The Cosmological Distance Ladder

The Cosmological Distance Ladder

It's not perfect,but it works!

First, we must determinehow big the Earth is.

This is done with principles used bysurveyors.

But first let’s talk about the stars at night. If you live in Texasand take a long exposure photograph towards the north, thestars will appear to rotate around the North Celestial Pole,which is close to the star Polaris.

The reason we have seasons is that the axis of rotation ofthe Earth is tilted to the plane of its orbit around the Sun.The northern hemisphere it tilted the most towards theSun on June 21st, which is the first day of summer. TheSun is highest in the sky at local noontime on that date.

In the 3rd century BCEratosthenes obtainedthe first estimate ofthe circumference ofthe Earth. He usedtwo observations ofthe Sun on the firstday of summer.

We can use a pointed column like the Luxor obelisk in Paris(or a stick) to cast a shadow of the Sun to determine our latitude.

Plots of the end of theshadow of a gnomon,as obtained in CollegeStation, TX, on the first day of winter(top), one day afterthe fall equinox (middle),and on the first day ofsummer (bottom).

After measuringthe length ofthe shadow ontwo key days,this is what we found.

Some simple geometry/trigonometry gives us theelevation angle of the Sun above the horizon:

On June 21st hmax = 82o 15.9 arcmin.

On December 21st hmax = 36o 06.6 arcmin.

90o minus the average of these two values gives us the latitude of College Station, namely 30o 48.7 arcmin. According to Google Earth, the right answer is 30o 37.2 arcmin. Knowing that we are in the Central time zone, measuring the time of themaximum height of the Sun gives us our longitude.We found 96o 06.5 arcmin west of Greenwich.

From a gnomon experiment done on September 3, 2006,I determine the latitude and longitude of a location inSouth Bend, IN. I found that South Bend and CollegeStation are 13.78 degrees apart along a great circle arc.

I drove from South Bend to College Stationand found that the two locations are 1263 miles apart.But, that’s a squiggly route, not the distance along agreat circle arc. Using a map and map tool, I determinedthat the great circle distance was 940 miles.

360/13.78 X 940 = 24,557 miles, our estimate for thecircumference of the Earth. The “right answer” is24,901 miles, so our value was 1.4% too small.

Determining the distance to the Moon

But first we must measure the angular size of the Moon.

Because the Moon’s orbit around the Earth is not circular,the angular size of the Moon varies over the course ofthe month. Likewise, the Sun’s angular size varies overthe course of the year.

A simple device for determining the angular size of the Moon.

Eyeball measuresof the angular sizeof the Moon over35 orbital periodsof the Moon. Themean angular sizeis 31.18 arcmin, a little over half adegree.

Aristarchus (310-230 BC)cleverly figured out how touse the geometry of a lunareclipse to determine thedistance to the Moon in terms of the radius of theEarth. Here is the angularradius of the Sun, is theangular radius of the Earth’sshadow at the distance ofthe Moon, PM is the “parallax”of the Moon, and PS is the“parallax” of the Sun.

It is not too difficult to show that PM + PS .

We know that the Sun has just about the angular size ofthe Moon, because when the new Moon occasionallyeclipses the Sun, a total solar eclipse only lasts a fewminutes. But we can also measure the angular size ofthe Sun when it near the horizon and we are looking through a lot of the Earth’s atmosphere. From two suchobservations I found = 15.4 +/- 0.4 arcmin.

Ptolemy (2nd century AD) asserted that the Earth’s shadow isabout 2.6 times the angular size of the Moon. Here you cansee that he is basically right.

On June 15, 2011, there was a total lunar eclipse visiblein Europe. I downloaded 6 images of the Moon and, usinga ruler and compass, determined that the Earth’s shadowwas 2.56 +/- 0.03 times the size of the Moon.

My value for the angular radius of the Earth’s shadow atthe distance of the Moon is (31.18/2) X 2.56 ~ 39.19 arcmin.

From the distance to the Sun (see later) we find that PS issmall, only 0.14 arcmin.

The “parallax” of the Moon turns out to be

PM = 15.4 + 39.19 – 0.14 = 55.17 arcmin .

Since sin(PM) = radius of Earth / distance to Moon, itfollows that the Moon’s distance in Earth radii is 1/sin(PM).On June 15, 2011, we found that the Moon was 62.3 Earthradii distant. The true range is 55.9 to 63.8 Rearth.

Next, we must determine the scale of the solar system.Copernicus (1543) correctly determined the relativesizes of the orbits of the known planets. But weneeded to know the Astronomical Unit in km.

One way to determine this is to observe a transitof Venus across the disk of the Sun from a varietyof locations spread out over the Earth. (Using the Earth as a baseline, we determine the distance.)

These transits have occurred in 1761, 1769, 1874, 1882, and 2004. The next one occurs in 2012.

The big problemwith these observationsis related to the thickatmosphere of Venus.

A much easier way to measure the scale of the solarsystem is to determine the orbit of an asteroid that comesclose to the Earth. Then, using two telescopes at differentlocations of the Earth, take two simultaneous images ofthe asteroid against the background stars. The asteroidill be in slightly different directions as seen from thetwo locations.

Asteroid 1996 HW1,imaged from Socorro, NM,on July 24, 2008 at08:27:27.8 UT.

The same asteroid imagedon the same date at thesame time, but 1130 kmto the west, in Ojai, CA.

An arc second is a very small angle. The thinnesthuman hair (diameter ~ 18 microns) at arm’s lengthsubtends an angle of just about 7 arc seconds.

From our two asteroid images, we found that the asteroidwas shifted 5.05 +/- 0.61 arcsec as viewd from the twolocations. This translates into a distance to the asteroidof 4.66 X 107 km. From a solution for the orbit of thisasteroid, we determined that it was 0.294 AstronomicalUnits distant. This means that the Earth-Sun distanceis roughly 1.59 +/- 0.19 X 108 km. The correct lengthof the AU is 1.496 X 108 km.

We determined the size of the Earth. Our value was1.4% smaller than the true value.

We determined the distance to the Moon. Our valuewas 3.3% larger that the known mean distance.

We determine the Earth-Sun distance. Our value was6% larger than the true value.

Rarely in science do we know the true value of something we are measuring. Otherwise it wouldn’tbe called research! But we think it was a usefulexercise to determine these first 3 rungs of thecosmological distance ladder, but distances throughoutthe cosmos depend on them.

By measuringstellar parallaxeswe can determinethe distances tothe nearby stars.The Hipparcossatellite measuredmany thousandsof parallaxes.

As the stars in the Hyades star cluster (in Taurus) movethrough space their proper motions appear to convergeat a particular point on the sky. This is an effect ofperspective. Think of a flock of birds all flying towardsthe horizon.

If the angular distance of the cluster from the apparent convergent point is , then there is simplerelationship between the radial velocity of a star inthe cluster and its transverse velocity:

vT = vR tan()

Since the transverse velocity vT = 4.74 dpc (whereis the proper motion in arcsec per year and d isthe distance in parsecs), we can use the moving cluster method to get estimates of the distance toall the Hyades stars with measured proper motions andradial velocities. The average would be the distanceto the cluster.

The method of main sequence fitting

Since main sequencestars of a particulartemperature allhave the “same”intrinsic brightness,a cluster with a mainsequence some number of magnitudesfainter than theHyades gives us its distance in termsof the distance to theHyades.

Pulsating stars of known intrinsic brightness

RR Lyrae stars have mean absolute magnitudes of aboutMV ~ +0.7. If you know the apparent magnitude andabsolute magnitude of a star, you can determine its distance using this formula

MV = mV + 5 – 5 log d ,

where the apparent magnitudes have been corrected forinterstellar extinction and d is the distance in parsecs.

If you find a Cepheid variable star with a period of30 days, it is telling you, “I am a star that is 10,000times more luminous than the Sun!”

Cepheid variable stars were used by Shapley to determinethe distances to globular clusters in our Galaxy. Thisallowed him to determine the distance to the center ofthe Galaxy.

In 1924 Edwin Hubble used Cepheids to show that theAndromeda Nebula is very much like our Milky WayGalaxy. Both are large ensembles of a couple hundredbillion stars. And the Andromeda Galaxy is a couplemillion light years away.

In the 1990's, using the Hubble Space Telescope, weused observations of Cepheids in somewhat more distantgalaxies to determine their distances.

Perhaps the best standard candles to use for extragalacticastronomy are white dwarf (Type Ia) supernovae. Thereis a relationship between their absolute magnitudes andthe shape of their light curves that allows us to determinetheir absolute magnitudes.

For Type Iasupernovae wedefine the “declinerate” as the numberof magnitudes theobject gets dimmerin the first 15 daysafter maximum light in the blue.Fast decliners arefainter objects.

On the x-axisis the number ofmagnitudes thata Type Ia super-nova gets fainterin the first 15 daysafter maximumlight.

Krisciunas et al. (2003)

In the near-infrared, TypeIa supernovaeare even betterstandard candles.Only the veryfastest declinersare fainter thanthe rest.

Similar to a figure in Krisciunas et al. (2004c)

A typical Type Ia supernova at maximum light is4 billion times brighter than the Sun. As a result,such an object can be detected halfway across theobservable universe, further than 8 billion light-yearsaway.

As with RR Lyr stars and Cepheids, if you know theabsolute magnitude and the apparent magnitude ofan object, you can calculate its distance, providingyou properly account for any effects of interstellardust.

Because galaxies exert gravitational attraction to eachother, galaxies have “peculiar velocities” on the orderof 300 km/sec. For galaxies at d ~ 42 Megaparsecs,their recessional velocities are roughly 3000 km/secfor a Hubble constant of 72 km/sec/Mpc. So onecan get an estimate the galaxy's distance using Hubble'sLaw (V = H0 d) with a 10% uncertainty due to thepeculiar velocity. At a redshift of 3 percent the speedof light (9000 km/sec), the effect of any perturbationson the galaxy's motion are correspondingly smaller(roughly 3 percent).

The bottom line is that from a redshift of z = 0.01 to0.1 (recessional velocity 3000 to 30,000 km/sec)one can use the radial velocity of a galaxy to determineits distance.

Beyond z = 0.1 one needs to know the mean densityof the universe and the value of the cosmologicalconstant in order to get the most accurate distances.In fact, it was the discovery that distant galaxies(with redshifts of about z = 0.5) are “too faint”that implied that they were too far away. This was the first observational evidence for the Dark Energythat is causing the universe to accelerate in itsexpansion.

For more information on the extragalactic distancescale, see:

www.astr.ua.edu/keel/galaxies/distance.html