The Cosmological Distance Ladder

1. The Cosmological Distance Ladder It's not perfect, but it works!

2. First, we must determine how big the Earth is. This is done with principles used by surveyors.

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

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

5. In the 3rd century BC Eratosthenes obtained the first estimate of the circumference of the Earth. He used two observations of the Sun on the first day of summer.

6. 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.

7. Plots of the end of the shadow of a gnomon, as obtained in College Station, TX, on the first day of winter (top), one day after the fall equinox (middle), and on the first day of summer (bottom).

8. After measuring the length of the shadow on two key days, this is what we found.

9. Some simple geometry/trigonometry gives us the elevation 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 the maximum height of the Sun gives us our longitude. We found 96o 06.5 arcmin west of Greenwich.

10. From a gnomon experiment done on September 3, 2006, I determine the latitude and longitude of a location in South Bend, IN. I found that South Bend and College Station are 13.78 degrees apart along a great circle arc. I drove from South Bend to College Station and found that the two locations are 1263 miles apart. But, that’s a squiggly route, not the distance along a great circle arc. Using a map and map tool, I determined that the great circle distance was 940 miles. 360/13.78 X 940 = 24,557 miles, our estimate for the circumference of the Earth. The “right answer” is 24,901 miles, so our value was 1.4% too small.

11. Determining the distance to the Moon But first we must measure the angular size of the Moon.

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

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

14. Eyeball measures of the angular size of the Moon over 35 orbital periods of the Moon. The mean angular size is 31.18 arcmin, a little over half a degree.

15. Aristarchus (310-230 BC) cleverly figured out how to use the geometry of a lunar eclipse to determine the distance to the Moon in terms of the radius of the Earth. Here s is the angular radius of the Sun, t is the angular radius of the Earth’s shadow at the distance of the Moon, PM is the “parallax” of the Moon, and PS is the “parallax” of the Sun.

16. It is not too difficult to show that s + t = PM + PS . We know that the Sun has just about the angular size of the Moon, because when the new Moon occasionally eclipses the Sun, a total solar eclipse only lasts a few minutes. But we can also measure the angular size of the Sun when it near the horizon and we are looking through a lot of the Earth’s atmosphere. From two such observations I found s = 15.4 +/- 0.4 arcmin.

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

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

19. My value for the angular radius of the Earth’s shadow at the 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 is small, only 0.14 arcmin. The “parallax” of the Moon turns out to be PM = 15.4 + 39.19 – 0.14 = 55.17 arcmin .

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

21. Next, we must determine the scale of the solar system. Copernicus (1543) correctly determined the relative sizes of the orbits of the known planets. But we needed to know the Astronomical Unit in km. One way to determine this is to observe a transit of Venus across the disk of the Sun from a variety of 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.

23. The big problem with these observations is related to the thick atmosphere of Venus.

24. A much easier way to measure the scale of the solar system is to determine the orbit of an asteroid that comes close to the Earth. Then, using two telescopes at different locations of the Earth, take two simultaneous images of the asteroid against the background stars. The asteroid ill be in slightly different directions as seen from the two locations.

25. Asteroid 1996 HW1, imaged from Socorro, NM, on July 24, 2008 at 08:27:27.8 UT. The same asteroid imaged on the same date at the same time, but 1130 km to the west, in Ojai, CA.

26. An arc second is a very small angle. The thinnest human hair (diameter ~ 18 microns) at arm’s length subtends an angle of just about 7 arc seconds. From our two asteroid images, we found that the asteroid was shifted 5.05 +/- 0.61 arcsec as viewd from the two locations. This translates into a distance to the asteroid of 4.66 X 107 km. From a solution for the orbit of this asteroid, we determined that it was 0.294 Astronomical Units distant. This means that the Earth-Sun distance is roughly 1.59 +/- 0.19 X 108 km. The correct length of the AU is 1.496 X 108 km.

27. We determined the size of the Earth. Our value was 1.4% smaller than the true value. We determined the distance to the Moon. Our value was 3.3% larger that the known mean distance. We determine the Earth-Sun distance. Our value was 6% larger than the true value. Rarely in science do we know the true value of something we are measuring. Otherwise it wouldn’t be called research! But we think it was a useful exercise to determine these first 3 rungs of the cosmological distance ladder, but distances throughout the cosmos depend on them.

28. By measuring stellar parallaxes we can determine the distances to the nearby stars. The Hipparcos satellite measured many thousands of parallaxes.

29. As the stars in the Hyades star cluster (in Taurus) move through space their proper motions appear to converge at a particular point on the sky. This is an effect of perspective. Think of a flock of birds all flying towards the horizon.

30. If the angular distance of the cluster from the apparent convergent point is , then there is simple relationship between the radial velocity of a star in the cluster and its transverse velocity: vT = vR tan() Since the transverse velocity vT = 4.74 dpc (where is the proper motion in arcsec per year and d is the distance in parsecs), we can use the moving cluster method to get estimates of the distance to all the Hyades stars with measured proper motions and radial velocities. The average would be the distance to the cluster.

31. The method of main sequence fitting Since main sequence stars of a particular temperature all have the “same” intrinsic brightness, a cluster with a main sequence some number of magnitudes fainter than the Hyades gives us its distance in terms of the distance to the Hyades.

32. Pulsating stars of known intrinsic brightness RR Lyrae stars have mean absolute magnitudes of about MV ~ +0.7. If you know the apparent magnitude and absolute 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 for interstellar extinction and d is the distance in parsecs.

33. If you find a Cepheid variable star with a period of 30 days, it is telling you, “I am a star that is 10,000 times more luminous than the Sun!”

34. Cepheid variable stars were used by Shapley to determine the distances to globular clusters in our Galaxy. This allowed him to determine the distance to the center of the Galaxy. In 1924 Edwin Hubble used Cepheids to show that the Andromeda Nebula is very much like our Milky Way Galaxy. Both are large ensembles of a couple hundred billion stars. And the Andromeda Galaxy is a couple million light years away. In the 1990's, using the Hubble Space Telescope, we used observations of Cepheids in somewhat more distant galaxies to determine their distances.

35. Perhaps the best standard candles to use for extragalactic astronomy are white dwarf (Type Ia) supernovae. There is a relationship between their absolute magnitudes and the shape of their light curves that allows us to determine their absolute magnitudes.

36. For Type Ia supernovae we define the “decline rate” as the number of magnitudes the object gets dimmer in the first 15 days after maximum light in the blue. Fast decliners are fainter objects.

37. On the x-axis is the number of magnitudes that a Type Ia super- nova gets fainter in the first 15 days after maximum light. Krisciunas et al. (2003)

38. In the near- infrared, Type Ia supernovae are even better standard candles. Only the very fastest decliners are fainter than the rest. Similar to a figure in Krisciunas et al. (2004c)

39. A typical Type Ia supernova at maximum light is 4 billion times brighter than the Sun. As a result, such an object can be detected halfway across the observable universe, further than 8 billion light-years away.

40. As with RR Lyr stars and Cepheids, if you know the absolute magnitude and the apparent magnitude of an object, you can calculate its distance, providing you properly account for any effects of interstellar dust.

41. Because galaxies exert gravitational attraction to each other, galaxies have “peculiar velocities” on the order of 300 km/sec. For galaxies at d ~ 42 Megaparsecs, their recessional velocities are roughly 3000 km/sec for a Hubble constant of 72 km/sec/Mpc. So one can get an estimate the galaxy's distance using Hubble's Law (V = H0d) with a 10% uncertainty due to the peculiar velocity. At a redshift of 3 percent the speed of light (9000 km/sec), the effect of any perturbations on the galaxy's motion are correspondingly smaller (roughly 3 percent).

42. The bottom line is that from a redshift of z = 0.01 to 0.1 (recessional velocity 3000 to 30,000 km/sec) one can use the radial velocity of a galaxy to determine its distance. Beyond z = 0.1 one needs to know the mean density of the universe and the value of the cosmological constant 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 Energy that is causing the universe to accelerate in its expansion.

44. For more information on the extragalactic distance scale, see: www.astr.ua.edu/keel/galaxies/distance.html