you.

This can happen in matter of couple of seconds if you look directly into Sun since the inside of the eye can be sun-burned in similar way like your skin and the damage may be permanent so watch out!

Dragallur