Session Aims

All images unless otherwise stated from www.nasa.gov

▪ Sharing ways of measuring space

▪ Looking at some of the maths behind exploring space

▪ Lo

▪ to add text

Eratosthenes – born in Libya c 276 BC.

Studied in Athens and Alexandria

Most known in schools for the Sieve of Eratosthenes

How big is the Earth?

Applied knowledge▪ Eratosthenes discovered that on the summer

solstice, there was no shadow in the middle

of the day in Syene.

▪ Eratosthenes knew he could calculate the

distance from Syene (Aswan) to Alexandria.

Image from google maps

And…. How???

Alexandria

Aswan

And….

Parallel sun rays

Angle = 7.2°

Angle = 7.2°

▪ In Alexandria the shadow

made an angle of 7.2°

▪ The distance is 5,000 stadia

▪ We can work out the

circumference of the Earth

▪ (in stadia….)

5000 stadia

How good were his calculations?

250,000 stadia?

We think:

176.4m ≤stadia ≤184.8m

How accurate do you think he was?

What assumptions did he make?

How did he measure the distance?

Using modern measurements with his method yields over 99.8% accuracy!

Parallel sun rays

Angle = 7.2°

14 –14.7 x 103 km diameter

Now known to be 12.7 x 103 km

What about The Solar System?

▪ How far away are the planets?

▪ How far away is our Sun?

Nicolaus Copernicus

Johannes Kepler

Galileo Galilei

Heliocentric Astronomy

These three Scientists described many rules that

describe the movements of our planets.

Their observations and laws earned them the

collective title as the “Fathers of Modern Astronomy”.

“Fathers of Modern Astronomy”

Kepler’s Third Law

▪ Kepler discovered three laws of planetary motion.

▪ His laws described how planets move in relation to the sun and each other.

▪ His Third Law states

“The square of the orbital period of a planet is proportional

to the cube of the semi-major axis of their orbit”

𝑇2 ∝ 𝑟3

Data

Planet r=

Distance to the sun

(AU)

T=

Period (days)

Mercury 0.39 87.8

Venus 0.72 225

Earth 1 365.25

Mars 1.52 687

Jupiter 5.2 4332

Saturn 9.5 10759

𝑇2 ∝ 𝑟3

Possible activities:

Plot a graph of 𝑇2 against 𝑟3 to

find the constant (could also

do a log graph for A Level)

Leave off some data to see if

you can read off Jupiter or

Saturn, or extrapolate to the

period of the icy planets.

Calculate 𝑇2

𝑟3for each planet

and make an equation

This data was collected by Kepler for him to

calculate his laws.

Derive the proportionality from Newton’s equations

Problems with the data

R² = 1

-20000000

0

20000000

40000000

60000000

80000000

100000000

120000000

140000000

0 100 200 300 400 500 600 700 800 900 1000

T s

qu

are

d

r cubed

Kepler's Third Law

How do we measure the distance

to the sun??

Planet r=

Distance to the

sun (AU)

T=

Period (days)

Mercury 0.39 87.8

Venus 0.72 225

Earth 1 365.25

Mars 1.52 687

Jupiter 5.2 4332

Saturn 9.5 10759

How far away is the sun??

1. Close one eye, stretch your arm in front

of you and line it up against Nelson’s

Column.

2. Open your eye and close the other one,

keeping your finger where it is.

3. What do you notice?

PARALLAX

Mars in ‘opposition’.

By measuring the angle to Mars from two known places on

Earth, they were able to estimate the distance.

𝜃 𝜃

𝑑

𝐿

𝜃

𝜃

𝐿

𝜃

▪ Can you write the relationship

between L, d and 𝜃?

▪ How would you make your

measurements as accurate as you

could be?

▪ What are your assumptions for the

model?𝐿 ≈

𝑑

2𝜃(assuming

𝜃 is in radians)

𝜃

𝜃

𝐿

The results

𝛼 = 9.5 𝑎𝑟𝑐𝑠𝑒𝑐𝑜𝑛𝑑𝑠

𝐿 = 7.27 × 107km (distance to Mars)

Calculating distancesWe can combine our distance with Kepler’s third law to calculate 1AU

(distance from the Sun to the Earth).

As 𝑇2 ∝ 𝑟3, then for all planets, 𝑇2

𝑟3must be the same value for all planets

1) We know that 𝑇𝑀

2

𝑇𝐸2 =

𝑟𝑀3

𝑟𝐸3

2) Using the data 𝑇𝑀 = 687 days, 𝑇𝐸 = 365 days we can show 𝑟𝑀

𝑟𝐸= 1.52 (3𝑠𝑓)

7.27× 107km

𝑟𝐸𝑟𝑀

𝑟𝐸 = 139 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑘𝑚 = 1𝐴𝑈

Calculating distancesCassini and Richer measured 1AU as 139 million km.

The distance is now known to be 149597871km

How accurate were Cassini and Richer in 1672?

Is this more or less accurate than you expected?

Image from www.nasa.gov

Distance to the Sun In AU

Archimedes (3rd Century

BCE)

0.426

Aristarchus (3rd Century

BCE)

0.016 – 0.065

Hipparchus (2nd Century

BCE)

0.021

Posidonius (1st Century

BCE)

0.426

Ptolemy (2nd Century) 0.052

Wendelin (1635) 0.597

Christiaan Hugyens

(1659)

1.023

Cassini & Richer 0.925

What next?

▪ 1716

▪ Halley

▪ “Transit of Venus can help!”

TRANSITS

Transit of Venus

The orbits of Earth and Venus are at a small angle so they only line up with the Sun

occasionally – although the synodic period is 1.6 years, transits happen very rarely.

What is the problem with the different latitudes?

Sun Dt

A1

Venus

Tv

Earth

Te De

A2

1AU = ?

De = diameter of Earth

Tv = Orbital time for Venus

Te = Orbital time for Earth

Small angle assumption

Simplified theory

1AU = 𝐷𝑒

2𝜋∆𝑡(1

𝑇𝑣−

1

𝑇𝑒)

Highgate, London Wellington, New Zealand

Get ready to measure the time between the

start of the two transits by counting seconds

1 counted second = 1 minute for real

Counting creates error just like they had

Keep

watching

Sun DtA

Venus

Tv = 224.7 dayE

Earth

Te = 365.25 dayE

De = 12756000 m

A

1AU

De = diameter of Earth

Tv = Orbital time for Venus

Te = Orbital time for Earth

Calculate

1AU = 𝐷𝑒

2𝜋∆𝑡(1

𝑇𝑣−

1

𝑇𝑒)

What?Values

IN SI Units

(metres or

seconds)

Diameter of Earth 1.28 × 107𝑚 1.28 × 107𝑚

Orbital period of

Earth(𝑇𝐸)365.25 days

Orbital period of

Venus (𝑇𝑣)224.7 days

Time between

observers (∆𝑡)Your guess

(minutes)

Answer using 11.5 minutes

1.4844 × 1011𝑚 (𝑎𝑟𝑜𝑢𝑛𝑑 150 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑘𝑚)

1AU =

31,557,600s

19,414,080s

690s11.5

Modern day Transits

Transition Region and Coronal Explorer (TRACE) was a NASA satellite that

was always looking at the sun

It was able to observe Venus and Mercury transits

ParallaxParallax was the only way of measuring distances in the galaxy until the early 20th

Century

In 1838 Bessel first successfully measured the distance to a star using parallax.

We still use parallax to measure distances, GAIA is a satellite measuring the

distance to 1 billion stars, predominantly using parallax.

Image from www.esa.int

What happens to the angle as the star gets

further away?

At some point the stars will be so far away they seem to not move at all

Cepheid Variables

Images from https://www.cfa.harvard.edu/

https://amsp.org.uk/resource/year-10-support - "Measuring Space"

Exploring space

▪ Go to student.desmos.com

▪ Enter code 88WBDT (the link is also in the chat box)

▪ Teacher link

https://teacher.desmos.com/activitybuilder/custom/5ef9be

9d21f7d778901daae7

Questions?

@CozensNicole

There was a question on women in Space – I feel I didn’t do the question justice so here’s some more brilliant role models:

Astronauts:

Valentina Tereshkova – first woman in Space (in 1963!) – she was an amazing engineer too.

Svetlana Savitskaya – first woman to do a spacewalk

Mae Jemmison – First African-American woman in space – there’s some great books about her for all ages

Samantha Cristoforreti – probably my personal favourite modern-day astronaut –she’s always so keen to use her space work

for education – has some great videos on ESA/youtube

Astronomers:

Dame Jocelyn Bell Burnell – Astrophycisist who first discovered pulsars whilst carrying out her PhD – the team won the nobel

prize (omitting Bell Burnell). She does a massive amount of outreach work, won a 2.3 million dollar prize which she then used to

set up a fund to help minority astronomers (female/refugee/BAME). A truly amazing woman.

Margaret Huggins – Jointly developed spectroscopy (1880s) for identifying chemical composition of stars. We now use this to

look at the chemical composition of exoplanets.

Professor Lucie Green – works on the Solar orbiter and is a presenter on Sky at Night

Professor Michele Dougherty – magnetometry specialist – worked on Cassini (Saturn mission), persuaded the mission to divert

to study the rings and moons in more detail due to unexpected magnetic field readings.

Dr Vera Rubin – part of the team who first suggested the existence of dark matter due to discovering the arms of spiral galaxies

rotating much too slowly.