The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion...
Transcript of The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion...
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The Cosmic Distance LadderTerence Tao (UCLA)
Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL
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Astrometry
Solar system montage, NASA/JPL
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Solar system montage, NASA/JPL
Astrometry is the study of positions and movements of celestial bodies
(sun, moon, planets, stars, etc.).
It is a major subfield of astronomy.
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Solar system montage, NASA/JPL
Typical questions in astrometry are:
• How far is it from the Earth to the Moon?
• From the Earth to the Sun?
• From the Sun to other planets?
• From the Sun to nearby stars?
• From the Sun to distant stars?
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These distances are far too vast to be measured directly.
D1
D2
D1 = ???
D2 = ???
Hubble deep field, NASA/ESA
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Nevertheless, there are several ways to measure these distances indirectly.
D1
D2
D1 / D2 = 3.4 ± 0.1
Hubble deep field, NASA/ESA
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The methods often rely more on mathematics than on technology.
D1
D2
v1 = H D1
v2 = H D2
v1 / v2 = 3.4 ± 0.1
D1 / D2 = 3.4 ± 0.1
Hubble deep field, NASA/ESA
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From “The Essential Cosmic Perspective”, Bennett et al.
The indirect methods control large distances in terms of smaller distances.
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From “The Essential Cosmic Perspective”, Bennett et al.
The smaller distances are controlled by even smaller distances...
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From “The Essential Cosmic Perspective”, Bennett et al.
… and so on, until one reaches distances that one can measure directly.
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From “The Essential Cosmic Perspective”, Bennett et al.
This is the cosmic distance ladder.
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1st rung: the Earth
Earth Observing System composite, NASA
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Nowadays, we know that the earth is approximately
spherical, with radius 6378 kilometers (3963 mi) at the equator and 6356 kilometers
(3949 mi) at the poles.
Earth Observing System composite, NASA
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These values have now been
verified to great precision by
many means, including modern
satellites.
Earth Observing System composite, NASA
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But suppose we had no advanced
technology such as spaceflight,
ocean and air travel, or even
telescopes and sextants.
Earth Observing System composite, NASA
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Could we still
calculate the radius
of the Earth?
Earth Observing System composite, NASA
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Could we even tell
that the Earth was
round?
Earth Observing System composite, NASA
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The answer is yes– if one knows some geometry!
Wikipedia
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Aristotle (384-322 BCE) gave a convincing indirect
argument that the Earth was round… by looking at the
Moon.
Copy of a bust of Aristotle by Lysippos (330 BCE)
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Aristotle knew that lunar eclipses only occurred when the Moon was
directly opposite the Sun.
Lunar Eclipse Phases, Randy Brewer
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He deduced that these eclipses were caused by the Moon falling into the
Earth’s shadow.
Lunar Eclipse Phases, Randy Brewer
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But the shadow of the Earth on the Moon in an
eclipse was always a circular arc.
Lunar Eclipse Phases, Randy Brewer
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In order for Earth’s shadows to always be
circular, the Earth must be round.
Lunar Eclipse Phases, Randy Brewer
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Aristotle also knew there were stars one could see
in Egypt but not in Greece.
Night Sky, Till Credner
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He reasoned that this was due to the curvature of
the Earth, so that its radius was finite.
Night Sky, Till Credner
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However, he was unable to get an accurate
measurement of this radius.
Night Sky, Till Credner
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Eratosthenes, Nordisk familjebok, 1907
Eratosthenes (276-194 BCE) computed the
radius of the Earth to be 40,000 stadia (6800 km,
or 4200 mi).
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Eratosthenes, Nordisk familjebok, 1907
This is accurate to within eight
percent.
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Eratosthenes, Nordisk familjebok, 1907
The argument was again indirect – but
now relied on looking at the Sun.
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Eratosthenes read of a well in Syene, Egypt which at noon on the summer solstice (June 21) would reflect the
overhead sun.
Syene
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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[This is because Syene lies almost directly on the
Tropic of Cancer.]
Syene
Sun directly
overhead
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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Eratosthenes tried the same experiment in his
home city of Alexandria.
Syene
Sun directly
overhead
Alexandria
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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But on the solstice, the sun was at an angle and did not reflect from the bottom of the well.
Syene
Sun directly
overhead
Alexandria
Sun not quite
overhead
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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Using a gnomon (measuring stick), Eratosthenes measured the deviation
of the sun from the vertical as 7o.
Syene
Sun directly
overhead
Alexandria
7o
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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From trade caravans and other sources, Eratosthenes knew Syene to be 5,000 stadia (740 km) south of Alexandria.
Syene
Sun directly
overhead
Alexandria
7o
5000 stadia
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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This is enough information to compute the radius of the Earth.
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
7o
5000 stadia7o
r
r
2π r * 7o / 360o
= 5000 stadia
r=40000 stadia
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[This assumes that the Sun is quite far away, but more on this later.]
7o
5000 stadia7o
r
r
2π r * 7o / 360o
= 5000 stadia
r=40000 stadia
Tropic of Cancer: Swinburne University, COSMOS Encyclopedia of Astronomy
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2nd rung: the
Moon
NASA
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• What shape is the Moon?
• How large is the Moon?
• How far away is the Moon?
NASA
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The ancient Greeks could answer these
questions also.
NASA
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Aristotle argued that the Moon was a sphere (rather than a disk) because the terminator (the boundary of the
Sun’s light on the Moon) was always an elliptical arc.
Wikipedia
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Aristarchus (310-230 BCE) computed the distance of the Earth to the Moon
as about 60 Earth radii.
[In truth, it varies from 57 to 63 Earth radii.]
Bust of Aristarchus - NASA
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Aristarchus also computed the radius of the Moon as 1/3 the radius of the
Earth.
[In truth, it is 0.273 Earth radii.]
Bust of Aristarchus - NASA
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The radius of the Earth was computed in the previous rung of the ladder, so we now know the size and location of the Moon.
Bust of Aristarchus - NASA
Radius of moon = 0.273 radius of Earth = 1,700 km = 1,100 mi
Distance to moon = 60 Earth radii = 384,000 km = 239,000 mi
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Aristarchus’s argument to measure the distance to the
Moon was indirect, and relied on the Sun.
Wikipedia
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Aristarchus knew that lunar eclipses were caused by the Moon passing through the
Earth’s shadow.
Wikipedia
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The Earth’s shadow is approximately two Earth radii wide*.
2r
Wikipedia* This is an oversimplification; there is a correction due to the angular width of the Sun.
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The maximum length of a
lunar eclipse is three hours*.
2r
v = 2r / 3 hours
Wikipedia* More precisely, the center of the Moon stays in the umbra for at most 3 hours.
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It takes one month* for the Moon to go
around the Earth.
2r
v = 2r / 3 hours
= 2π d / 1 month d
Wikipedia* Here, “month” means “synodic lunar month”, which is approximately 29.5 days.
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This is enough information* to work out the distance to the Moon in Earth radii.
2r
d
v = 2r / 3 hours
= 2π d / 1 month
d = 60 r
Wikipedia* The actual calculation is more complicated than the above equations suggest, due to the previous footnotes.
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Also, the Moon takes about 2 minutes to
set.
V = 2R / 2 min2R
Moonset over the Colorado Rocky Mountains,
Sep 15 2008, Alek Kolmarnitsky
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The Moon takes 24 hours to make a full (apparent) rotation around the Earth.
2RV = 2R / 2 min
= 2π d / 24 hours
Moonset over the Colorado Rocky Mountains,
Sep 15 2008, Alek Kolmarnitsky
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This is enough information to determine the radius of the Moon, in terms of the distance to the Moon…
2RV = 2R / 2 min
= 2π D / 24 hours
R = D / 180
Moonset over the Colorado Rocky Mountains,
Sep 15 2008, Alek Kolmarnitsky
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… which we have just computed.
2RV = 2R / 2 min
= 2π D / 24 hours
R = D / 180
= r / 3
Moonset over the Colorado Rocky Mountains,
Sep 15 2008, Alek Kolmarnitsky
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[Aristarchus, by the way, was handicapped by not having an
accurate value of π, which had to wait until Archimedes (287-
212BCE) some decades later!]
2RV = 2R / 2 min
= 2π D / 24 hours
R = D / 180
= r / 3
Moonset over the Colorado Rocky Mountains,
Sep 15 2008, Alek Kolmarnitsky
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EIT-SOHO Consortium, ESA, NASA
3rd rung: the Sun
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EIT-SOHO Consortium, ESA, NASA
• How large is the Sun?
• How far away is the Sun?
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EIT-SOHO Consortium, ESA, NASA
Once again, the ancient Greeks could answer these questions (but with imperfect accuracy).
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EIT-SOHO Consortium, ESA, NASA
Their methods were indirect, and relied on the Moon.
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Zimbabwe Solar Eclipse 4 Dec 2002, Murray Alexander
Aristarchus already computed that the radius of the Moon was 1/180 of the distance to
the Moon.
![Page 61: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/61.jpg)
Zimbabwe Solar Eclipse 4 Dec 2002, Murray Alexander
He also knew that during a solar eclipse, the Moon covered the Sun almost
perfectly.
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Zimbabwe Solar Eclipse 4 Dec 2002, Murray Alexander
Using similar triangles, he concluded that the radius of
the Sun was also 1/180 of the distance to the Sun.
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Zimbabwe Solar Eclipse 4 Dec 2002, Murray Alexander
So his next task was to compute the distance
to the Sun.
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Zimbabwe Solar Eclipse 4 Dec 2002, Murray Alexander
For this, he turned to the Moon again for
help.
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He knew that new Moons occurred when the Moon was between the Earth and Sun…
BBC
New moon
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… full Moons occurred when the Moon was directly opposite the
Sun…BBC
New moon
Full moon
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… and half Moons occurred when the Moon made a right angle
between Earth and Sun.BBC
New moon
Full moon
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This implies that half Moons occur slightly closer to new Moons than to full Moons.
BBC
θ
θ<π/2
New moon
Full moon
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Aristarchus thought that half Moons occurred 6 hours before the
midpoint of a new and full Moon.BBC
θ
θ = π/2 – 2 π * 6
hours/1 month
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From this and trigonometry, he concluded that the Sun was 20
times further away than the Moon.BBC
θ
θ = π/2 – 2 π * 6
hours/1 month
cos θ = d/D
d
DD = 20 d
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Unfortunately, with ancient Greek technology it was hard to time a
new Moon perfectly.BBC
θ
θ = π/2 – 2 π * 6
hours/1 month
cos θ = d/D
d
DD = 20 d
![Page 72: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/72.jpg)
The true time discrepancy is ½ hour (not 6 hours), and the Sun is 390
times further away (not 20 times).BBC
θ
θ = π/2 – 2 π * 6
0.5 hour/1 month
cos θ = d/D
d
DD = 20 390 d
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Nevertheless, the basic method was correct.
BBC
θ
θ = π/2 – 2 π /2
hour/1 month
cos θ = d/D
d
DD = 390 d
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And Aristarchus’ computations led him to an
important conclusion…BBC
θ
d = 60 r
D/d = 20
R/D = 1/180
d
D
r
R
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… the Sun was much larger than the Earth.
BBC
θ
d = 60 r
D/d = 20
R/D = 1/180
d
D
r
R
R ~ 7 r
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[In fact, it is much, much larger.]
BBC
θ
d = 60 r
D/d = 20 390
R/D = 1/180
d
D
r
R
R = 7 r 109 r
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He then concluded it was absurd to think the Sun
went around the Earth…
NASA/ESA
Earth radius = 6371 km = 3959 mi
Sun radius = 695,500 km = 432,200 mi
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… and was the first to propose the heliocentric
model that the Earth went around the Sun.
NASA/ESA
Earth radius = 6371 km = 3959 mi
Sun radius = 695,500 km = 432,200 mi
![Page 79: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/79.jpg)
[1700 years later, Copernicus would credit
Aristarchus for this idea.]
NASA/ESA
Earth radius = 6371 km = 3959 mi
Sun radius = 695,500 km = 432,200 mi
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Ironically, Aristarchus’ theory was not accepted
by the other ancient Greeks…
NASA/ESA
Earth radius = 6371 km = 3959 mi
Sun radius = 695,500 km = 432,200 mi
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… but we’ll explain why later.
NASA/ESA
Earth radius = 6371 km = 3959 mi
Sun radius = 695,500 km = 432,200 mi
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The distance from the Earth to the Sun is known as the Astronomical Unit (AU).
Wikipedia
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It is an extremely important rung in the cosmic distance ladder.
Wikipedia
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Aristarchus’ original estimate of the AU was inaccurate…
Wikipedia
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… but we’ll see much more accurate ways to measure the AU later on.
Wikipedia
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4th rung: the
planets
Solar system montage, NASA/JPL
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The ancient astrologers knew that all the planets lay on a plane (the
ecliptic), because they only moved through the Zodiac.
Solar system montage, NASA/JPL
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But this still left many questions unanswered:
Solar system montage, NASA/JPL
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• How far away are the planets (e.g. Mars)?
• What are their orbits?
• How long does it take to complete an orbit?
Solar system montage, NASA/JPL
![Page 90: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/90.jpg)
Ptolemy (90-168 CE) attempted to answer these questions, but
obtained highly inaccurate answers…
Wikipedia
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... because he was working with a geocentric model rather than a heliocentric one.
Wikipedia
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The first person to obtain accurate answers was Nicolaus
Copernicus (1473-1543).
Wikipedia
![Page 93: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/93.jpg)
Copernicus started with the records of the ancient Babylonians, who knew
that the apparent motion of Mars (say) repeated itself every 780 days (the
synodic period of Mars).
Babylonian world map, 7th-8th century BCE, British Museum
ωEarth – ωMars = 1/780 days
![Page 94: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/94.jpg)
Using the heliocentric model, he also knew that the Earth went around the Sun once a year.
Babylonian world map, 7th-8th century BCE, British Museum
ωEarth – ωMars = 1/780 days
ωEarth = 1/year
![Page 95: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/95.jpg)
Subtracting the implied angular velocities, he found that Mars went around the Sun every 687 days (the sidereal period of
Mars).
Babylonian world map, 7th-8th century BCE, British Museum
ωEarth – ωMars = 1/780 days
ωEarth = 1/year
ωMars = 1/687 days
![Page 96: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/96.jpg)
Assuming circular orbits, and using measurements of the location of Mars in
the Zodiac at various dates...
Babylonian world map, 7th-8th century BCE, British Museum
ωEarth – ωMars = 1/780 days
ωEarth = 1/year
ωMars = 1/687 days
![Page 97: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/97.jpg)
…Copernicus also computed the distance of Mars from the Sun to
be 1.5 AU.
Babylonian world map, 7th-8th century BCE, British Museum
ωEarth – ωMars = 1/780 days
ωEarth = 1/year
ωMars = 1/687 days
![Page 98: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/98.jpg)
Both of these measurements are accurate to two decimal places.
Babylonian world map, 7th-8th century BCE, British Museum
ωEarth – ωMars = 1/780 days
ωEarth = 1/year
ωMars = 1/687 days
![Page 99: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/99.jpg)
Tycho Brahe (1546-1601) made extremely detailed and long-term measurements of the position of
Mars and other planets.Wikipedia
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Unfortunately, his data deviated slightly from the predictions of the Copernican
model.
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Johannes Kepler (1571-1630) reasoned that this was because the orbits of the Earth and Mars
were not quite circular.
Wikipedia
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But how could one use Brahe’s data to work out the orbits of
both the Earth and Mars simultaneously?
![Page 103: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/103.jpg)
That is like solving for two unknowns using only one
equation – it looks impossible!
![Page 104: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/104.jpg)
To make matters worse, the data only shows the declination
(direction) of Mars from Earth. It does not give the distance.
![Page 105: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/105.jpg)
So it seems that there is insufficient information
available to solve the problem.
![Page 106: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/106.jpg)
Nevertheless, Kepler found some ingenious ways to solve the
problem.
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He reasoned that if one wanted to compute the orbit of Mars
precisely, one must first figure out the orbit of the Earth.
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And to figure out the orbit of the Earth, he would argue
indirectly… using Mars!
![Page 109: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/109.jpg)
To explain how this works, let’s first suppose that Mars is fixed,
rather than orbiting the Sun.
![Page 110: The cosmic distance ladder - What's new · The Cosmic Distance Ladder Terence Tao (UCLA) Orion nebula, Hubble & Spitzer telescopes, composite image, NASA/JPL. Astrometry Solar system](https://reader036.fdocuments.in/reader036/viewer/2022062917/5ecc6a57bbbc2c3ed8293a36/html5/thumbnails/110.jpg)
But the Earth is moving in an unknown orbit.
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At any given time, one can measure the position of the Sun and Mars from Earth, with respect
to the fixed stars (the Zodiac).
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Assuming that the Sun and Mars are fixed, one can then triangulate to determine the position
of the Earth relative to the Sun and Mars.
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Unfortunately, Mars is not fixed; it also moves, and along an
unknown orbit.
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So it appears that triangulation does not
work.
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But Kepler had one additional piece of
information:
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he knew that after every 687 days…
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Mars returned to its original position.
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So by taking Brahe’s data at intervals of 687 days…
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… Kepler could triangulate and compute Earth’s orbit relative to any position of Mars.
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Once Earth’s orbit was known, it could be used to compute more positions of Mars by taking other
sequences of data separated by 687 days…
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… which allows one to compute the orbit of Mars.
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Kepler’s laws of planetary motion
1. Planets orbit in ellipses, with the Sun as one of the foci.
2. A planet sweeps out equal areas in equal times.
3. The square of the period of an orbit is proportional to the cube of its semi-major axis.
Using the data for Mars and other planets,Kepler
formulated his three laws of planetary motion.
NASA
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Newton’s law of universal gravitation
Any pair of masses attract by a force proportional to the masses, and inversely proportional to the
square of the distance.
|F| = G m1 m2 / r2
This helped Isaac Newton (1643-1727) in formulating and
confirming his law of gravity.
NASA
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NASA
Kepler’s methods allowed for very precise measurements of planetary
distances in terms of the AU.
Mercury: 0.307-0.466 AU
Venus: 0.718-0.728 AU
Earth: 0.98-1.1 AU
Mars: 1.36-1.66 AU
Jupiter: 4.95-5.46 AU
Saturn: 9.05-10.12 AU
Uranus: 18.4-20.1 AU
Neptune: 29.8-30.4 AU
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NASA
Conversely, if one had an alternate means to compute
distances to planets, this would give a measurement of the AU.
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NASA
One way to measure such distances is by parallax– measuring the same object from two different locations on the
Earth.
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NASA
By measuring the parallax of the transit of Venus across the Sun simultaneously in
several locations (including James Cook’s voyage!), the AU was computed reasonably
accurately in the 18th century.
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NASA
With modern technology such as radar and interplanetary satellites, the AU and the
planetary orbits have now been computed to extremely high precision.
1 AU = 149,597,871 km = 92,955,807 mi
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NASA
Incidentally, such precise measurements of Mercury revealed a precession that was not
explained by Newtonian gravity…
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NASA
… , and was one of the first experimental verifications of general relativity (which is
needed in later rungs of the ladder).
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5th rung: the speed
of light
Lucasfilm
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Technically, the speed of light, c, is not a
distance.
Lucasfilm
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However, one needs to know it in order to ascend higher
rungs of the distance ladder.
Lucasfilm
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The first accurate measurements of c were by Ole Rømer
(1644-1710) and ChristiaanHuygens (1629-1695).
Ole Rømer
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Their method was indirect… and used a moon of Jupiter,
namely Io.
Christaan Huygens
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Io has the shortest orbit of all the major moons of Jupiter. It orbits Jupiter once every 42.5
hours.
NASA/JPL/University of Arizona
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Rømer made many measurements of this orbit by timing when Io entered and
exited Jupiter’s shadow.
NASA/JPL/University of Arizona
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He noticed* that when Jupiter was aligned with the Earth, the orbit
advanced slightly; when Jupiter was opposed, the orbit lagged.
NASA/JPL/University of Arizona* As Romer could not observe Io in the daytime, his actual calculations occurred
when Earth nade an acute angle with Jupiter.
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The difference was slight; the orbit lagged by about 20 minutes when
Jupiter was opposed.
NASA/JPL/University of Arizona
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Huygens reasoned that this was because of the additional distance (2AU) that the light from Jupiter
had to travel.
NASA/JPL/University of Arizona
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Using the best measurement of the AU available to him, he then
computed the speed of light as c = 220,000 km/s = 140,000 mi/s.
[The truth is 299,792 km/s = 186,282 mi/s.]
NASA/JPL/University of Arizona
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This computation was important for the future development of physics.
NASA/JPL/University of Arizona
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James Clerk Maxwell (1831-1879) observed that the speed of light almost matched the speed his
theory predicted for electromagnetic radiation.
Wikipedia
c ~ 3.0 x 108 m/s
0 ~ 8.9 x 10-12 F/m
0 ~ 1.3 x 10-6 H/m
0)-1/2 ~ 3.0 x 108 m/s
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He then reached the important conclusion that light was a form
of electromagnetic radiation.
Science Learning Hub, University of Waikato, NZ
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This observation was instrumental in leading to Einstein’s theory of
special relativity in 1905.
Wikipedia
x = vt ↔ x’ = 0
x = ct ↔ x’ = ct’
x = -ct ↔ x’ = -ct’
x’ = (x-vt)/(1-v2/c2)1/2
t’= (t-vx/c2)/(1-v2/c2)1/2
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It also led to the development of spectroscopy.
Ian Short
First spectroscope: 1814 (Joseph von Fraunhofer)
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Both of these turn out to be important tools for climbing higher rungs of the ladder.
Ian Short
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6th rung: nearby
stars
Wikipedia
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We already saw that parallax from two locations on the Earth could
measure distances to other planets.
Wikipedia
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This is not enough separation to use optical parallax* to discern distances to even the next closest star (which
is about 270,000 AU away!)
Wikipedia
270,000 AU
= 4.2 light years
= 1.3 parsecs
= 4.0 x 1016 m
= 2.5 x 1013 mi 2 Earth radii / 270,000 AU = 0.000065 arc seconds
* However, one can sometimes use gravitational microlensing parallax (rather than optical parallax) for such distant objects.
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However, if one takes measurements six months apart, one gets a distance separation of
2AU...
From “The Essential Cosmic Perspective”, Bennett et al.
2 Earth radii = 12,700 km
2 AU = 300,000,000 km
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… which gives enough parallax to easily measure all stars within
about 100 light years (30 parsecs) with ordinary optical telescopes.
From “The Essential Cosmic Perspective”, Bennett et al.
1 light year = 9.5 x 1015 m
1 parsec = 3.1 x 1016 m
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This gives the distances to tens of thousands of stars - lots of very useful data for the next rung of
the ladder!
Wikipedia
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These parallax computations, which require accurate
telescopy, were first done by Friedrich Bessel (1784-1846) in
1838.
Wikipedia
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Nowadays, with modern telescopes (such as the satellite
telescope Hipparcos), the parallax method can measure distances up to about 10,000
light years away.
Wikipedia
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Ironically, when Aristarchus proposed the heliocentric model, his contemporaries dismissed it, on the grounds that they did not observe any parallax effects…
Wikipedia
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… so the heliocentric model would have implied that the stars were an absurdly large distance away.
Wikipedia
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[Which, of course, they are.]
Wikipedia
Distance to Proxima Centauri
= 40,000,000,000,000 km
= 25,000,000,000,000 mi
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7th rung: the
Milky Way
Milky Way, Serge Brunier
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One can use detailed observations of nearby stars to provide a
means to measure distances to more distant stars.
Milky Way, Serge Brunier
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Using spectroscopy, one can measure precisely the colour of a nearby star; using photography,
one can also measure its apparent brightness.
Milky Way, Serge Brunier
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Using the apparent brightness, the distance, and inverse square law,
one can compute the absolute brightness of these stars.
Milky Way, Serge Brunier
M = m – 5( log10 DL – 1)
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Ejnar Hertzsprung (1873-1967) and Henry Russell (1877-1957) plotted this absolute brightness against color for thousands of nearby stars in 1905-1915…
Leiden Observatory University of Chicago/Yerkes Observatory
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… leading to the famous Hertzprung-Russell diagram.
Richard Powell
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Once one has this diagram, one can use it in reverse to measure
distances to more stars than parallax methods can reach.
Richard Powell
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Indeed, for any star, one can measure its colour and its
apparent brightness…
Richard Powell
Spectroscopy Colour
Photography Apparent brightness
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and from the Hertzprung-Russell diagram, one can then infer the
absolute brightness.
Richard Powell
Spectroscopy Colour Absolute brightness
Photography Apparent brightness
HR Diagram
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From the apparent brightness and absolute brightness, one
can solve for distance.
Richard Powell
Spectroscopy Colour Absolute brightness
Photography Apparent brightness Distance
HR Diagram
Inverse square law
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This technique (main sequence fitting) works out to about
300,000 light years (covering the entire galaxy!)
Milky Way, Serge Brunier
300,000 light years = 2.8 x 1021 m = 1.8 x 1018 mi
Diameter of Milky Way = 100,000 light years
Distance to galactic center = 25,000 light years
= 2.4 x 1020 m = 1.5 x 1017 mi
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Beyond this distance, the main sequence stars are too faint to be
measured accurately.
Milky Way, Serge Brunier
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8th rung: Other
galaxies
Hubble deep field, NASA
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Henrietta Swan Leavitt (1868-1921) observed a certain class of stars (the Cepheids) oscillated in
brightness periodically.
American Institute of Physics
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Plotting the absolute brightness against the periodicity, she
observed a precise relationship.
Henrietta Swan Leavitt, 1912
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This gave yet another way to obtain absolute brightness, and
hence observed distances.
Henrietta Swan Leavitt, 1912
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Because Cepheids are so bright, this method works up to 100,000,000 light years!
Diameter of Milky Way = 100,000 light years
Most distant Cepheid detected (NGC 4604, HST) : 108,000,000 light years
Diameter of universe > 76,000,000,000 light years
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Most galaxies are fortunate to have at least one Cepheid in them, so
we know the distances to all galaxies out to a reasonably
large distance.
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Cepheid measurements also allowed Harold Shapley (1885-1972) to map the Milky Way, showing that the solar system was not at the
centre of its own galaxy.
Schematic side view of Milky Way (NASA/CXC/M. Weiss)
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Similar methods, using supernovae instead of Cepheids, can sometimes work to even
larger scales than these, and can also be used to independently confirm the Cepheid-based
distance measurements.
Supernova remnant, NASA, ESA, HEIC, Hubble Heritage Team
Diameter of Milky Way = 100,000 light years
Most distant Cepheid detected (NGC 4604, HST) : 108,000,000 light years
Most distant Type 1a supernova detected (1997ff) : 11,000,000,000 light years
Diameter of universe > 76,000,000,000 light years
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9th rung: the
universe
Simulated matter distribution in universe, Greg Bryan
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Edwin Hubble (1889-1953) noticed that distant galaxies had
their spectrum red-shifted a greater amount from those of
nearby galaxies.
NASA
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With this data, he formulated Hubble’s law: the red-shift of an object was proportional
to its distance.
NASA
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As explained by Christian Doppler (1803-1853), the red-shift of galaxies indicated that
distance to these galaxies increased with time.
NASA, WMAP
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This led to the famous Big Bang model of the expanding universe, which has now been confirmed by many other cosmological
observations.
NASA, WMAP
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But it also gave a way to measure distances even at extremely large scales… by first measuring the
red-shift and then applying Hubble’s law.
Hubble deep field, NASA
Spectroscopy Red shift Recession velocity Distance
Speed of light Hubble’s law
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These measurements have led to accurate maps of the universe at
very large scales…
Two degree field Galaxy red-shift survey, W. Schaap et al.
1,000,000,000 light years
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which have led in turn to many discoveries of very large-scale structures, such as the Great
Wall.
Two degree field Galaxy red-shift survey, W. Schaap et al.
1,000,000,000 light years
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For instance, our best estimate (as of 2004) of the current diameter of the entire universe is that it is
at least 78 billion light-years.
Cosmic microwave background fluctuation, WMAP
Most distant object detected (gamma ray burst) : 13 billion light years
Diameter of observable universe = 28 billion light years
Diameter of entire universe > 78 billion light years (4.6 x 1023 mi)
Age of universe = 13.7 billion years
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The mathematics becomes more advanced at this point, as the effects of general relativity have highly influenced the data we have at
this scale of the universe.
Artist’s rendition of a black hole, NASA
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Cutting-edge technology (such as the Hubble space telescope (1990-) and WMAP (2001-2010)) has also been
vital to this effort.
Hubble telescope, NASA
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Climbing this rung of the ladder (i.e. mapping the universe at its very large scales) is still a very active
area in astronomy today!
WMAP, NASA
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Celestial object Distance (metres) First (relative)
measurement
Earth 1.2 x 107 (diameter) Eratosthenes (~240BCE)
Moon 3.6 x 108 Aristarchus (~270BCE)
Sun 1.5 x 1011 Aristarchus (~270BCE)
Cook etc. (1761,1769)
Mars 2.3 x 1011 (from Sun) Copernicus (1543)
Saturn 1.5 x 1012 (from Sun) Copernicus (1543)
Pluto 7.4 x 1012 (from Sun) Tombaugh (1930)
Proxima Centauri 4.0 x 1016 Alden (1928)
61 Cygni 1.1 x 1017 Bessel (1838)
Hyades cluster 1.4 x 1018 Smart (1939)
Pleiades cluster 4.2 x 1018 Detweiler et al. (1984)
Galactic center 2.6 x 1020 Shapley (1914)
Large Magellanic Cloud 1.5 x 1021 Arp (1967)
Andromeda Galaxy 2.4 x 1022 Hubble (1923)
NGC 4603 1.0 x 1024 HST (1999)
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Celestial object Distance (metres) First (relative)
measurement
Sloan Great Wall 1.3 x 1025 (diameter) Gott et al. (2003)
1997ff Type Ia supernova 1.0 x 1026 HST (1997)
GRB (Gamma Ray Burst)
090423
1.2 x 1026 Swift satellite (2009)
UDFy-38135539
(Galaxy)
1.2 x 1026 Lehnert et al. (2010)
Observable universe 2.8 x 1026 (diameter) Hubble (1929)
Entire universe >7.2 x 1026 Cornish et al. (2004)
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Image credits• 1: Chaos at the Heart of Orion – NASA/JPL-Caltech/STScl
• 2-4, 86-89: Solar System Montage - NASA/JPL
• 5-7, 170,181: Hubble digs deeply – NASA/ESA/S. Beckwith (STScl) and the HUDF team
• 8-11: BENNETT, JEFFREY O.; DONAHUE, MEGAN; SCHNEIDER, NICHOLAS; VOIT, MARK, ESSENTIAL
COSMIC PERSPECTIVE, THE, 3rd Edition, ©2005. Electronically reproduced by permission of Pearson
Education, Inc., Upper Saddle River, New Jersey. p. 384, Figure 15.16.
• 12-17: Earth – The Blue Marble - NASA
• 18: Trigonometry triangle – Wikipedia
• 19: Bust of Aristotle by Lysippus – Wikipedia
• 20-23: Lunar Eclipse Phases – Randy Brewer. Used with permission.
• 24-26: Night Sky – Till Credner: AlltheSky.com. Used with permission.
• 27-29: Eratosthenes, Nordisk familjebok, 1907 - Wikipedia
• 30-37: Tropic of Cancer – Swinburne University, COSMOS Encyclopedia of Astronomy
http://astronomy.swin.edu.au/cosmos . Used with permission.
• 38-40: The Moon - NASA
• 41: Moon phase calendar May 2005 – Wikipedia
• 42-44: Bust of Aristarchus (310-230 BC) - Wikipedia
• 45: Geometry of a Lunar Eclipse – Wikipedia
• 51-55: Moonset over the Colorado Mountains, Sep 15 2008 – Alek Komarnitsky – www.komar.org
• 56-59: Driving to the Sun – EIT – SOHO Consortium, ESA, NASA
• 60-64: Zimbabwe Solar Eclipse – Murray Alexander. Used with permission.
• 65-76: The Earth – BBC. Used with permission.
• 77-81: Earth and the Sun – NASA Solarsystem Collection.
• 82-85: Solar map - Wikipedia
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• 90: Claudius Ptolemaeus – Wikipedia
• 91:Ptolemaeus Geocentric Model – Wikipedia
• 92: Nicolaus Copernicus portrait from Town Hall in Thorn/Torun – 1580 - Wikipedia
• 93-98: Babylonian maps – Wikipedia
• 99: Tycho Brahe – Wikipedia
• 100, 102-108: Tycho Brahe – Mars Observations – Wikipedia
• 101: Johannes Kepler (1610) – Wikipedia
• 122-130: Our Solar System – NASA/JPL
• 131-133: Millenium Falcon – Courtesy of Lucasfilm, Ltd. Used with permission.
• 134: Ole Roemer – Wikipedia
• 135: Christaan Huygens – Wikipedia
• 136-142: A New Year for Jupiter and Io – NASA/JPL/University of Arizona
• 143: James Clerk Maxwell – Wikipedia
• 144: Electromagnetic spectrum – Science Learning Hub, The University of Waikato, New Zealand
• 145: Relativity of Simultaneity – Wikipedia
• 146-147: The Spectroscopic Principle: Spectral Absorption lines, Dr. C. Ian Short
• 148 -150, 153: Nearby Stars – Wikipedia
• 151-152: BENNETT, JEFFREY O.; DONAHUE, MEGAN; SCHNEIDER, NICHOLAS; VOIT, MARK,
ESSENTIAL COSMIC PERSPECTIVE, THE, 3rd Edition, ©2005. Electronically reproduced by permission of
Pearson Education, Inc., Upper Saddle River, New Jersey. p. 281, Figure 11.12.
• 154-157: Friedrich Wilhelm Bessel - Wikipedia
• 158-161, 168-169: Milky way - Serge Brunier. Used with permission.
• 162: Ejnar Hertzprung – Courtesy Leiden University. Used with permission.
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• 162: Henry Russell – The University of Chicago / Yerkes Observatory. Used with permission.
• 163-167: Richard Powell, http://www.atlasoftheuniverse.com/hr.html, Creative Commons licence.
• 171: Henrietta Swan Leavitt - Wikipedia
• 172-173: Leavitt’s original Period-Brightness relation (X-axis in days, Y-axis in magnitudes) – SAO/NASA
• 174-175: Refined Hubble Constant Narrows Possible Explanations for Dark Energy – NASA/ESA/ A. Riess
(STScl/JHU)
• 176: Rampaging Supernova Remnant N63A – NASA/ESA/HEIC/The Hubble Heritage Team (STScl/AURA)
• 177: Large-scale distribution of gaseous matter in the Universe – Greg Bryan. Used with permission.
• 178: Edwin Hubble (1889-1953) – NASA
• 179: Hubble’s law – NASA
• 180: Big Bang Expansion - NASA
• 182-183, 188: Sloan Great Wall – Wikipedia
• 184: Full-Sky Map of the Oldest light in the Universe – Wikipedia
• 185: Spinning Black Holes and MCG-6-30-15 – XMM-Newton/ESA/NASA
• 186: Hubble Space Telescope – NASA
• 187: WMAP leaving Earth/Moon Orbit for L2 - NASA
• 188: Atlas Of Ancient And Classical Geography, J. M. Dent And Sons, 1912, Map 26;
• 188: Rotating Earth - Wikipedia/
Many thanks to Rocie Carrillo for work on the image credits.
Thanks also to Richard Brent, Ford Denison, Estelle, Daniel Gutierrez, Nurdin
Takenov, Andreas Thom, Dylan Thurston and several anonymous contributors
to my blog for corrections and comments.