Unit 4.3 - Orbits and the wider universe Flashcards
What do we apply in this unit?
What we learned about gravitational fields to the behaviour of objects in orbit
What happens if the initial speed of a cannonball is increased? Why?
The path becomes longer since it travels further
If a cannonball was fired from the top of a mountain when would it be in orbit? Explain
If it was fired at a certain speed at a certain speed, which is when it’s travelling long the equipotential since no work is done nd so no energy is lost
Is there friction in the atmosphere?
No
When is something in orbit? Explain
When it’s travelling along the equipotential since no work is done and so no energy is lost
What would you feel if you were inside a cannonball fired in space? Explain
You would constantly be falling but in addition the air around you would be falling at what same speed - you would feel an absence from a gravitational field and so feel weightless
What did Kepler based his work on?
The observations of Tycho Brahe
What are all of Kepler’s laws - why and what does this mean?
Empirical
He described what he saw based on observation
They weren’t agreed physically, only agreed with observation
How did Newton explain Keplers laws?
Using his laws of motion and the law of general gravitation
When was Kepler’s 3rd law no longer empirical?
When newton explained his laws using his laws of motion and the law of general gravitation
Kepler’s first law
The planets move in elliptical orbits with the sun at one focus
Kepler’s second law
The vector radius (an imaginary line connecting the planet and the sun) sweeps out an equal area in an equal time
Kepler’s third law
The square of the period of orbit of a planet is directly proportional to the cube of its average distance from the sun
T^2 ∝ r^3
How do we know that Kepler’s 1st law is true?
The sum of the distance between 2 foci is always constant
How would we describe an eclipse if the 2 foci are far away?
Eccentric (stretched out)
When do planets travel fastest in their orbit and why?
When closer to the sun
Angular momentum is conserved and is constant
Explain why planets travel faster when closer to the sun in terms of Kepler’s 2nd law
Close to sun = small distance to sun but moves faster so larger radius to make up for it
What shaped orbits are studied in this course?
Circular
Why do we use circular orbits in this course?
We will have a constant velocity since planets won’t get further/closer to the sun
What kind of motion do we have if we only study circular orbits?
Circular motion
Explain why Kepler’s first 2 laws are trivial for circular orbits?
An ellipse has two foci - a circle has both at the centre (where the mass causing the orbit lies)
The orbiting body moves at a constant speed and therefore the area swept out in a given time
Why were Kepler’s laws empirical?
Since they described the motions but didn’t give a theoretical underpinning to them
Why were Newton’s theories required for Kepler?
To explain why the planets moved in the way described by Kepler
How did Newton derive an equation for Kepler’s third law?
Using his universal law of gravitation
Derive Kepler’s third law
This is a derivation for circular orbits (where the eccentricity of the ellipse is zero)
If a planet orbits in a circle, there must be a centripetal force - the force of gravity
F = M2v^2/r
This force is provided by the gravitational attraction of M1 on M2
F = GM1M2/r^2
Therefore, Fgrav=Fcent
M2v^2/r = GM1M2/r^2
Where m1 is the mass of the sun and m2 is the mass of the planet
v^2 = GM1/r
The distance travelled in one orbit is the circumference of the orbit (2pir) and so the speed is the expression for v in circular motion:
v = 2pir/T
Placing this in the previous equation
(2pir/T)^2 = GM1/r
Which gives:
4pi^2r^2/T^2 = GM1/r
And by rearranging we get
T^2 = 4pi^2/GM1 x r^3
Which is Kepler’s third law
How do we get to T^2 ∝ r^3 from T^2 = 4pi^2/GM1 x r^3?
Since 4pi^2/GM1 is constant
T^2 ∝ r^3 meaning
The square of the period of orbit is proportional to the cube of the radius of the orbit
What is Kepler’s law proved by?
The laws of motion and gravitation
What does Kepler’s third law show?
That the further away a planet is, the longer its period of orbit (but not in a linear way)
What was Kepler’s law originally discovered for but what does it work equally well for?
Originally for the motion of the planets
Works equally well for all satellite motion
When using Kepler’s third law for calculations about artificial satellites around the earth, what do we need to remember?
The M in the constant is the mass of the earth in that case (or the mass of whichever body is being orbited)
Only force acting on a single planet orbiting a star
Gravity
What is gravitational force given by?
F = GM1M2/r^2
How do geostationary satellites orbit?
So that they are above the same point on earth at all times
Orbital period of geostationary satellites
24 hours
How do geostationary satellites orbit?
Directly above the equator
What are geostationary satellites used for?
GPS and communication
How do we work out a satellites height above the earth surface?
r - r(earth)
Can work out r from rearranging T^2 = 4pi^2/GM x r^3 and get r^3 as the subject and remember that M is the mass of the earth and T is 24 hours
How do we work out how elliptical an orbit is?
More elliptical = bigger difference between the maximum and minimum height
How do we know if an orbit is more elliptical?
Bigger difference between the maximum and minimum height
How do we work out the mass of an orbited body?
Use Kepler’s law but rearrange to have M as the subject
M = 4pi^2/GT^2 x r^3
When do include the radius of a planet in a calculation?
If its r is given in the question
Centre of gravity
The point where all the weight of an object can be considered to be acting
The sum of the moments about that point is zero when the system is in equilibrium
Why doesn’t make sense to use the centre of gravity when discussing stars?
Since the field strength varies with distance
How do a system of masses orbit?
They orbit about a common point rather than on around the other
Centre of mass
A common point which a system of masses orbit around rather than around each other
When does the centre of mass lie within one of the masses? Give an example
When there is a great difference between the masses
e.g - between the sun and earth
Why does it seem that the earth orbits the sun?
Because there is a great difference between the masses and the centre of mass lies within the sun
When is it okay to assume that a planet is orbiting the sun?
When the sun is a lot bigger than the planet so it’s centre of mass is within the sun
When will the centre of mass between two masses lie quite far from the centre of each individual mass?
When the masses are nearly equal
Where will the centre of mass lie when the masses of a system are nearly equal?
Quite far from the centre of each individual mass
In what type of star systems is it true that the centre of mass is quite far from the centre of each individual mass?
Binary star systems
Where will the centre of mass be if the masses of 2 stars are equal in a binary star system?
Halfway between both stars
Equation for working out where a centre of mass is + explanation of symbols
r1 = M2/M1 + M2 x d
r1 = distance from mass M1
d = separation of the masses
Derive the equation for working out the location of a centre of mass
Considering moments:
Mg x r1 = M2g x r2
d = r1 + r2
r2 = d - r1
M1 x r1 = M2 x r2
M1r1 = M2(d-r1)
M1r1 = M2d - M2r1
M1r1 + M2r1 = M2d
r1(M1 + M2) = M2d
r1 = M2/M1 + M2 x d
Binary systems
Two stars orbit round a common centre of mass
When do we modify Kepler’s third law when working out centres of mass?
When considering supermassive orbiting orbits where the centre of mass lies outside the radius of the larger body (e.g - binary star systems, extremely large planets)
In a binary system where we have 2 big masses, what is the same and what is different?
Same —> period of orbit around the centre of mass
Different —> radial velocities due to their different distances to the centre of mass
What are we ignoring with Kepler’s third law usually and why?
The mass of the planet
Since the mass of the sun is so much bigger
How is Kepler’s third law modified hen we have 2 big masses?
The total mass is not just M1 but rather (M1 + M2) and the co-radius of orbit is given by d (separation)
T^2 = (4pi^2/G(M1+M2)) xd^3 (which is in the data book)
When do we use T^2 = (4pi^2/G(M1+M2)) xd^3 as Kepler’s law?
When we have 2 big masses
Explain the Doppler effect in terms of sound waves with an example
When an ambulance approaches and passes you, the pitch of the siren changes
Describe the wavelength of higher pitches
Shorter wavelengths
How does the Doppler effect occur with stars?
The wavelength of light waves changes as a radiating body (a star) moves towards or away from the observer
How does the wavelength of light form a star change as it moves towards the observer?
The wavelength is decreased by Δλ
Why is the wavelength of light decreased by Δλ as a star moves towards the observer?
Because as it moves towards you, the light waves have a higher frequency
What do different frequency light waves cause?
Different colours
What does the colour change of stars depend on?
Their relative motion to us
What is used as the reference wavelength λ in Doppler effect equations?
The hydrogen alpha line in the absorption spectrum of the stars (this is a characteristic line)
When a star is moving away from us, describe the:
Δλ
Whether the wavelength has increased or decreased
The colour shift
Positive Δλ
The shift has increased the wavelength
Redshift
When a star is moving towards us, describe the:
Δλ
Whether the wavelength has increased or decreased
The colour shift
Negative
The shift has decreased the wavelength
Blueshift
What is v in the Doppler effect equation?
The radial velocity (the linear velocity vector)
How is the radial velocity (the linear velocity vector) of a star calculated?
Δλ/λ = v/c
What does a bigger shift mean in terms of velocity?
Faster velocity
What’s the reason for the changing variable motion of a star?
The star is in orbit around another object which causes the relative motion to us to vary
How do we work out time periods (in terms of the Doppler shift)?
Maximum red-shift to maximum red-shift
(Or the same with blue-shift)
What can we work out for a star if we have V and T?
Its mass
When working out the radial velocity of a star, what do we also need to consider and why?
That there is a recession velocity since the system itself is moving away from us (since the Big Bang)
Binary star
A variable star. It’s formed of two stars in mutual orbit. When the dimmer star is in front of the brighter star, the intensity is at a minimum. When they are side by side, the intensity is at a maximum.
What can we use the Doppler shift to work out?
The relative speed of stars in a system
Describe frequency if we have a higher wavelength
Lower
Frequency for blueshift
Higher frequency
Frequency for redshift
Lower frequency
What does a bigger shift in frequency mean for a star?
Faster
Which astronomer found evidence for the hypothetical type of matter now called dark matter?
Vera Rubin
When was dark matter discovered?
In the 1970s
What does the centre of a rotation curve represent?
The core region of our solar system (i.e - the sun) or the core region of a spiral galaxy
Where is most of the mass of our solar system concentrated?
In the sun
Where is the most mass in spiral galaxies and why?
The core region
The core region has the highest concentration of visible stars
In which direction is the gravity of a spiral galaxy concentrated towards and why?
The centre
This is where most of its mass is concentrated
Described the expected orbital speed of a star the further away it is from the centre of either a solar system or a spiral galaxy
Slower expected orbital speed the further out it is
Why is a star’s orbital speed expected to be slower when it’s further from the centre of a solar system or spiral galaxy?
Most of the mass is at the centre of the solar system or spiral galaxy (the sun, or the most visible stars) and this means that the gravity is concentrated towards this centre
How did the stars at the edges of galaxies more compared to what Vera Rubin expected?
They moved faster than expected
What did gravity calculations come out with for stars at the edges of galaxies? Why?
Showed that he stars should have been moving more slowly
The gravity calculations used only the visible matter in the galaxies
What made Vera Rubin think that stars at the edges of galaxies should have been moving more slowly?
Gravity calculations that only used the visible matter in the galaxies
What was the predicted cause of the discrepancy between the expected and actual speed of stars in galaxies?
Unseen dark matter
What is it causing the higher than expected speeds of the stars at the edges of spiral galaxies?
Extra mass from dark matter
Who has previously predicted the existence of invisible matter in the 1930s and how?
The astronomer Fritz Zwicky
Following his observations of the Coma galaxy cluster
How did Vera Rubin look at the velocity of the outermost visible stars in the andromeda galaxy?
Used the Doppler effect
Why is dark matter called this?
It doesn’t reflect light
Which forces doesn’t dark matter interact with and which does it interact with and how do we know this?
Doesn’t interact with e.m or the strong force
Interacts with the gravitational force (that’s how we know about it)
Why were neutrinos rejected as a candidate for dark matter?
There aren’t enough of them in the galaxy for this to be plausible
Another candidate for dark matter?
WIMPs (weakly interacting massive particles)
What is the main candidate for dark matter?
Higgs boson
What was suspected to be Higgs boson when it was discovered and turned out to be exactly that? When?
4th July 2012, the discovery of a new particle with a mass between 125 and 127 GeV/c^2 was announced
Where was Higgs boson discovered?
In the Hadron collider
Explain how, possibly, dark matter forms
- Two quarks approach each other in a collider. The quarks are in a hadron.
- In the interaction, two W bosons are exchanged
- The W bosons annihilate each other producing a Higgs boson (observed in 2012)
- Possibly, the Higgs could decay after a very short time to a pair of dark matter particles
What is the estimated mass of stars based on in calculations?
Observations of the visible radiation given off by the galaxy and knowing the mass of stars of a particular brightness
What do the measured velocities in a galaxy depend on?
The mass of the galaxy
Explain why dark matter is thought to be responsible for the difference in measured and estimated velocities in a galaxy
- the measured velocity is greater than the estimated velocity
- the estimated mass is based on observations of the visible radiation given off by the galaxy and knowing the mass of stars of a particular brightness
-since the measured velocity depends on the mass of the galaxy, and that this is greater, hence there must be more “unseen” mass in the galaxy which is themed dark matter
How do we calculate a maximum and minimum red shift if we also have a recession velocity?
Work out the minimum velocity (subtracting the rotational from the recession velocity)
Work out the maximum velocity (adding the rotational to the recession velocity)
Use the Doppler shift equation for both of these velocities
Maximum will be the wavelength given + the highest shift
Minimum will be the wavelength given + the smallest shift
Who demonstrated that there were other galaxies and when?
Edwin Hubble
1925
What did Edwin Hubble demonstrate?
That there were other galaxies aside from the Milky Way - thousands, perhaps millions of them, and many of them huge distances away from our own
Are other galaxies moving away from us or towards us?
Away from us
How did Hubble work out that other galaxies in the universe are moving away from us?
With improves telescopes, Hubble notices that the light coming from these galaxies was shifted a little towards the red end of the spectrum due to the Doppler effect (known as “redshift”) which indicated that the galaxies were moving away from us
Which shift proved that galaxies are moving away from us?
The redshift
How did Hubble measure the Doppler effect in other galaxies?
By measuring the hydrogen alpha line in all of the galaxies
How many electrons come from a star and why?
An infinite number
Because a star is so dense
How are galaxies and clusters of galaxies flying apart from each other and how was this figured out?
At great speeds
After a detailed analysis of the redshifts of a special class of stars called Cepheids (which have specific properties making them useful as “standard candles” or distance markers)
What specific properties to Cepheid stars have?
Properties that make them Seoul as “standard candles” or distance markers
What type of stars have specific properties that make them useful as “standard candles” or distance markers for working out redshifts?
Cepheids
What was concluded from the fact that galaxies and clusters of galaxies are flying apart from each other at great speed?
The universe is definitely growing in size
What colour are all of the galaxies we see and why?
Slightly red
Due to the redshift
What did Hubble plot on a graph to give a straight line grandient of the Hubble constant?
The velocity at which the galaxies were receding against distance
What is the speed at which every galaxy is rushing away from us in our expanding universe in direct proportion to?
It’s distance
Hubbles law
The speed a which galaxies are rushing away from us in our expanding universe is in direct proportion to its distance
Explain Hubble’s law
Since the speed at which galaxies are rushing away from us is in direct proportion to its distance, a galaxy that is twice as far away as another is receding twice as fast, and one ten times as far away is receding ten times as fast etc
Hubble’s law in an equation
v = H0D
Define the symbols in Hubble’s law equation
v = H0D
V = velocity of recession
D = distance of the galaxy from the observer
H0 = Hubble constant
The exact value of which constant has been subject to much controversy?
The Hubble constant
Value of the Hubble constant from Hubble’s initial estimates v.s using the Hubble telescope and WMAP probe
500kms-1Mpc-1
72kms-1Mpc-1
What is the parsec?
A unit of distance
Why is the Hubble constant a hard parameter to measure?
There’s a lot of uncertainties
Why is the Hubble constant not technically a constant, and what is it really?
It’s technically a parameter, because it will actually change over long periods of time. It’s only constant within our time scale.
How do we convert from Kms-1Mpc-1 to s-1 (the Hubble constant unit)?
Get the top and bottom in metres so that we can cancel m out to only leave s-1
Top = value and 10^3 to get rid of km
Bottom = 1 parsec value AND remember to include the “Mega” multiplier, so it will be 3.09x10^22
Does the expansion of space overcome the gravitational force? How do we know?
No
It’s only the distance between galaxies increasing
What word is used to explain the expansion of space?
“Metric expansion” of space
What type of effect is the expansion of space? Explain
A broad-brush effect
Individual galaxies themselves are not expanding, but the clusters of galaxies into which the matter of the universe has become divided are becoming more widely separated and more thinly spread throughout space
Explain how the galaxy is actually expanding
It’s not expanding “outwards” into pre-existing space, space itself is expanding, defined by the relative separation of parts of the universe
What’s the relationship between the distance between galaxies and how fast they move apart?
The larger the distance between galaxies, the faster they move apart
How fast do galaxies move away from each other when they’re very far apart?
Faster than the speed of light
How can we imagine the universe expanding?
Tiny dots painted on a balloon to represent galaxies, and as the balloon expands, the distance between the dots increases and the further apart the dots, the faster they move apart
Why are we not necessarily at the centre of the universe even though it might appear like that?
In the way the universe expands (space itself is expanding, defined by the relative separation of parts of the universe), the universe continues to look more or less the same from every galaxy, so the fact that we see all the galaxies receding from us does not necessarily mean that we are at the very centre of the universe: observers in all other galaxies would also see all the other galaxies flying away according to the same law, and the pattern of galactic dispersal would appear very much the same form anywhere in the cosmos
What appears the same form anywhere in the cosmos and why?
The pattern of galactic dispersal
Because observers in all galaxies would see the other galaxies flying away form each other according to the same law
Is the way in which the universe is expanding the same as the Doppler shift?
Not quite, but we still use the same equation
What is the Doppler effect due to?
The relative motion between the source and the observer
What is the cosmological red shift due to?
Space itself expanding
What proves that the universe is expanding - the Doppler effect or the cosmological red-shift? Explain
The cosmological red-shift
The Doppler effect is due to the relative motion between the source and the observer
The cosmological red shift is due to the universe itself expanding
Best forms of evidence for the Big Bang
CMBR (cosmic microwave background radiation)
The uniformity of the universe (hydrogen and helium everywhere)
What can be used to calculate the age of the universe?
The Hubble constant
Explain how the Hubble constant can be used to calculate the age of the universe
Hubble equation: V = H0D
Time as subject: D/V = t
D/V = 1/H0
t = 1/H0
Convert seconds into years by dividing by 60^2, 24 and then 365
How do we convert from seconds to years?
Divide by 60^2, 24 and then 365
Redshift equation + explain
v = zc
z = redshift
What is the fate of the universe determined by?
The struggle between the momentum of expansion and the pull of gravity
What is the rate of expansion of a universe determined by?
The Hubble constant
What does the strength of gravity depend on?
The density and pressure of the matter in the universe
In what situation is the fate of the universe governed by the density?
If the pressure of the matter is low (as is the case with most forms of matter of which we know)
Describe the pressure of most forms of matter of which we know
Low
What does most of the matter of which we know having a low density mean we can figure out and why?
The fate of the universe evince the fate of the universe will be governed by the density in this situation
In which situation will the universe expand forever?
If the density of the universe is less than the “critical density”, which is proportional to the square of the Hubble constant
Critical density
Square of the Hubble constant
What will the universe do if the density of the universe is less than the critical density?
It will expand forever
What happens if the density of the universe is greater than the critical density?
Gravity will eventually win and the universe will collapse back in on itself, the so called “Big Crunch”
Under which circumstance would the universe collapse back in on itself for the “Big Crunch”?
If the density of the universe is greater than the critical density
For a flat universe, when does the radial (recessional) velocity of galaxies become zero? Explain
When the time is infinite i.e - the radial velocity of galaxies ie equal to the escape velocity of the universe
In what type of universe is the radial (recessional) velocity of galaxies become zero when time is infinite?
Flat
What is the radial velocity of galaxies the same as in flat universes?
The escape velocity of the universe
What is the geometry of our universe predicted to be?
Flat
Why is the geometry of our universe predicted to be flat?
The simplest version of the inflationary theory (an extension of the Big Bang theory) predicts that the density of the universe is every close to the critical density and that the geometry of the universe is flat like a sheet of paper
Describe the density of our universe?
Very close to the critical density
Which theory proves that our universe is flat?
The inflationary theory (an extension of the Big Bang theory)
What are the possible space curvatures of the universe?
Closed
Flat
Open
What does the density of a universe also determine as well as its fate?
Its geometry
In which situation would a universe be closed?
If the density of the universe is high enough and exceeds he critical density
Describe a closed universe
Positively curved like the surface of a sphere
What does a closed universe imply in terms of photons? Explain
This implies that initially parallel photon paths converge slowly, eventually cross, and return back to their starting point (if the universe lasts long enough) - a straight path around the universe would eventually reach back to the start
Under which situation would photons return to their starting point in a closed universe
If the universe lasted long enough
Under which conditions is a universe open?
If the density of the universe is low enough and is less than the critical density, then the geometry of space is open (infinite)
Describe an open universe
Negatively curved like the surface of a sphere
Under which conditions would a universe be flat?
If the density of the universe exactly equals the critical density (density between closed and open), then the geometry of the universe is flat like a sheet of paper
Which universes are infinite?
Open and flat
Derive the critical density equation
M = mass of the universe
m = mass of a distant object (e.g - galaxy)
V = velocity that the galaxy is receding at
D = distance between M and m
If the universe is flat and the total density of the universe is exactly the same as the critical density, then the current KE of a distant object will reach zero at time infinity and become wholly PE
In other words, as the universes age approaches infinity, all the KE of the galaxies approaches zero, what this means is that the PE has increased to its maximum value
1/2mv^2 = GMm/r ——> 1/2v^2 = GM/r
From the Hubble equation v = H0D and also r=D
1/2(H0D)^2 = GM/RD
From density equation p = m/v —> M=pV
1/2(H0D)^2 = GpV/D
From volume of a sphere formula V = 4/3piD^3
1/2(H0D)^2 = Gp4piD^3/3D
1/2H0^2D^2 = Gp4piD^2/3 ——> 1/2H0^2 = Gp4pi/3
p = 3H0^2/8piG
Describe the KE of a distant object if the universe is flat and the total density of the universe is exactly the same as the critical density
It will reach zero at time infinity and become wholly PE
What happens to the KE of galaxies as the universe’s age approaches infinity? What does this mean for the PE?
All the KE of the galaxies approaches zero
The PE has increased to its maximum value
What can the WMAP spacecraft can measure?
The basic parameters of the Big Bang theory, including the geometry of the universe
Describe the brightest microwave background fluctuation (“spots”) if the universe were flat
Would be about one degree across
Describe the brightest microwave background fluctuation (or “spots”) if the universe were open
Would be less than one degree across
What would be less than one degree across if the universe were closed?
The brightest microwave background fluctuation (or “spots”)
What can measure the basic parameters of the Big Bang theory including the geometry of the universe?
The WMAP spacecraft
Describe the brightest microwave background fluctuation (or “spots”) if the universe were closed
Greater than one degree across
What type of experiments have shown that the brightest spots are about 1 degree across?
Measurements by a number of ground-based and balloon-based experiments, including MAT/TOCO, boomerang, maxima and DASI
How far across are the brightest spots in the universe?
About 1 degree across
What do we think the shape of the universe is?
Flat
Accuracy of the claim that the universe is flat prior to WMAP results
15%
What has WMAP confirmed?
That the universe is flat with very high accuracy and precision
To what level of accuracy do we know that the universe is flat? Why?
WMAP has confirmed it, 0.4% margin of error
What does the universe being flat suggest?
That the universe is infinite
Why can we only observe a finite volume of the universe?
Since the universe has a finite age
What is the only thing we can truly conclude about the universe?
The universe is much larger than the volume we can directly observe
How do we prove Kepler’s third law with two sets of data?
Use r^3/T^2 for both sets and get the same answer
What is it we look at for stars when working out Doppler shifts?
The stars spectral line
+ measure the Doppler shift in this spectral line
Describe the velocity in the Hubble equation
Constant
What is the relationship between red shift and velocity?
Directly proportional
What do we need to refer to when explaining how we can work out the age of the universe using Hubble’s constant?
That we have a constant velocity
What do we always need to talk about when discussing dark matter?
Mass
What was used to work out that dark matter existed?
Large telescope and spectrometer
What do we need to do when we have a question about something that isn’t electrons (e.g - hydrogen)?
Use u as mass not e
What do we need to remember to state when using Hubble’s constant to work out the age of the universe?
That it’s a constant velocity
What are orbital speeds measured using?
The Doppler effect
