Modelling StarsModelling the planets

The Astrolabe

Mathematical Modelling Summer School 2009

25th May 2009

Modelling StarsModelling the planets

Astrolabe

Used in medieval times toI Showing the night sky will look likeI Telling the timeI Calculating rising and setting times of celestial objects

Modelling StarsModelling the planets

Placing the planets

The sun and planets lie in a plane called the ecliptic, denotedby the the numerals I to XII on the star plate. (These numeralsgive the position of the sun at the start of each month.)

Sun � VIIMercury ' VII–VIVenus ♀ VI – VMars ♂ VI–VJupiter X IIISaturn Y X-IXMoon (dark) $ IX-VIII

Modelling StarsModelling the planets

Showing the night sky

To show the night sky at 2200GMT(=2300IST) line up todaysdate on the star plate with 22 on the overlay. Visible objects lieinside the thick distorted circle.Visible objects include

I Saturn YI CassiopeaI CygnusI Lyra

Modelling StarsModelling the planets

Rising and setting timeTo calculate the time at which an object will rise (set) rotate thestar plate until the object is on the eastern (western) side. Therise time (GMT) on the overlay corresponds to today’s date onthe star plate. Add one hour for IST.Rise times

Rise time GMT ISTX Jupiter 2300 0000♀ Venus 0200 0300♂ Mars 0200 0300' Mercury 0300 0400� Sun 0400 0500

EWDateTime

Modelling StarsModelling the planets

Telling the time

To tell the time measure the angular altitude of two stars.Rotate the star plate until the two stars are at the correctaltitudes on the overlay. As before today’s date on the star plategives the time on the overlay.

θ1

star 1

θ2

star 2

θ1θ2

star 1

star 2

DateTime

Modelling StarsModelling the planets

Planetary orbitsAssume planets move in regular circular orbits around the sunin the ecliptic plane. (This takes us up to Copernicus, butignores improvements by Kepler, Newton and Einstein.) Orbithas fixed radius r , and angle in plane increases linearly withtime θ − θ0 = ωt . In cartesian coordinates

x = r cos ωt (1)y = r sin ωt (2)

Or more elegantly

x = Rz (3)y = Iz (4)

z = reiωt (5)

Modelling StarsModelling the planets

The Maya and 584

Dresden codex.

Modelling StarsModelling the planets

Orbital parameters

r♁ 0.723 aur♀ 1.000 aur♂ 1.523 auT♀ 224.7 daysT♁ 365.2 daysT♂ 687.0 days

ω = 2π/T

Modelling StarsModelling the planets

Earth and Venus

� Ari

es

19th

Apri

l] Tau

rus

20th

May

^Gemini

21stJune

_Cancer

22ndJuly

�Leo

23rdA

ugust

V̀irg

o

22nd

Sep

tember aL

ibra

23rd

Octo

ber

bSco

rpio

22ndN

ovember

cSagittarius

21st December

dCapricornus

20th January

eAquariu

s

18th

Febua

ry

fPis

ces 20th

Mar

ch

� ♀0 ♁ 0

♀ 100

♁

100

♀200♁200

♀

300♁300

♀400

♁ 400♀

500

♁

500

♀ 584

♁584

Modelling StarsModelling the planets

EpicyclesThings are simpler with a heliocentric solar system

Motion of mars as seen from Earth

θ♁♂ = arg (z♁ − z♂) (6)

Alternatively

θ♁♂ = arctany♁ − y♂

x♁ − x♂(7)

Modelling StarsModelling the planets

Retrograde motion

� Ari

es

19th

Apr

il] Taur

us

20th

May

^Gemini

21stJune

_Cancer

22ndJuly

�Leo

23rdAugust

`Virgo

22ndSeptem

ber

aLibra

23rdO

ctober

bScorpio

22nd November

cSagittarius

21st December

dCapricornus

20th January

eAquarius

18th

Febua

ry

fPi

sces 20

thM

arch

From time to time Mars appears to move backwards (retrogrademotion). Modelled by epicycles: a circle within a circle.

Modelling StarsModelling the planets

An astronomical insult

Helen Monsieur Paroles, you were born under a charitable starParoles Under Mars, I. . . . When he was predominant.Helen When he was retrograde I think rather.Paroles Why think you so?Helen You go so much backwards when you fight.

Alls Well That Ends Well. William Shakespeare.