QP2 Flashcards
(178 cards)
1
Q
Nucleus of the atom diameter
A
approx. 10^-14m
2
Q
Electrons orbit the nucleus at a distance of
A
approx. 10^-10m
3
Q
The electrons and the nucleus should have been drawn to each other due to them being opposite charges. What did Rutherford suggest to this
A
Rutherford suggested that the electrons orbited the nucleus, but in doing so they should have lost energy and slowly circled down into the nucleus.
4
Q
What was a consequence of Rutherford’s model of the atom
A
That as the electrons orbited down, their angular speeds would change continuously. However, observed spectra consisted of discrete energy lines, which disagreed with Rutherford’s suggestion.
5
Q
What was Bohr’s solution for the model of the atom?
A
Bohr suggested stable orbits, meaning electrons would orbit the nucleus at a fixed distance and would not radiate energy.
6
Q
Bohr’s model of the atom
A
Bohr’s model stated that there was a definite energy associated with each available stable orbit, and an electron only emits energy if it moves from one orbit to another.
7
Q
What is the energy emitted with an electron moving from one orbit to another?
A
The energy emitted in such a transition is in the form of a photon, and the energy of that photon is given by the equation: E = hf = E_initial - E_final
8
Q
What is the angular momentum of an electron?
A
The angular momentum of the electron is a multiple of ℏ, i.e its quantised.
We know: L = mvr = nℏ.
Where each value of n (1, 2, 3… ) corresponds to a valid orbit of radius r_n and corresponding velocity v_n.
9
Q
Principle Quantum Number definition
A
The Principle Quantum Number is the value of n for each orbit.
10
Q
Hydrogen atom composition
A
The hydrogen atom is the simplest in the periodic table. With one electron orbiting around a nucleus containing one proton.
11
Q
Force for an electron in a given orbit (for Hydrogen Atom)
A
For the radius to remain constant in the Bohr model, the force must provide exactly the radial motion force. So, for the hydrogen atom, where both charges involved carry the same magnitude of charge e…
F = mv^2 / r = 1/4pƐ . e^2/r^2
12
Q
Kinetic Energy for an electron in a given orbit
A
Kn = 0/5mv^2 = 1/Ɛ^2 . me^4 / 8n^2h^2
13
Q
Potential Energy for an electron in a given orbit
A
Un = -1/4pƐ . e^2/r = -1/Ɛ^2 . me^4 / 4n^2h^2
14
Q
The total Energy for an electron in a given orbit
A
En = Kn + Un = -1/Ɛ^2 . me^4 / 8n^2h^2
15
Q
What do we apply the Schrodinger equation for?
A
As Bohr models isn’t perfect, we apply the Schrodinger equation to find the wave functions for states with definite energy values for the hydrogen atom.
16
Q
Reduced Mass
A
The electron does not orbit the proton, they both orbit their common centre of mass. Thus, we use the reduced mass, which is when you have two masses orbiting each other, then the reduced mass is given by:
m_r = m1m2 / (m1+ m2)
17
Q
The Schrodinger equation for the hydrogen atom solutions:
A
The solutions are obtained by separating the variables involved. This means we express the wave function as a product of three function, one associated with each of the coordinates: y(r,q,f) = R(r)Q(q)F(f)
18
Q
Boundary equation for the Schrodinger equation for the hydrogen atom.
A
-R(r) tends to zero as r increases: this is because we are dealing with bound states that are close to the nucleus of the atom.
-F(f) must be periodic, i.e. it must repeat every 360 degrees or 2p radians: for example, (r,q,f) and (r,q,f + 2p) describe the same point, so F(f + 2p) = F(f). Also, the solutions to the functions must be finite – i.e. cannot take zero or infinity as solutions.
19
Q
Solving the boundary conditions produces a relation for the energy levels as follows:
A
En = - 1/(4pƐ_o)^2 . me^4 / 2n^2h^2 = -13.60 eV / n^2
Where n is the principle quantum number of the energy level En. The negative sign indicates that the electrons are in bound states, as you need to provide a positive energy to remove the electron.
20
Q
Angular momentum formula
A
L = ( l( l+1) )^0.5
21
Q
Orbital angular momentum quantum number (l)
A
Given by: l = 0, 1, 2, … , n - 1
22
Q
There are n different possible values of …
A
L for the nth energy level. The Bohr model said there was one value: Ln = nℏ
23
Q
Z - component of angular momentum is given by:
A
Lz = mlℏ
24
Q
Orbital magnetic quantum number
A
ml also called the orbital angular momentum projection quantum number. It can take values m = 0, ±1, ±2, … , ±l.
25
Condition for L and Lz magnitudes
The magnitude of the z - component of the orbital angular momentum is always less than the magnitude of the total orbital angular momentum vector is a requirement of the uncertainty principle. E.g the component Lz can never be quite as big as L itself, unless both of them are zero.
26
The angle between L and Lz is given by;
θ = arccos ( Lz / L)
27
Maximum angle between L and Lz
This occurs of the smallest value of Lz
28
Minimum angle between L and Lz
This occurs for the largest value of Lz
29
We will never know the precise direction of the orbital angular momentum because ...
If we knew the direction of the Lz then we could define that direction to be the z - axis and so L = Lz. Comparing to L = mvr, all the motion of the particle would be in the x-y plane, then the z-component of the linear momentum would have to be zero and carry no uncertainty. But this would require that z has an infinite uncertainty by the uncertainty principle, which is impossible for a localised state.
30
Instead of defining the precise directions of L...
We end up drawing cones of direction of the vector L.
31
In quantum mechanics we can describe the interaction of charge particles in terms...
of the emission and absorption of photons. For example, two electrons repelling each other can be thought of as one throwing out a photon, which the second one then catches.
32
The uncertainty principle (for energy) states:
That a state that exists for a short time Δt has an uncertainty in its energy ΔE such that ΔEΔt ≥ ℏ/2
33
The uncertainty principle (for energy) allows the creation of ...
A photon with energy ΔE, provided that it lives no longer than the time Δt. This type of photon is called a virtual photon.
34
Photons mediate ...
electromagnetic forces
35
Two nucleons can mediate...
nuclear forces
36
The nuclear force between two nucleons ( protons or neutrons) can be described by...
Potential energy U(r) which is given in the formula sheet with the weird exponential of r / ro. In this equation, f represents the strength of the interaction and r0 its range. r is the distance at which the potential is measured.
37
The Heisenberg uncertainty principle states...
that the exact position of a particle and its momentum cannot be known at the same time. This is due to the that that in quantum theory, quantum particles do not have exact properties only probabilities.
38
The four types of particle interactions are...
1. The strong interaction
2. The electromagnetic interaction
3. The weak interaction
4. The gravitational interaction
39
The mediating particle for the electromagnetic interaction is ... with spin ...
the photon with spin 1s
40
The particle for the gravitational interaction is the ... with spin ...
Graviton with spin 2. Note that is it currently a hypothetical particle, since the force is very weak
41
The strong interaction is responsible for ...
the nuclear force and also for the production of pions and several other particles in high - energy collisions.
42
The mediating particle for the strong force is...
the gluon, however the force between nucleons is more easily described in terms of mesons as the mediators.
43
The Potential energy U(r) function is also a possible potential energy function for the ...
nuclear force. The strength of the interaction is described by the constant f^2, which has units of energy times distance.
44
A better basis for comparison with other forces is the dimensionless ratio...
f^2 / ℏc
called the coupling constant for the interaction. The observed behaviour of nuclear forces suggests that this ratio is roughly equal to 1.
45
The dimensionless coupling constant for electromagnetic interactions is ...
1/4πε . e^2 / ℏc = 7.297 x 10^-3 = 1/137.0
(in formula sheet)
46
The weak interaction is responsible for...
beta decay , such as the conversion of a neutron into a proton, an electron and an anti-neutrino.
It is also responsible for the decay of many unstable particles (e.g. pions into muons).
47
The force mediating particles for the weak force are
W+ , W-, Z^0
They are short-lived particles.
They have spin 1.
48
Charge of and mass of the weak mediating force particles
W± have mass 80.4 GeV/c^2 and charge ±e
Z^0 have mass 91.2 GeV/c^2 and charge 0
49
Mass, charge and spin of force mediating particles not including weak force
Gluon: 0 , 0 , 1
Photon: 0 , 0, 1
Graviton: 0 , 0, 2
50
Strength relative to Strong for interactions
Strong: 1
Electromagnetic: 1/137
Weak: 10^-9
Gravitational: 10^-38
51
Range of interactions
Strong: Short (~1 fm)
Electromagnetic: Long (1/r^2)
Weak: Short (~0.001fm)
Gravitational: Long (1/r^2)
52
Which particles are known as the "light ones" ?
The leptons e.g electrons
53
What makes up the hadrons
Mesons and Baryons (these consist of quark)
54
What makes up a meson
A quark and anti quark
55
What makes up a Baryon
3 quarks
(most common for protons and neutrons)
56
What are the different types of quark?
Up (u)
Down (d)
Top (t)
bottom (b)
charm (c)
strange (s)
57
what is a boson
A force mediating particle such as the photon, graviton, gluon and [W^ {+}, W^ {-}, Z] bosons.
58
What spin do fermions have
half - integer spins
59
What is spin?
A property of a particle relating to its orientation.
Spin is an intrinsic property of particles that mathematically behaves like angular momentum.
60
What spin do bosons have?
Zero or integer spins.
61
What types of particles obey the exclusion principle?
Fermions obey the exclusion principle, bosons don't.
62
Types of Lepton
The electron, muon and tau particles, and their associated neutrinos.
63
Types of lepton numbers
we have 3 lepton numbers
Le
Lμ
Lt
64
What do leptons obey
Leptons obey a conservation principle
Each lepton number is separately conserved.
65
What numbers are the lepton numbers assigned with
The electron and the electron neutrino are
assigned
Le = 1, and their anti-particles are given Le = -1.
66
The wave functions for the hydrogen atom are determined by the values of the three
quantum numbers what are these and meaning
The quantum numbers are: n, l, ml.
Where n determines the energy values En
l sets the magnitude of the orbital angular momentum
ml fixes the values of the (arbitrarily
chosen) z-component of the angular momentum.
67
What is degeneracy?
The existence of more than one distinct state with the same energy.
68
States with various values of the orbital quantum number l are often labelled with letters, according to the following scheme …
l = 0: s states
l =1: p states
l = 2: d states
l = 3: f states
l = 4: g states
l = 5: h states
69
What is a shell
A shell is the area of space associated with a given n shell.
70
Types of shells:
n =1: K shell
n = 2: L shell
n = 3: M shell
n = 4: N shell
and so on alphabetically
71
Angular momentum has two components:
Spin and orbital angular momentum
72
The spin angular momentum of an electron is given by:
S and it is quantised, e.g it can only take specific values
73
The z-component of the vector S can by found by...
Sz = ± ℏ / 2 and Sz = msℏ
74
The magnitude of spin angular momentum [S] is given by the expression:
S = [S] = (0.5(0.5+1))^0.5 . ℏ = (3/4)^0.5 ℏ
(similar to orbital angular momentum, but instead of l its the spin quantum number, which is s = 0.5)
75
The quantum number ms specifies...
the electron spin orientation. It can take values 0.5 or -0.5 .
76
The spin angular momentum vector S can have only two orientations in space relative to the x-axis:
- Spin up - the z-component is + ℏ / 2
- Spin down - the z-component is - ℏ / 2
77
The total angular momentum J is defined as:
J = L + S
78
The magnitude of the total angular momentum J is given by:
J = (j(j+1))^0.5 .ℏ
Where j is another quantum number
79
States of the quantum number j:
We can then have states in which
j = [l ±1/2]. The l + 1/2 states correspond to the case in which the vectors L and S have parallel z - components, while for the l - 1/2 states, L and S have anti-parallel z components.
80
Explain spectroscopic notation: Example 2^P_1/ 2
The superscript is the number of possible spin orientations, in this example, there are spin 2 states.
The subscript is the value of j, this example, j = 1/2
81
Quarks carry charges that have magnitudes of...
1/3e or 2/3e
82
Quarks obey the conservation of...
baryon number
83
Baryon numbers for quarks
Each quark has a fractional value of 1/3 and each antiquark has a baryon-number of -1/3.
Baryon number is conserved in all interactions.
84
Spin angular momentum of a meson
they can have spin angular momentum components parallel to form a spin - 1 meson or antiparallel to form a spin - 0 meson.
85
What is the baryon number of a meson
In a meson, a quark and antiquark combine with net baryon number 0
86
Baryon number of a baryon
The 3 quarks combine to make a net baryon number 1
87
What is the spin of a baryon
the quarks combine to make a spin-half baryon or spin 3/2 baryon
88
Which quarks have charge Q/e of 2/3 ?
Up , charm, top
89
Which quarks have charge Q/e of -1/3 ?
Down, strange, bottom
90
All quarks have spin...
a half
91
All quarks have a baryon number of
1/3
92
The antiquarks take ... values of Q, B, S, C, B’ and T.
opposite
93
Strangeness, charm, Bottomness and Topness take values of...
S = -1
C = 1
B' = 1 or -1 (depends)
T = 1
94
Quarks come in three colours...
Red , blue and green
a baryon will always contain one of each
95
A gluon contains a colour...
-anticolour pair
96
A gluon transmits colour when exchanged and colour is always ...
conserved during emission and absorption of a gluon by a quark
97
The gluon-exchange process changes the colours of the quarks in such a way that...
there is always one quark of each colour in every baryon
98
The colour of an individual quark changes continually as...
gluons are exchanged
99
Mesons have ... net colour
no
100
Mesons have spin ...
0 or 1
101
are mesons stable?
no, there are no stable mesons they all decay
102
Baryons include the nucleons and several particles called hyperons, including ...
sigma Σ
lambda Λ
doublet of xi Ξ
omega Ω
103
A hyperon is ...
any baryon containing one or more strange quarks, but no charm, bottom, or top quark.
104
Baryons have a baryon number of...
B = 1 or B = -1
105
Baryons have a spin of ...
a half
106
The only stable baryon is
the proton
107
how do you know if a particle is its own antiparticle?
if its quark content is the same , means same particle and anti particle
108
Example, A Σ decays into a Λ and a photon. Assuming that the kinetic energy of the Λ is negligible, determine the energy of the emitted photon.
As the kinetic energy of Λ is negligible, the kinetic energy of the photon is just the difference of the energy of the masses. Apply E = mc^2 to both Σ and Λ, the difference of mass is the energy of the photon.
109
wavelength and momentum formula
λ = ℎ / p
110
energy and wavelength formula
𝐸 = ℎ𝑓 = ℎ𝑐 /λ
111
energy and momentum formula
𝐸 = ℎ𝑐 / λ =ℎ𝑐 / ℎ/𝑝
𝐸 = 𝑝𝑐
112
is it reasonable to ignore the kinetic energy of Λ ?
We can check this by finding the kinetic energy of Λ. Results shows that it is a lot less than the energy of the photon, so we can ignore it.
113
Hyperons Λ^0 and Σ^±0 were assigned a strangeness quantum number of
S = -1 and their associated K^0 and K^+ mesons were assigned S = +1.
114
Strangeness is conserved in production processes, for example ...
π− + p -> Σ + K+
115
When strange particles decay individually, strangeness is...
not usually conserved
116
The standard model includes the 3 families of particles:
(1) the 6 leptons, which don’t interact via the strong interaction
(2) the 6 quarks, from which all hadrons are made
(3) the particles that mediate the various interactions. These mediators are gluons for the strong interaction among quarks, photons for the electromagnetic interaction, the W± and Z0 particles for the weak interaction. Plus the Higgs boson (needed for non-zero masses).
Note that the standard model does not include gravity.
117
The electroweak unification - unite the electromagnetic and weak interactions
The idea is that at low energies, the electromagnetic and weak interactions behave quite differently due to the mass difference between the photons (which have no mass) and the bosons (which have masses of around 100 GeV/c^2). This difference, though, disappears at high energy, and the two merge into a single interaction.
118
Grand unified theories
Some of these theories predict that protons are not, in fact, stable, but decay, violating the conservation of baryon number. The predictions put the lifetime of the proton at 1028 years. Compare this with the age of the universe, thought to be around 1.5 × 10^10 years. So to observe any such decays, you’d need 6 tonnes of protons to see a decay rate of 1 a day.
119
Supersymmetric theories and TOEs - The theory of everything
The ultimate goal is to combine all four interactions under one theory. However, to do this it seems you may need a space-time continuum with more than 4 dimensions. The extra dimensions are “rolled up” into extremely tiny structure that we ordinarily do not notice.
120
Many-electron atoms
Schrodinger’s equation can be applied to the general atom, however the complexity of the analysis of this is so extreme that it has not been solved exactly for even the helium atom, which has only 2 electrons. The problem arises because each of the Z electrons in a many-electron atom interacts with not only every proton in the nucleus, but also with every other electron. The number of variables is immense.
The simplest approximation is to assume that when an electron moves, it ignores the effects of all other electrons and only feels the influence of the nucleus it orbits, which is taken to be a point charge.
121
The central-field approximation
Here we think of all the electrons together as making up a charge cloud that is, on average, spherically symmetric. We can then think of each individual electron as moving in the total electric field due to the nucleus and this averaged out cloud of all the other electrons.
There is then a corresponding spherically symmetric potential energy function, U(r).
The energy of a state now depends on both n and l, rather than just on n as with hydrogen, due to the change in U(r).
122
restrictions on values of the quantum numbers:
𝑛 ≥ 1
0 ≤ 𝑙 ≤ 𝑛 − 1
|𝑚𝑙| ≤ 𝑙
𝑚𝑠 = ±1/2
123
We might expect that in the ground state of a complex atom, all the electrons should be in this lowest state. If so we would only see ...
gradual changes in physical and chemical properties when we look at the behaviour of atoms with increasing numbers of electrons, Z.
Such gradual changes are not seen though. Instead properties of elements vary widely from one to the next.
124
This is where Wolfgang Pauli came in. His exclusion principle states:
“No two electrons can occupy the same quantum-mechanical state in a given system”
“No two electrons in an atom can have the same values of all four quantum numbers n,l,ml,ms”.
125
the exclusion principle puts a limit on ...
the amount that electron wave functions can overlap
126
For each state ms can be ...
1/2 or -1/2
127
electrons with larger values of n are concentrated at ...
larger distances from the nucleus
128
the exclusion principle forbids multiple occupancy of a state, this means that when an atom has more than 2 electrons...
they can’t all huddle down in the low-energy n = 1
states nearest to the nucleus because there are only
two of these states
129
Each value of n corresponds roughly to a region of space around the nucleus in the form of a
spherical shell
130
States with the same n but different l form
subshells
131
(forming periodic table from exclusion principle)
To obtain the ground state of the atom as a whole we ...
we fill the lowest energy electron states – those closest to the nucleus with the smallest values of n and l – first, and we use successively higher states until all the electrons are in place
132
For hydrogen, the ground state is
1s, what are the other quantum numbers?
the single electron is in a state
n =1, l = 0, ml = 0 and ms = ±1/2.
133
For helium, Z=2, the ground state is 1s^2, where the superscript 2 means that
there are two electrons in the 1s subshell. Both electrons in the 1s states have opposite spins, one has ms = - 1/2 and the other ms = +1/2.
134
Helium is a noble gas meaning,
it has no tendency to gain or lose electrons and it forms no compounds.
135
Lithium, with Z = 3, has three electrons. What are the ground states?
In its ground states, two are in 1s states and one is in a 2s state, so we denote the lithium ground state as
1s^2 2s.
On average, the 2s electron is considerably further from the nucleus than the 1s electrons are.
136
lithium is an alkali metal meaning ...
It forms ionic compounds in which each lithium atom loses an electron and has a valence of +1
137
What is valence?
the property of an element that determines the number of other atoms with which an atom of the element can combine
in an electron, the outermost energy level of an atom helps to form chemical bonds, this outer shell contains the valence electron
138
What is Z (for atoms)?
The atomic number
139
Beryllium with an atomic number of Z = 4. Its ground state is 1s^2 2s^2, with its two valence electrons filling the s subshell of the L shell.
Beryllium is the first of the alkaline
earth elements, forming ionic compounds in which the valence of the atoms is +2.
140
each column in the periodic table represents a...
specific group. And the similarity of the elements in these groups is explained by the similarity in outer-electron configuration
141
All noble gases (helium, neon, …, radon) have...
filled shell or filled shell plus filled p subshell configurations.
142
All alkali metals (lithium, sodium, …, francium) have...
the “noble gas plus 1” pattern
143
All alkali Earth metals (beryllium, magnesium, …, radium) have
the “noble gas plus 2” pattern
144
All halogens (fluorine, chlorine, …, astatine) have...
the “noble gas minus 1” pattern
145
Doppler shift relationship
Relationship between the wavelength λ0 of light measured now from a source receding at a speed
v and the wavelength λ𝑠 measured in the rest frame of the source when it was emitted:
λo = λ𝑠√𝑐+𝑣 / 𝑐−v
146
What is redshift
Wavelengths from receding sources are always shifted toward longer wavelengths – this increase in wavelength is called redshift.
147
The speed or recession is found by...
rearranging the doppler shift formula for v
148
Hubbles law
Analysis of redshifts from many distant galaxies led
Edwin Hubble to a remarkable conclusion: The
speed of recession v of a galaxy is proportional to
its distance r from us.
This relationship is now called Hubble’s Law, and is
expressed by the following equation:
v = Ho r
Where H0 is known as the Hubble constant.
149
Determining this constant has been a key goal
of the Hubble Space Telescope, which can ...
measure distances to galaxies with extreme accuracy.
150
Astronomical distances are often measures in parsecs.
1 parsec = 1 pc = 1/3600 = 1 arcsecond
151
One parsec is the distance at which there is one arcsecond angular separation
between two objects that are 1.5 × 1011m apart, such as
the average distance between the
Earth and the Sun.
152
One parsec is equal to ... lightyears
3.26
153
The cosmological principle
on average the universe looks the same from all locations (as distant galaxies appear to be receding from us in all directions)
matter is distributed uniformly
154
based off the cosmological principle, it appears that the Hubble constant is constant in …
space although not necessarily constant in time, and
the laws of physics are the same everywhere
155
Big bang theory - what is it
the theory states that around 14 billion years ago all matter and energy in the universe was at a point of infinite density and temperature. It then expanded rapidly. Eventually stars, galaxies and planets formed. This expansion was the beginning of time and continues to this day.
156
When did the expansion of the universe begin?
v = H0r and r = vt
𝑡 = 𝑟 / 𝑣 = 𝑟 / 𝐻0𝑟 = 1/𝐻0 = 1.4 × 10^10 years
So this law predicts an age of the universe at 14 billion years
157
The expanding space - what is it
An alternative interpretation of the way the universe is expanding distance is that distance objects increase in distance which comes from the expansion of space itself, and everything in intergalactic space, including the wavelengths of light travelling to us from distant sources.
158
How to understand the expanding space
Consider a two-dimensional world. You could plot
any position on this two-dimensional plane in terms of x and y co-ordinates, and if it extended indefinitely then we could describe this plane as being indefinite or unbounded. No matter how far you travelled in any particular direction you would never reach an edge or boundary.
159
Expanding space - expanding balloon analogy
Suppose you have two coordinates on the surface of a balloon. As the balloon gets bigger and bigger the radius gets bigger but the specific coordinates of a point on the surface don’t change. The distance between two points, though, would get bigger. Further, as the radius increase, so the rate of change of distance between two points – their recession speed – is proportional to their separation, just as in the Hubble Law.
You can see that whilst the quantity R isn’t one of the co-ordinates giving the position of a point on the balloon’s surface, it does play an important role in the discussion of distance. It is the radius of curvature of our 2-dimensional space, and it
is also a varying scale factor that changes as this 2-dimensional universe expands.
160
Generalising the expanding space to 3 dimensions
Generalising to 3 dimensions we have to picture our 3-dimensional universe as being embedded in a space with 4 or more dimensions.
Our real 3-dimensional space is not Cartesian – to describe its characteristics in any small region requires at least one additional parameter – the curvature of space.
This is analogous to the radius of our balloon. In a sense the scale factor, which we’ll still call R, describes the size of the universe, just as the radius of the sphere described the size of our 2 dimensional spherical universe.
161
Any length that is measured in intergalactic space is proportional to R, so...
the wavelength of light travelling to us from a distant galaxy increases along with every other dimension as the universe expands.
This is given by:
λo / λ = 𝑅o / R
The zero subscripts refer to the values of wavelength and scale factor today – just as H0 refers to the Hubble constant’s value today. The R and λ without subscripts are the values that these quantities take at any time in the past, present or future.
162
Cosmological redshift
This increase of wavelength with time as the scale factor increases in our expanding universe is called the cosmological redshift. The farther away the object is, the longer its light takes to get to us, and the greater the change in R and λ.
163
Critical density - Whether the universe continues to expand indefinitely depends on the average
density of the matter in the universe. If matter is relatively dense ...
then there is a lot of gravitational attraction to slow and eventually stop the expansion and make the
universe contract again. If not, the expansion should continue forever. There is therefore a critical density, ρ𝑐, which is needed to just stop the expansion continuing indefinitely
164
The total energy when a projectile of mass m and speed v is at a distance r from the centre of the Earth, which has mass mE is given by …
Total energy = Kinematic energy + Potential energy
𝐸 = 𝐾 + 𝑈
𝐸 = 1/2𝑚𝑣^2 − 𝐺𝑚𝑚𝐸 / 𝑟
where G is the gravitational constant of the universe.
165
Spherically symmetric implies
same gravitational potential (for two objects if M and m were both points)
166
For total energy of our galaxy, the different conditions of E imply that:
-If E is positive, then our galaxy has enough energy to escape from the gravitational attraction of the mass M inside the sphere. The universe should expand forever.
-If E is negative, then our galaxy cannot escape and the universe should eventually collapse.
We can find the critical density (pc) at the point where E = 0.
167
What are two main ways of working out the density of matter in the universe
1 - Count the number of galaxies in a patch of sky, then use the average mass of a star and the average number of stars in an average galaxy. This gives you
an estimate of the average density of the matter we can “see” – luminous matter aka matter that emits electromagnetic radiation
2 - Study the motion of galaxies within galaxy clusters (by monitoring redshift) to get an idea of their speeds. These speeds are related to the gravitational
force exerted on each galaxy by the other members of the cluster, which in turn is depend on the cluster’s mass. And from this we get the average density of all the matter in the cluster, whether it emits in the electromagnetic spectrum or not.
168
where does the idea of dark matter come from
Studies of the type (2) show that the average density of all matter in the universe is around 26 % of the critical density. However, the average density of luminous matter is only 4 %. In other words, the majority of matter in the universe does not emit
electromagnetic radiation – we call this dark matter.
169
What is important about dark matter
For Dark matter it appears that its average density is less than the critical level needed to prevent continuing expansion. Gravitational attraction will slow that expansion, but never enough to stop it.
170
The expansion of the universe is in fact speeding up not slowing down. We know this because
If the expansion of the universe was slowing, then the expansion must have been happening faster in this distant past. We would expect very distant galaxies to have greater redshifts than predicted by the Hubble Law. However, measurements made show that the redshifts of the most distance galaxies are actually smaller than the Hubble Law prediction. This implies that the expansion of the universe is in fact speeding up , not slowing down.
171
How can the universe be expanding if gravity should be acting to slow it down?
The current theory states that space is suffused with a kind of energy that has no gravitational effect and emits no electromagnetic radiation. Rather it acts as a kind of “anti-gravity” producing a universal repulsion. This invisible, immaterial energy is called dark energy. We have no idea what this is.
172
The energy density of this dark energy is ...
the amount of energy per cubic metre
74% of the critical density multiplied by c^2
173
The average density of all matter has been found to be...
26% of the critical density
From E = mc^2, the average density of matter in the universe is therefore 0.26ρ𝑐𝑐^2
174
The energy density of dark energy is nearly 3 times greater than that of matter this means that...
The expansion of the universe will continue to accelerate, the expansion will never stop and the universe will never contract.
175
How do particles become progressively uncoupled?
Over time, our universe has become larger, less dense and colder. The average particle energy has therefore decreased, and as this happened so the basic interactions of those particles became progressively uncoupled.
176
It is assumed that at sufficiently high energies and short distances, gravitation unites with the others. The distance at which this happens is called...
the Planck length, lp
(in formula sheet)
177
The plank time is
the time required for light to travel planks length, found by tp = lp / c
178
If we go back in time, 10^-43 is as far back as we can go because
before then all interactions were united, and we have no theories to describe how the universe behaved then. Meaning that, earlier than the Plank time, when the universe was smaller than the Plank length.
