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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. 11 Astronomical Unit (AU) - 1.496x10 m 15 Light Year (ly) – 9.461x10 m / 63 239 AU 16 Parsec (pc) – 3.086x10 m / 3.26 ly Radius of the Earth Eratosthenes worked out the size of the Earth around 240 BCE Radius of the Earth Eratosthenes used an observation 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 The Hubble Constant relates the distance to an object and its velocity as determined by its red shift. 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. Geometry – 1-2 AU Recap Earth parallax – 30-40 AU Trigonometric Parallax – 100pc Secular/Statistical Parallax – 500pc RR Lyrae variables – 700 kpc TRGB – 1 Mpc PNLF – 20 Mpc Cepheid Variables – 26 Mpc Tully-Fisher Relation – 100 Mpc Type 1a Supernovae – 2200 Mpc Hubble Constant – edge of observable universe – 3800+ Mpc.
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