Cosmic Distance Ladder

How do we know the distances to objects in space?

Jason Nishiyama

Cosmic Distance Ladder

Space is vast and the techniques of the cosmic distance ladder help us measure that vastness.

Units of Distance

Metre (m) – base unit of SI. Astronomical Unit (AU) - 1.496x1011m Light Year (ly) – 9.461x1015m / 63 239 AU Parsec (pc) – 3.086x1016m / 3.26 ly

Radius of the Earth

Eratosthenes worked out the size of the Earth around 240 BCE

Radius of the Earth

Eratosthenes used anobservation and simple geometry to determine the

Earth's circumference

He noted that on the summer solstice that the bottom of wells in Alexandria were in shadow

While wells in Syene were lit by the Sun

Radius of the Earth

From this observation, Eratosthenes was able to

● Deduce the Earth was round.

● Using the angle of the shadow, compute the circumference of the Earth!

Out to the Solar System

In the early 1500's, Nicholas Copernicus used geometry to determine orbital radii of the planets.

Planets by Geometry

By measuring the angle of a planet when at its greatest elongation, Copernicus solved a triangle and worked out the planet's distance from the Sun.

Kepler's Laws

Johann Kepler derived three laws of planetary motion in the early 1600's. One of these laws can be used to determine the radii of the planetary orbits.

Kepler III

Kepler's third law states that the square of the planet's period is equal to the cube of

their distance from the Sun. In other words:

p2=a3

Problem

Though now we know the orbital radii of the planets in AU, we still don't know what the AU is.

Geometry can't tell us that.

To work out the AU, we need another technique!

PARALLAX!

To the Moon!

Circa 150 BCE, Hipparchus used parallax to work out the distance from the Earth to the Moon. He also confirmed his distance with clever geometry.

To the Moon

Hipparchus used geometry to work out that the Moon was at most ~67 Earth radii (~425 000km) from the Earth.

To the Moon

Hipparchus also used a solar eclipse to determine the parallax of the Moon and

hence it's distance. With this method ~71 Earth Radii (~454 400 km).

Planetary Parallax

It is possible to use the diameter of the Earth as a baseline. This is enough to provide

parallax to the planets.

This was first done by Cassini in 1673.

To the Stars

The stars are too far away for parallax based on the diameter of the Earth to be seen.

Parallax can be based on the Earth's orbit though.

Stellar Parallax

Even with the Earth's orbit as a baseline, stellar parallax can't be seen with the unaided eye. The stars are just too far away.

This was used as an argument against the Earth going around the sun.

Stellar Parallax

To see stellar parallax you need a telescope.

Stellar parallax was first measured by Bessel, Struve and Henderson (independently) around 1840.

Stellar Parallax

More recently, the HIPPARCOS satellite measured the parallax to over 100 000 nearby stars with unprecedented accuracy providing parallax distances up to 300 pc!

Stellar Parallax

The direct measurement of stellar parallax, also called trigonometric parallax, has it's limits. Even with our best Earth based equipment, the distance can only be measured to stars within 100 pc of the Earth.

Secular Parallax

The Sun moves about 4 AU per year. Over time this creates a longer baseline.

Secular Parallax is the parallax measured from this motion.

Multiple years can be used to increase this baseline.

Secular Parallax

Secular parallax works best on a cluster of stars.

Using a cluster averages the motion of the stars, which should be random, and hence cancel out the stellar motion leaving the parallax.

Statistical Parallax

Statistical parallax assumes the average radial velocity is equal to the average transverse velocity. Thus the average proper motion can be computed in linear units and the distance computed from that.

Secular and statistical parallax are good to a distance of about 500 pc and only work on star clusters.

Beyond Parallax

Standard Candles and Standard Rods

At this point we can no longer directly measure distance to an object. We must find another way of determining distance.

Standard Rods

If an object has a property that is measurable and independent of its distance but related to the object's size, then we have a standard rod.

Standard Rods

The expanding shell method is a standard rod.

If a shell of matter (or light) is expanding at a known rate, it is possible to determine the physical diameter of the shell and hence its distance.

Standard Candles

If we find a measurable property of a star that is independent of its distance but related to its intrinsic (actual) brightness, that makes that star a standard candle.

Then when we find a star with that property, we can then work out the distance to it using the inverse square law of light.

Interstellar extinction and reddening can cause errors.

Spectroscopic "Parallax"

Despite the name this is actually a standard candle method.

All stars have a spectral class (e.g. G2v) and stars of the same spectral class have similar brightness.

Once the spectral class and apparent brightness have been worked out, an approximate distance can be found.

Main Sequence Fitting

Similar to spectroscopic parallax, only with clusters.

By plotting the cluster's HR diagram against a "master", one can work out the difference between apparent and absolute magnitude for the cluster.

Cepheid Variables

In 1908 Henrietta Leavitt discovered that the period of a particular type of variable known as a Cepheid variable is directly related to it's maximum brightness.

Cepheid Variables

This discovery allowed Harlow Shapley to work out the distance to the centre of our galaxy in 1920.

It also allowed Edwin Hubble to determine the distance to M31 in the 1930's proving that the spiral "nebulae" were not part of our galaxy.

Cepheid Variables

Cepheid variables are our most important standard candle. They are very bright so are visible out to about 24 Mpc!There are some things to be careful of though:

● There are two types of Cepheid variables that have different period-luminosity functions.● Not all galaxies contain Cepheids.

RR-Lyrae Variables

Another type of distinctive variable star is the RR-Lyra variable.

This is a complement to the Cepheid variable as often a galaxy that doesn't have Cepheids will have RR-Lyrae Variables.

Unfortunately RR-Lyrae variables are 50 times less luminous than Cepheids so are really only good for the local group.

Tip of the Red Giant Branch

The TRGB method uses the fact that dying stars undergoing their helium flash cause a break in the red giant branch. These are generally the brightest stars in a galaxy and are easy to observe.

Planetary Nebula Luminosity Function

Planetary nebulae are formed by a narrow range of stars based on their mass. This creates a function that can be measured and compared to a master.

This method is good to about 20 Mpc.

Tully-Fisher Relation

R. B. Tully and J.R. Fisher discovered a relationship between the brightness of a spiral galaxy and the speed at which it rotates.

Tully-Fisher Relation

● Only works with spiral galaxies inclined to ours

● Luminosity now often done in IR to reduce error

● Good for distances up to 100 Mpc

Supernovae

Type 1a supernovae are excellent standard candles:● Due to their cause they have consistent peak brightness of around -19.5● Their high peak brightness make them useful as a standard candle out to about 2200 Mpc

The main drawback to supernovae is that they are rare.

Hubble Constant

The universe is expanding giving far away objects red shifts. The further an object is away, the greater the red shift.

Hubble Constant

Though this step can measure things to the edge of the observable universe. The exact value of the Hubble constant is uncertain and hence this method is not very accurate.

The Hubble Constant relates the distance to an object and its velocity as determined by its red shift.

RecapGeometry – 1-2 AU

Earth parallax – 30-40 AU

Trigonometric Parallax – 100pc

Secular/Statistical Parallax – 500pc

RR Lyrae variables – 700 kpc

Cepheid Variables – 26 Mpc

TRGB – 1 Mpc

PNLF – 20 Mpc

Tully-Fisher Relation – 100 Mpc

Type 1a Supernovae – 2200 MpcHubble Constant – edge of observable universe – 3800+ Mpc