Answer key · Unit 27

Galaxies and cosmology.

Each lesson’s practice questions, their correct answers and the mistakes they spot. Answers read off the Scene use the same setting the learner sees.

Misconception catalogue

Answer key 3

Lesson 1 of 3 · About 20 minutes

Galaxies

Islands of billions of stars: why a spiral galaxy’s arms are blue and why they don’t wind up, what an elliptical galaxy’s colour tells, and how long the Sun takes to go round the Milky Way.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Islands of stars
Practice · Choose
Question
In a spiral galaxy the middle is yellowish and the arms bluish. Why are the arms blue?

Correct answer
- Correct New stars are born in the arms, and the blue ones, hot and heavy, burn out before they leave the arm
Setup, choices and feedback
The Scene at this step
Galaxy type
Spiral
Spiral arms
2
Arm winding
4.2 radians
Show the dark matter
Off
Choices
Shown in a random order.
- Correct New stars are born in the arms, and the blue ones, hot and heavy, burn out before they leave the arm
- Wrong The arms are moving towards us, so their light is shifted to blue
  Feedback
  Such a shift is far too small to change a colour: at a few hundred kilometres a second the wavelength changes by less than one per cent, and half the arms are moving away.
- Wrong The dust in the arms tints the light blue
  Feedback
  Dust actually reddens light passing through it, like the setting Sun. The dark lanes in the arms are dust.
Feedback for a right answer
The arms are blue because they are young: blue stars do not live long enough to leave them.

Worked solution
1. In the Stars and the universe Unit you saw that a heavy star is hotter, and so bluer, and that it lives only millions of years.
2. In the arms gas is squeezed, and new stars are born there, heavy blue ones among them.
3. The blue stars die before they have time to leave the arm, so they mark it out. The middle holds mostly old stars, lighter and redder.
Step 2 of 5
Open this step
Four shapes
Practice · Choose
Question
Galaxy type is now Elliptical: a smooth cloud of stars with no arms, yellow-orange in colour. What does that tell you about it?

Correct answer
- Correct It has almost no gas left, so hardly any new stars are born, and most of its stars are old
Setup, choices and feedback
The Scene at this step
Galaxy type
Elliptical
Spiral arms
2
Arm winding
4.2 radians
Show the dark matter
Off
Choices
Shown in a random order.
- Correct It has almost no gas left, so hardly any new stars are born, and most of its stars are old
- Wrong Its stars are especially hot
  Feedback
  Hot stars glow blue-white. Yellow-orange marks fairly cool stars, like the Sun and cooler.
- Wrong It is a young galaxy that has not yet grown arms
  Feedback
  The opposite: big elliptical galaxies mostly form from mergers of galaxies, and they are among the oldest in the universe.
Feedback for a right answer
A “red and dead” galaxy: old stars, with no gas for new ones.

Worked solution
1. The colour of a population of stars tells its age: heavy blue stars die fast, and with no new stars, the yellow and red ones are left, living billions of years.
2. An elliptical galaxy has almost no cold gas, the raw material of new stars.
3. So it is yellow-orange, and its stars move on orbits in every direction, with no disc and no arms.
Step 3 of 5
Open this step
Arms that don’t wind up
Practice · Choose
Question
Stars near the centre complete a lap faster than distant ones. So why don’t the arms wind up and vanish after a few turns?

Correct answer
- Correct An arm is a pattern, not a structure: a region where stars and gas crowd together as they pass, like a traffic jam that stays put while cars drive through
Setup, choices and feedback
The Scene at this step
Galaxy type
Spiral
Spiral arms
2
Arm winding
4.2 radians
Show the dark matter
Off
Choices
Shown in a random order.
- Correct An arm is a pattern, not a structure: a region where stars and gas crowd together as they pass, like a traffic jam that stays put while cars drive through
- Wrong The whole galaxy turns as one body, so the arms keep their shape
  Misconception: Thinks a galaxy turns like a solid wheel, every star taking the same time to go round.
  Feedback from the catalogue
  A galaxy is not a rigid wheel. Each star circles the centre on its own orbit, at roughly the same speed at every distance. An inner orbit is shorter, so a star near the centre completes a lap in far less time than a distant one. If the arms were fixed chains of stars, they would wind round the centre and vanish after a few turns.
- Wrong Galaxies are too young for their arms to have wound up yet
  Feedback
  The Sun has gone round the centre of the Milky Way about twenty times since it formed. Arms of fixed stars would have wound up long ago.
Feedback for a right answer
A density wave: the arm stays, and the stars pass through it.

Worked solution
1. If an arm were a fixed chain of stars, its inner part would pull ahead of its outer part, and the arm would wind up like a clock spring.
2. In density-wave theory, an arm is a crowded region that turns more slowly than the stars. The stars and gas pass into it and out again.
3. In the squeeze, gas collapses into new stars, so the arm glows, though the stars in it keep changing.
Slide Arm winding from loose to tight, 2.5 to 7.5 radians: real galaxies have both kinds.
Step 4 of 5
Open this step
The galactic year
Practice · Calculate
Question
The Sun is about $26{,}000$ ⁠ light-years from the centre of the Milky Way and goes round it at about 230 km/s. How many years does one lap take? A light-year is $9.46 \times 10^{15}\ \mathrm{m}$ ⁠, and a year has $3.16 \times 10^7$ ⁠ seconds. (You can type $2 \times 10^8$ ⁠ as 2e8.)

Correct answer
213000000 yr ± 3%
Setup, choices and feedback
The Scene at this step
Galaxy type
Barred spiral
Spiral arms
2
Arm winding
4.2 radians
Show the dark matter
Off
Accepted within
± 3%
A fixed value in the question
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
About $2.1 \times 10^8$ ⁠ years: the Sun has gone round the Galaxy only about twenty times since it was born.

Hint
$T = \dfrac{2\pi r}{v}$ ⁠, with $r$ ⁠ in metres and $v$ ⁠ in metres per second.

Worked solution
1. $r = 26{,}000 \cdot 9.46 \times 10^{15} \approx 2.46 \times 10^{20}\ \mathrm{m}$ ⁠.
2. $T = \dfrac{2\pi r}{v} = \dfrac{2\pi \cdot 2.46 \times 10^{20}}{2.30 \times 10^5} \approx 6.7 \times 10^{15}\ \mathrm{s}$ ⁠.
3. $\dfrac{6.7 \times 10^{15}}{3.16 \times 10^7} \approx 2.1 \times 10^8$ ⁠ years.
A “galactic year” of about $210$ ⁠ million years. One lap ago, the first dinosaurs were just appearing.
Step 5 of 5
Open this step
Too fast at the edge
Practice · Choose
Question
Show the dark matter is On: the rotation curve shows stars and gas at the galaxy’s edge going round about as fast as the inner stars, although nearly all the light comes from the middle. What follows?

Correct answer
- Correct There is a lot of mass that gives no light, spread in a wide halo round the galaxy
Setup, choices and feedback
The Scene at this step
Galaxy type
Spiral
Spiral arms
2
Arm winding
4.2 radians
Show the dark matter
On
Choices
Shown in a random order.
- Correct There is a lot of mass that gives no light, spread in a wide halo round the galaxy
- Wrong The stars at the edge are heavier, so they move faster
  Misconception: Thinks a star’s orbital speed depends on its own mass, so fast outer stars could just be heavy ones.
  Feedback from the catalogue
  On a circular orbit $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$ ⁠, and the star’s own mass $m$ ⁠ cancels: $v = \sqrt{\dfrac{GM}{r}}$ ⁠. The speed depends only on the mass inside the orbit and the distance, not on the star’s own mass. A light cloud of gas and a giant star at the same distance move at the same speed, just as a feather and a hammer fall together in a vacuum.
- Wrong The galaxy turns like a wheel, so the edge is fast
  Misconception: Thinks a galaxy turns like a solid wheel, every star taking the same time to go round.
  Feedback from the catalogue
  A galaxy is not a rigid wheel. Each star circles the centre on its own orbit, at roughly the same speed at every distance. An inner orbit is shorter, so a star near the centre completes a lap in far less time than a distant one. If the arms were fixed chains of stars, they would wind round the centre and vanish after a few turns.
Feedback for a right answer
A halo of dark matter, several times heavier than all the stars together.

Worked solution
1. The speed on a circular orbit depends only on the mass inside it: $v = \sqrt{\dfrac{GM}{r}}$ ⁠.
2. If all the mass were in the middle, like the light, the speeds at the edge would fall like $\dfrac{1}{\sqrt{r}}$ ⁠.
3. A speed that stays flat means $M$ ⁠ grows along with $r$ ⁠: the farther out you go, the more mass lies inside the orbit, though there is almost no light there. That is the dark matter halo, blue in the Scene.

Lesson 2 of 3 · About 25 minutes

Dark matter and rotation curves

Weighing a galaxy by its stars’ speeds: the speeds do not fall at the edge, so it holds far more mass than shines, spread in a halo round it.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
What was expected
Practice · Predict, then look
Question
The stars and gas are concentrated in the galaxy’s middle, and a marked hydrogen cloud circles far beyond the edge of the light. What will the measured orbital speeds show, going out from the centre?

Correct answer
- Correct They rise and then stay nearly flat, even far beyond the light
Setup, choices and feedback
The Scene at this step
Dark matter in the halo
0 billion Suns
Stars and gas
65 billion Suns
Show the speeds
Starlight
Choices
- Correct They rise and then stay nearly flat, even far beyond the light
- Wrong They fall at the edge, as for the outer planets
  Misconception: Expects the stars and gas far out in a galaxy to orbit more slowly, as the outer planets do round the Sun.
  Feedback from the catalogue
  That is exactly what was thought until the 1970s. Most of a galaxy’s light comes from its middle, and if most of the mass is there too, then farther out $v = \sqrt{\dfrac{GM}{r}}$ ⁠ should fall, as in the Solar System. But Vera Rubin and Kent Ford measured, and the speeds do not fall: they stay nearly flat far beyond the edge of the light. So there is mass out there we cannot see.
- Wrong They keep rising in a straight line, as on a wheel
  Misconception: Thinks a galaxy turns like a solid wheel, every star taking the same time to go round.
  Feedback from the catalogue
  A galaxy is not a rigid wheel. Each star circles the centre on its own orbit, at roughly the same speed at every distance. An inner orbit is shorter, so a star near the centre completes a lap in far less time than a distant one. If the arms were fixed chains of stars, they would wind round the centre and vanish after a few turns.
Once the prediction is locked in, the Scene changes
Show the speeds
Speeds
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Flat. Vera Rubin found it again and again, galaxy after galaxy.

Worked solution
1. Near the centre the speed rises, because the farther out you go, the more mass lies inside the orbit.
2. Outside most of the light, if there were no more mass there, the speed would have to fall like $\dfrac{1}{\sqrt{r}}$ ⁠.
3. The measurements on the chart stay near 220 km/s out to the farthest cloud. The curve is “flat”, and that is the first evidence for dark matter.
Step 2 of 5
Open this step
How much is missing
Practice · Calculate
Question
With Stars and gas alone, at 65 billion Suns from the starlight, the mass can hold the far hydrogen cloud on a circle only at about 124 km/s. The cloud is measured moving at about 218 km/s. Assuming the extra mass is spread like the mass already there, by how many times must the mass inside the orbit grow?

Correct answer
3.08 ± 3%
Setup, choices and feedback
The Scene at this step
Dark matter in the halo
0 billion Sunsfixed
Stars and gas
65 billion Sunsfixed
Show the speeds
Speeds
Accepted within
± 3%
Computed from the Scene at this step’s setting
How many times the hypothesised mass inside the outermost marked cloud must grow, spread as it is, to hold the cloud at its measured speed: (measured ÷ held speed)²/nodes/galaxy/outputs/massFactor
Wrong answers it recognises
- 1.76 ± 3%
  Misconception: Thinks the mass needed to hold an orbit grows in proportion to the orbital speed, not its square.
  Feedback from the catalogue
  From $\dfrac{GM}{r} = v^2$ ⁠ comes $M = \dfrac{v^2 r}{G}$ ⁠: the mass needed grows as the square of the speed. A speed $1.75$ ⁠ times higher needs $1.75^2 \approx 3$ ⁠ times the mass, not $1.75$ ⁠ times.
Feedback for a right answer
About $3$ ⁠ times: two thirds of the mass inside the cloud’s orbit are missing from the light.

Hint
$v = \sqrt{\dfrac{GM}{r}}$ ⁠, so $M \propto v^2$ ⁠.

Worked solution
1. At the same distance, $M \propto v^2$ ⁠.
2. $\dfrac{M_{\text{needed}}}{M_{\text{visible}}} = \left(\dfrac{218}{124}\right)^2 = 1.76^2 \approx 3.1$ ⁠.
It takes three times the mass the stars and gas provide. Answering $1.76$ ⁠ assumes the mass grows like the speed rather than its square.
Step 3 of 5
Open this step
Weighing with an orbit
Practice · Calculate
Question
The far hydrogen cloud is $20$ ⁠ kiloparsecs from the centre, $6.17 \times 10^{20}\ \mathrm{m}$ ⁠, and moves at 218 km/s. Using $M = \dfrac{v^2 r}{G}$ ⁠, what is the mass inside its orbit, in kilograms? $G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}$ ⁠.

Correct answer
4.404 × 10⁴¹ kg ± 3%
Setup, choices and feedback
The Scene at this step
Dark matter in the halo
0 billion Suns
Stars and gas
65 billion Suns
Show the speeds
Speeds
Accepted within
± 3%
Computed from the Scene at this step’s setting
Mass inside the outermost marked cloud’s orbit as its measured speed weighs it, M = v² r / G (the spherical estimate)/nodes/galaxy/outputs/enclosedMass
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
About $4.4 \times 10^{41}$ ⁠ kg, some $220$ ⁠ billion Suns: more than three times what shines.

Hint
Convert the speed to metres per second: $2.18 \times 10^5\ \mathrm{m/s}$ ⁠.

Worked solution
1. $v^2 = (2.18 \times 10^5)^2 \approx 4.75 \times 10^{10}\ \mathrm{m^2/s^2}$ ⁠.
2. $M = \dfrac{4.75 \times 10^{10} \cdot 6.17 \times 10^{20}}{6.67 \times 10^{-11}} \approx 4.4 \times 10^{41}\ \mathrm{kg}$ ⁠.
3. In solar masses: $\dfrac{4.4 \times 10^{41}}{1.99 \times 10^{30}} \approx 2.2 \times 10^{11}$ ⁠, about $220$ ⁠ billion.
The stars and gas weigh only about $65$ ⁠ billion solar masses. The formula assumes the mass is spherical, so it is only an estimate for a flat galaxy, but the gap is far bigger than any such correction.
Step 4 of 5
Open this step
A halo
Practice · Reach the target
Question
Stars and gas is locked at 65 billion Suns, the mass from the starlight, and the marked bodies’ dashed paths fly outwards. Change Dark matter in the halo until every path closes into the circle the body really moves on, and press Check.

Goal
- 2 ± 0.1
  At the start of the step: 1
Setup, choices and feedback
The Scene at this step
Dark matter in the halo
0 billion Suns
Stars and gas
65 billion Sunsfixed
Show the speeds
Speeds
Read from the Scene at
1 too little mass (bodies drift out), 2 holds every marked body, 3 too much (they fall in), 4 mass in the wrong places/nodes/galaxy/outputs/regime
One setting that reaches it
Dark matter in the halo
200 billion Suns
Feedback for a right answer
Every path closes: the halo holds every body on its circle.

Feedback for a wrong answer
Not yet. Check whether the paths fly out (add more) or dive in (take some away).

Hint
Too little mass: the paths fly outwards. Too much: they dive inwards.

Worked solution
1. Without dark matter, the bodies move too fast for the mass, and the paths fly outwards.
2. Add dark matter little by little. At about $200$ ⁠ to $300$ ⁠ billion solar masses within $30$ ⁠ kiloparsecs every path closes: that is just enough.
3. Beyond that, the pull is too strong: the paths dive inwards.
In such a solution most of the mass inside the far cloud, two thirds or more, is dark matter.
Step 5 of 5
Open this step
Why not more stars
Practice · Choose
Question
With no dark matter, Stars and gas at 195 billion Suns, three times as much: the far cloud is held at last, but the inner stars plunge towards the middle. Why can’t more stars and gas set every path right?

Correct answer
- Correct The stars and gas are concentrated in the middle, so adding them strengthens the pull mostly inside. The missing mass has to be spread far out
Setup, choices and feedback
The Scene at this step
Dark matter in the halo
0 billion Suns
Stars and gas
195 billion Suns
Show the speeds
Speeds
Choices
Shown in a random order.
- Correct The stars and gas are concentrated in the middle, so adding them strengthens the pull mostly inside. The missing mass has to be spread far out
- Wrong Heavier stars move more slowly, so they fall in
  Misconception: Thinks a star’s orbital speed depends on its own mass, so fast outer stars could just be heavy ones.
  Feedback from the catalogue
  On a circular orbit $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$ ⁠, and the star’s own mass $m$ ⁠ cancels: $v = \sqrt{\dfrac{GM}{r}}$ ⁠. The speed depends only on the mass inside the orbit and the distance, not on the star’s own mass. A light cloud of gas and a giant star at the same distance move at the same speed, just as a feather and a hammer fall together in a vacuum.
- Wrong It just needs still more stars, four or five times as many
  Misconception: Thinks adding more ordinary stars and gas where we see them could explain the flat rotation curve.
  Feedback from the catalogue
  Stars and gas are concentrated towards the galaxy’s middle, which is how we see them. Doubling or tripling them adds pull mostly near the centre: the inner stars get too much and dive inwards, while the far cloud is still not held. For the speeds to stay flat, the missing mass must be spread far out, in a halo round the whole galaxy.
Feedback for a right answer
The question is not only how much mass, but where it is.

Worked solution
1. The speed on an orbit is set by the mass inside it.
2. Extra stars and gas, spread as we see them, add mass mostly inside the inner stars’ orbits, and those get too strong a pull.
3. For both the inner and the far bodies to be held, the missing mass has to grow with distance, in a wide halo. That is just what a dark-matter halo does.

Lesson 3 of 3 · About 30 minutes

The cosmic microwave background

The oldest light we can see: a glowing fog that filled the young universe, cleared about 380,000 years after the Big Bang, and has stretched into microwaves since.

Open the lesson

The 8 steps and their answers

Step 1 of 8
Open this step
A glow from every direction
Practice · Choose
Question
A microwave telescope pointed anywhere in the sky picks up a faint glow, almost equally strong in every direction. Where does it come from?

Correct answer
- Correct From all round: from the glowing fog that filled the whole young universe
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
10.14 log years
Look with
Microwave telescope
Show the ripples
Off
Choices
Shown in a random order.
- Correct From all round: from the glowing fog that filled the whole young universe
- Wrong From one direction, the place where the Big Bang happened
  Misconception: Thinks the microwave background comes from one place in the sky, where the Big Bang happened.
  Feedback from the catalogue
  The Big Bang was not an explosion at one point in empty space: it happened everywhere, and all of space expanded. The glowing fog filled the whole universe, so its light reaches us from every direction, almost exactly as strong. Every observer, in every galaxy, sees this glow all round them.
- Wrong From all the stars in our Galaxy together
  Feedback
  Starlight crowds into the band of the Milky Way and peaks in visible light. This glow is even across the whole sky, far from the Milky Way too.
Feedback for a right answer
From all round: the cosmic microwave background is the glow of the whole young universe.

Worked solution
1. In 1965 Arno Penzias and Robert Wilson picked up a stubborn radio hiss that came from every direction, day and night.
2. The Big Bang did not happen at one point: all of space was hot and dense, and all of space expanded.
3. So in every direction you look, far enough away, you reach a region where the universe was glowing fog. Its light reaches us now from all round.
Step 2 of 8
Open this step
A universe of glowing fog
Practice · Choose
Question
Time since the Big Bang is at Glowing fog, about a hundred thousand years after the Big Bang. The light in the Scene zigzags and gets nowhere. Why?

Correct answer
- Correct Free electrons scatter the light, and each ray is knocked into a new direction almost at once
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
5 log years
Look with
Eyes
Show the ripples
Off
Choices
Shown in a random order.
- Correct Free electrons scatter the light, and each ray is knocked into a new direction almost at once
- Wrong Hydrogen atoms absorb the light
  Feedback
  There are no atoms yet: it is too hot, and electrons do not stay on the nuclei. When atoms form, the fog actually clears.
- Wrong The dense universe’s gravity bends the light back
  Feedback
  Gravity does not scatter light into zigzags like these. They come from photons bouncing off electrons.
Feedback for a right answer
A fog of electrons: like the Sun behind a cloud, the light gets out, but scattered.

Worked solution
1. In the young universe the temperature is above about 3000 K, and energetic enough photons break up every hydrogen atom that forms.
2. So the electrons are free, and a free electron scatters light very easily.
3. Each ray hits an electron and changes direction again and again, like headlights in thick fog. The universe glows, but you cannot see through it.
When the universe cools below about 3000 K, the electrons settle onto the nuclei, and the light is set free. That is why no telescope can see anything earlier by light.
Step 3 of 8
Open this step
The fog clears
Practice · Calculate
Question
Time since the Big Bang is at Fog clears, about $380{,}000$ ⁠ years. Since then the universe and every light wave in it have stretched about $1080$ ⁠ times. Today the glow is at 2.7255 K, and the temperature falls in the same ratio as the waves stretch. What was the glow’s temperature then, in kelvin?

Correct answer
2936 K ± 2%
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
5.58 log yearsfixed
Look with
Eyes
Show the ripples
Off
Accepted within
± 2%
Computed from the Scene at this step’s setting
Temperature of the glow at that moment, T₀ (1 + z)/nodes/sky/outputs/temperature
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
About 2940 K: an orange glow that eyes could have seen.

Hint
$T_{\text{then}} = T_{\text{now}} \cdot (1 + z)$ ⁠.

Worked solution
1. By Wien’s law, temperature is inversely proportional to the peak wavelength. Waves stretched $1080$ ⁠ times match a temperature $1080$ ⁠ times lower.
2. $T_{\text{then}} = 2.7255 \cdot 1080 \approx 2940\ \mathrm{K}$ ⁠.
About 3000 K, much like an old light bulb’s filament: the fog glowed orange then.
Step 4 of 8
Open this step
Stretched into microwaves
Practice · Predict, then look
Question
Look with is set to Eyes. What will eyes see when Time since the Big Bang moves on to Dark ages, about $30$ ⁠ million years after the Big Bang, before any stars formed?

Correct answer
- Correct A black sky: the glow is still there, but in the infrared, beyond what eyes can see
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
5.58 log years
Look with
Eyes
Show the ripples
Off
Choices
- Correct A black sky: the glow is still there, but in the infrared, beyond what eyes can see
- Wrong A dim red glow from all round
  Feedback
  At about 180 K the peak is already near 16 µm, in the infrared, with almost no light at the wavelengths eyes can see.
- Wrong A brighter sky, because the fog has cleared
  Misconception: Multiplies Wien’s constant by the temperature instead of dividing, so a colder body peaks at a shorter wavelength.
  Feedback from the catalogue
  Wien’s law is $\lambda_{\max} = \dfrac{b}{T}$ ⁠: the peak wavelength is inversely proportional to the temperature. A colder body glows at longer wavelengths: red before blue, infrared before red, and in the end microwaves. That is why, as the universe’s glow cooled $1100$ ⁠ times, its peak stretched $1100$ ⁠ times.
Once the prediction is locked in, the Scene changes
Time since the Big Bang (powers of ten)
7.5 log years
The Scene then shows: 179.3 K · Temperature of the glow at that moment, T₀ (1 + z)/nodes/sky/outputs/temperature
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
A black sky: the “dark ages”, until the first stars lit up.

Worked solution
1. The universe keeps expanding and the glow keeps cooling: about $30$ ⁠ million years after the Big Bang it is already about 180 K.
2. By Wien’s law the peak is at $\dfrac{b}{T} \approx 16\ \mu\mathrm{m}$ ⁠, in the infrared, and almost nothing is left in visible light.
3. No stars have formed yet, so the sky is black to the eye. Switch to Microwave telescope, and the glow is still there.
Step 5 of 8
Open this step
The glow today
Practice · Calculate
Question
Time since the Big Bang is at Today: the glow is at 2.7255 K. At what wavelength is its peak, in millimetres? $b = 2.90 \times 10^{-3}\ \mathrm{m \cdot K}$ ⁠.

Correct answer
1.063 mm ± 2%
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
10.14 log yearsfixed
Look with
Microwave telescope
Show the ripples
Off
Accepted within
± 2%
Computed from the Scene at this step’s setting
Wien peak of the glow at that moment, b / T/nodes/sky/outputs/peakWavelength
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 7.9 mm ± 3%
  Misconception: Multiplies Wien’s constant by the temperature instead of dividing, so a colder body peaks at a shorter wavelength.
  Feedback from the catalogue
  Wien’s law is $\lambda_{\max} = \dfrac{b}{T}$ ⁠: the peak wavelength is inversely proportional to the temperature. A colder body glows at longer wavelengths: red before blue, infrared before red, and in the end microwaves. That is why, as the universe’s glow cooled $1100$ ⁠ times, its peak stretched $1100$ ⁠ times.
Feedback for a right answer
About 1.06 mm: that is why it is called the microwave background.

Hint
$\lambda_{\max} = \dfrac{b}{T}$ ⁠.

Worked solution
1. $\lambda_{\max} = \dfrac{b}{T} = \dfrac{2.90 \times 10^{-3}}{2.7255} \approx 1.06 \times 10^{-3}\ \mathrm{m}$ ⁠.
2. About $1.06$ ⁠ millimetres: microwaves.
Back when the Fog clears, the peak was $1080$ ⁠ times shorter, about $1$ ⁠ micrometre. Answering about $7.9$ ⁠ multiplies $b$ ⁠ by the temperature instead of dividing.
Step 6 of 8
Open this step
Ripples in the glow
Practice · Choose
Question
Show the ripples is On: the contrast is turned up thousands of times, and warmer and cooler spots show that differ by only about one part in $100{,}000$ ⁠. What are they?

Correct answer
- Correct Regions a little denser or thinner, which gravity grew over billions of years into galaxies and clusters of galaxies
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
10.14 log years
Look with
Microwave telescope
Show the ripples
On
Choices
Shown in a random order.
- Correct Regions a little denser or thinner, which gravity grew over billions of years into galaxies and clusters of galaxies
- Wrong Noise in the telescope
  Feedback
  The same pattern has been measured again and again, by different telescopes, from the ground and from space, and it matches the predictions closely.
- Wrong The first stars
  Feedback
  At $380{,}000$ ⁠ years there were no stars yet. They formed hundreds of millions of years later, in these denser regions.
- Wrong The place where the Big Bang began
  Misconception: Thinks the microwave background comes from one place in the sky, where the Big Bang happened.
  Feedback from the catalogue
  The Big Bang was not an explosion at one point in empty space: it happened everywhere, and all of space expanded. The glowing fog filled the whole universe, so its light reaches us from every direction, almost exactly as strong. Every observer, in every galaxy, sees this glow all round them.
Feedback for a right answer
The seeds of all the structure in the universe: our Galaxy began as such a ripple too.

Worked solution
1. The glow is almost perfectly even, but not quite: it has tiny differences of about $10^{-5}$ ⁠.
2. A warm or cool spot marks a region that was slightly denser or thinner than average.
3. A denser region pulls in more matter by gravity, and grows. After billions of years those regions became the galaxies, clusters and giant filaments of today’s universe.
Step 7 of 8
Open this step
Catching electrons
Practice · Reach the target
Question
At Glowing fog every electron is free, and today almost all are bound to nuclei. Move Time since the Big Bang (powers of ten) to the moment when exactly half the electrons have been caught in atoms, and press Check.

Goal
- 0.5 ± 0.05
  At the start of the step: 1
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
5 log years
Look with
Eyesfixed
Show the ripples
Off
Read from the Scene at
Share of electrons still free, 0..1/nodes/sky/outputs/freeElectrons
One setting that reaches it
Time since the Big Bang (powers of ten)
5.48 log years
Feedback for a right answer
At about $10^{5.48}$ ⁠ years, some 300,000 years after the Big Bang: a little before the fog clears.

Hint
It happens a little before Fog clears. Try values a little below $5.58$ ⁠.

Worked solution
1. At $5$ ⁠, about a hundred thousand years, the universe is too hot for atoms to hold together: any electron caught is knocked out again at once.
2. Once the universe cools below about 3000–4000 K, the nuclei catch electrons fast.
3. Half have been caught at about $5.48$ ⁠, that is $10^{5.48} \approx 300{,}000$ ⁠ years.
With each electron caught there is less for the light to scatter off. Soon after the fog clears, and that light is what reaches us today.
Step 8 of 8
Open this step
The first galaxies
Practice · Reach the target
Question
Light that set off at some moment has been stretched $1 + z$ ⁠ times since, where $z$ ⁠ is the redshift of that moment. The most distant galaxies the James Webb Space Telescope sees are at about $z = 10$ ⁠. Move Time since the Big Bang (powers of ten) to the moment when $z = 10$ ⁠, and press Check.

Goal
- 10 ± 0.4
  At the start of the step: 1.776 × 10⁻¹⁵
Setup, choices and feedback
The Scene at this step
Time since the Big Bang (powers of ten)
10.14 log years
Look with
Eyes
Show the ripples
Off
Read from the Scene at
Redshift z of that moment: its light has been stretched 1 + z times by today/nodes/sky/outputs/redshift
One setting that reaches it
Time since the Big Bang (powers of ten)
8.66 log years
Look with
Eyes
Feedback for a right answer
At about $10^{8.67}$ ⁠ years, some 470 million years after the Big Bang: the universe was $11$ ⁠ times smaller.

Hint
At Fog clears $z$ ⁠ is about $1100$ ⁠, and today $0$ ⁠. Look among hundreds of millions of years, about $8.5$ ⁠ to $9$ ⁠.

Worked solution
1. The earlier the moment, the more the light has stretched on its way to us, and the bigger $z$ ⁠.
2. $z = 10$ ⁠ means the universe was then $11$ ⁠ times smaller than today.
3. That happens at about $8.67$ ⁠, that is $10^{8.67} \approx 470$ ⁠ million years after the Big Bang.
Blue and ultraviolet light from the first stars reaches us stretched $11$ ⁠ times, into the infrared. That is why the James Webb Space Telescope is built to see infrared.

Across the unit · 7 misconceptions

Misconception catalogue

Each common mistake this unit recognises, in a few words. Open one for what the learner reads when an answer matches it, and where it is asked.

Index

Galaxies

Islands of stars

Question

Correct answer

The Scene at this step

Choices

Worked solution

Four shapes

Question

Correct answer

The Scene at this step

Choices

Worked solution

Arms that don’t wind up

Question

Correct answer

The Scene at this step

Choices

Worked solution

The galactic year

Question

Correct answer

The Scene at this step

Hint

Worked solution

Too fast at the edge

Question

Correct answer

The Scene at this step

Choices

Worked solution

Dark matter and rotation curves

What was expected

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

How much is missing

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Weighing with an orbit

Question

Correct answer

The Scene at this step

Hint

Worked solution

A halo

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

Why not more stars

Question

Correct answer

The Scene at this step

Choices

Worked solution

The cosmic microwave background

A glow from every direction

Question

Correct answer

The Scene at this step

Choices

Worked solution

A universe of glowing fog

Question

Correct answer

The Scene at this step

Choices

Worked solution

The fog clears

Question