Night 3:In an Unknown Land
Models, hypotheses and data
Models, hypotheses and data
For many years now we have been observing the grouping of Perseus galaxies. In fact, we’ve actually got more data from Perseus than from CasA.
Perseus is about 65 million light years in size, but since it’s 250 million light years away, its apparent size in the sky is a few degrees and we get all of it in one shot using the camera on our telescopes.
Why don’t you analyse the data we have from Perseus now you know how to do it? Look for the Theta2 plot of the accumulated data for Perseus in the scientific notebook and try to interpret the result.
We're a team of many scientists looking to solve the subject of dark matter. And we all help each other. Cosmologists, theorists, particle physicists... we all contribute and look the problem from different angles.
Cosmologists show us how dark matter should be distributed in Perseus. They’ve studied it with optical telescopes and can make maps of dark matter from different areas of the Universe.
And theoretical physicists use models to calculate the amount of gamma rays we would expect to find. It’s with this information that we’re able to go hunting.
We have our hypotheses and we have some telescopes at our disposal: we’ll be looking for data to determine if our hypotheses are right.
The theory I am testing claims that the number of gamma rays we should observe is determined by the following logic:
Ngamma_ray ( r ) = N_0 (si r<0.1 deg)
Ngamma_ray ( r ) = N_0/(100 r^2) (si r>= 0.1 deg)*
In the notebook on the right, you can see how I use this and the fact that I can’t see gamma rays from Perseus to define the maximum of dark matter in Perseus.
# All hope is not lost…the dark side may still be there
Let’s see what it means to not see a gamma-ray signal when I look at the Perseus cluster. That doesn’t mean that they don’t reach us, only that, if they do, they do so in a quantity that is smaller than a specific number. Let’s have a look!
%matplotlib inline
%pylab inline
import math
import numpy as np
import matplotlib.pyplot as pl
import random as rnd
Populating the interactive namespace from numpy and matplotlib
The first thing to do is to implement the theoretical model which tells us how many gamma rays come from the Perseus cluster. That depends on what part of Perseus we’re looking at and it can be simplified with a parameter indicating the radius (r) of the circumference we’re observing. It also depends on a normalization factor that we’ll call N0.
To find out how many gamma rays reach us for each parameter value, it’s best to define a function. You’ve already used a lot of Python functions. Let’s now see how can define one…it’s easy:
def NumeroRayosGamma( No, r ):
if r <= 0.1 :
Num = No* math.pi*r**(2.0)
if r > 0.1 :
Num = No* math.pi*(0.1**(2.0)+1e-4*(0.1**(-2.0)-r**(-2.0)))
return Num
Yup! A piece of cake! First define the function itself: what it’s called and what parameters we put through it.
def NumeroRayosGamma( No, r ):
Then, we define what the function does. As I said before, the number of gamma rays that come from the Perseus cluster depends on which part of the cluster we look at. In fact, the value is constant if we look at less than 0.1 degrees from the centre, but it decreases rapidly (with the radius to the fourth power) if we look further away. To take into account these two options we need to use the if condition:
if r <= 0.1 :
if r > 0.1 :
And within each condition we calculate how many gamma rays we see by looking from the centre of the cluster to the value of radius (r) and the normalization (No) that we include in the function.
Finally, we tell the function to return the number of calculated gamma rays:
return Num
Watch out, though! Where each line of code starts is important…for me, that’s the most annoying thing about Python.
Now we just have to write a line of code to know how many gamma rays should reach us. For example, if we look inside 0.2 degrees and assume a normalization factor of 1000:
NumeroRayosGamma( 1000, 0.2 )
NumeroRayosGamma( 1000, 0.2 )
54.97787143782138
In fact, before we saw that if I count events in the first two divisions of my theta plot, I have:
Events ON = 18477.0 Events OFF = 18338.0
Since I have 40 divisions in theta square between 0 and 0.40, that means that the first two divisions cover 0.02 square degrees in the theta square, which translates to a square root of 0.02 (0.1414) degrees. That means that for a normalization of 1000, we’d hope to receive.
print (NumeroRayosGamma( 1000, 0.1414 ), "rayos gamma")
47.1191445659 rayos gamma
Now we have to do something similar to what Alba did to understand what the significance means. Do you remember what that was?
We know that, on average, there are 18,338 noise events and that, on average, 47.1 events reach us. But those are just the average values. If we observe only once for the more than 100 hours that we have carried out, the values may be different. This is what happened with the observations simulated by Alba assuming there was no signal. Only now we’ll assume that the signal is on average 47.1 gamma rays.
Now that you know how to define a function, you no longer need to use it :D. So, let’s see how we simulate those observations.
First we define where we want to put the results of the simulation and initialize the function rnd that generates random numbers:
sigma = np.zeros(10000)
exceso = np.zeros(10000)
rnd.seed(1975)
So, I have created a variable sigma and another exceso (excess) which for now are a set of 10000 zeros.
With “rnd.seed (1975)” I give the seed to the random number generator so that it starts generating them. That is necessary because computers stil don’t know how to do 100% random things (well, I’m not sure I know how to do it either), they are pseudo-random and the series is defined by the seed. But what we’re doing is more than enough for what we need.
Now we simulate 10000 observations for which we have an average of 18338.0 events OFF and 18338.0 + 47.1 events ON and we see how many excesses there are for each simulated observation.
test = NumeroRayosGamma(1000,0.02**(0.5))
for x in range(0, 10000):
EventosON=rnd.gauss(18338, (18338)**(0.5))+rnd.gauss(test, (test)**(0.5));
EventosOFF=rnd.gauss(18338, 18338**(0.5));
sigma[x] = (EventosON-EventosOFF)/(EventosON+EventosOFF)**(0.5)
exceso[x] = EventosON - EventosOFF
To avoid having to write 47.1 a load of times, I’ve created a variable where I put the value of the gamma rays that I’m expecting using the function that we defined earlier. That also lets me see what happens with a different value of expected gamma rays just by changing a line of code:
test = NumeroRayosGamma(1000,0.02**(0.5))
Alba used the variable sigma, we’ll use the variable exceso (excess). Let’s see what distribution it has.
veces, excesos, _ = pl.hist(exceso, bins=100, histtype='stepfilled',alpha=0.2, normed=False)
pl.xlabel('Excesos')
pl.ylabel('Numero de Veces')
pl.show()
Now let’s see what percentage of these simulated observations would give us more excesses than we see in the data (18477-18338 = 139). For that, we need to use the number of times that an excess occurs and that I have saved in veces (times) and excesos using the graph that I just generated. Do you remember how this is done? With “veces, excesos, _ = pl.hist (exceso, bins=100, histtype=‘stepfilled’, alpha=0.2, normed=False)” I keep the Y axis values of the graph in veces and those of the X axis in excesos
Now I can make a loop giving values to x between 0 and 100 (“for x in range (0.100):”) and then check when we have more than what we observed in our actual observation. It’s basically the same as what Alba did to see the probability of having 2.7 sigmas.
VecesAcumuladas = np.cumsum(veces)
Probabilidad = 1.0-VecesAcumuladas/(VecesAcumuladas.max())
for x in range(0,100):
if excesos[x] > (18477.0-18338):
print ("La Probabilidad de tener más de ", 18477.0-18338, "eventos es: ", Probabilidad[x]*100, "%")
break
The probability of having more than 139.0 events is: 28.83%
When we have no signal in our observations, what we do is put an upper limit on gamma rays that are getting to us. One way to do this is to look for the gamma-ray value for which 95% of the time we’d have more excesses than we’ve observed. With the hypothesis of N0 = 1000, we’re far away.
Now is when it’s good for me to use the variable test and to have defined the function NumberRaysGamma (No, r). It makes it easier to find for what value of N0 the condition is met.
test = NumeroRayosGamma(10000,0.02**(0.5))
for x in range(0, 10000):
EventosON=rnd.gauss(18338, (18338)**(0.5))+rnd.gauss(test, (test)**(0.5));
EventosOFF=rnd.gauss(18338, 18338**(0.5));
sigma[x] = (EventosON-EventosOFF)/(EventosON+EventosOFF)**(0.5)
exceso[x] = EventosON - EventosOFF
veces, excesos, _ = pl.hist(exceso, bins=100, histtype='stepfilled',alpha=0.2, normed=False)
pl.xlabel('Excesos')
pl.ylabel('Numero de Veces')
VecesAcumuladas = np.cumsum(veces)
Probabilidad = 1.0-VecesAcumuladas/(VecesAcumuladas.max())
for x in range(0,100):
if excesos[x] > (18477.0-18338):
print ("La Probabilidad de tener más de ", 18477.0-18338, "eventos es: ", Probabilidad[x]*100, "%, cuando esperamos ", test, "rayos gamma, que se obtienen con N0 = ",10000)
break
The probability of having more than 139.0 events is: 94.64%, when we expect 471.238898038 gamma rays, which is what we get with N0 = 10000
N0 is related to the amount of dark matter in the Perseus cluster. So, my data doesn’t tell me that there’s no dark matter there, only that there’s less than a certain amount…all hope is not lost.
Dictionary of the gamma ray hunter
Active Galactic Nuclei
There's party going on inside!
This type of galaxy (known as AGN) has a compact central core that generates much more radiation than usual. It is believed that this emission is due to the accretion of matter in a supermassive black hole located in the centre. They are the most luminous persistent sources known in the Universe.
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Black Hole
We love everything unknown. And a black hole keeps many secrets.
A black hole is a supermassive astronomical object that shows huge gravitational effects so that nothing (neither particles nor electromagnetic radiation) can overcome its event horizon. That is, nothing can escape from within.
Blazar
No, it's not a 'blazer', we aren't going shopping
A blazar is a particular type of active galactic nucleus, characterised by the fact that its jet points directly towards the Earth. In other words, it’s a very compact energy source associated with a black hole in the centre of a galaxy that’s pointing at us.
Cherenkov Radiation
It may sound like the name of a ames Bond villain, but this phenomenon is actually our maximum object of study
Cherenkov radiation is the electromagnetic radiation emitted when a charged particle passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. When a very energetic gamma photon or cosmic ray interacts with the Earth’s atmosphere, a high-speed cascade of particles is produced. The Cherenkov radiation of these charged particles is used to determine the origin and intensity of cosmic or gamma rays.
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Cherenkov Telescopes
Our favourite toys!
Cherenkov telescopes are high-energy gamma photon detectors located on the Earth’s surface. They have a mirror to gather light and focus it towards the camera. They detect light produced by the Cherenkov effect from blue to ultraviolet on the electromagnetic spectrum. The images taken by the camera allow us to identify if the particular particle in the atmosphere is a gamma ray and at the same time determine its direction and energy. The MAGIC telescopes at Roque de Los Muchachos (La Palma) are an example.
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Cosmic Rays
You need to know how to distinguish between rays, particles and sparks!
Cosmic rays are examples of high-energy radiation composed mainly of highly energetic protons and atomic nuclei. They travel almost at the speed of light and when they hit the Earth’s atmosphere, they produce cascades of particles. These particles generate Cherenkov radiation and some can even reach the surface of the Earth. However, when cosmic rays reach the Earth, it is impossible to know their origin, because their trajectory has changed. This is due to the fact that they have travelled through magnetic fields which force them to change their initial direction.
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Dark Matter
What can it be?
How can we define something that is unknown? We know of its existence because we detect it indirectly thanks to the gravitational effects it causes in visible matter, but we can’t study it directly. This is because it doesn’t interact with the electromagnetic force so we don’t know what it is composed of. Here, we are talking about something that represents 25% of everything known! So, it’s better not to discount it, but rather try to shed light on what it is …
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Duality Particle Wave
But, what is it?
A duality particle wave is a quantum phenomenon in which particles take on the characteristics of a wave, and vice versa, on certain occasions. Things that we would expect to always act like a wave (for example, light) sometimes behave like a particle. This concept was introduced by Louis-Victor de Broglie and has been experimentally demonstrated.
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Event
These really are the events of the year
When we talk about events in this field, we refer to each of the detections we make via telescopes. For each of them, we have information such as the position in the sky, the intensity, and so on. This information allows us to classify them. We are interested in having many events so that we can carry out statistical analysis a posteriori and draw conclusions.
Gamma Ray
Yes, we can!
Gamma rays are extreme-frequency electromagnetic ionizing radiation (above 10 exahertz). They are the most energetic range on the electromagnetic spectrum. The direction from which they reach the Earth indicates where they originate from.
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Lorentz Covariance
The privileges of certain equations.
Certain physical equations have this property, by which they don’t change shape when certain coordinates changes are given. The special theory of relativity requires that the laws of physics take the same form in any inertial reference system. That is, if we have two observers whose coordinates can be related by a Lorentz transformation, any equation with covariant magnitudes will be written the same in both cases.
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Microquasar
Below you will learn what a quasar is...well a microquasar is the same, but smaller!
A microquasar is a binary star system that produces high-energy electromagnetic radiation. Its characteristics are similar to those of quasars, but on a smaller scale. Microquasars produce strong and variable radio emissions often in the form of jets and have an accretion disk surrounding a compact object (e.g. a black hole or neutron star) that’s very bright in the range of X-rays.
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Nebula
What shape do the clouds have?
Nebulae are regions of the interstellar medium composed basically of gases and some chemical elements in the form of cosmic dust. Many stars are born within them due to condensation and accumulation of matter. Sometimes, it’s just the remains of extinct stars.
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Particle Shower
The Niagara Falls of particles!
A particle shower results from the interaction between high-energy particles and a dense medium, for example, the Earth’s atmosphere. In turn, each of these secondary particles produced creates a cascade of its own, so that they end up producing a large number of low-energy particles.
Pulsar
Now you see me, now you don't
The word ‘pulsar’ comes from the shortening of pulsating star and it is precisely this, a star from which we get a discontinuous signal. More formally speaking, it’s a neutron star that emits electromagnetic radiation while it’s spinning. The emissions are due to the strong magnetic field they have and the pulse is related to the rotation period of the object and the orientation relative to the Earth. One of the best known and studied is the pulsar of the Crab Nebula, which, by the way, is very beautiful.
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Quantum Gravity
A union of 'grave' importance ...
This field of physics aims to unite the quantum field theory, which applies the principles of quantum mechanics to classical systems of continuous fields, and general relativity. We want to define a unified mathematical basis with which all the forces of nature can be described, the Unified Field Theory.
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Quasar
A 'quasi' star
Quasars are the furthest and most energetic members of a class of objects called active core galaxies. The name, quasar, comes from ‘quasi-stellar’ or ‘almost stars’. This is because, when they were discovered, using optical instruments, it was very difficult to distinguish them from the stars. However, their emission spectrum was clearly unique. They have usually been formed by the collision of galaxies whose central black holes have merged to form a supermassive black hole or a binary system of black holes.
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Supernova Remnant
A candy floss in the cosmos
When a star explodes (supernova) a nebula structure is created around it, formed by the material ejected from the explosion along with interstellar material.
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Theory of relativity
In this life, everything is relative...or not!
Albert Einstein was the genius who, with his theories of special and general realtivity, took Newtonian mechanics and made it compatible with electromagnetism. The first theory is applicable to the movement of bodies in the absence of gravitational forces and in the second theory, Newtonian gravity is replaced with more complex formulas. However, for weak fields and small velocities it coincides numerically with classical theory.
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