The first three rungs of the cosmological distance ladder
The astronomical triangle. The North Celestial Pole is labeled NCP. The zenith is at Z. The latitude of the site is . An object in the western sky is marked by an asterisk. The hour angle of the object is t, which is positive in the western sky, negative in the eastern sky. The declination of the object is . Its elevation angle above the horizon is h, so the zenith angle is 90° − h. The azimuth A of an object is measured clockwise around the horizon, with north = 0°.
A gnomon consists of a wooden base with a hole drilled through it, and a vertical stick which fits tightly. It can be a pointed stick. We have fashioned a small sphere at the top. It is easier to measure the center of the elliptical shadow of the sphere than the end of the darker part of the shadow of a pointed stick.
The X-Y positions of the end of the shadow of a 632 mm gnomon used at College Station, Texas. For the December 21, 2010 observations we scaled the coordinates by 632/550, because a 550 mm gnomon was used on that date. The curvature of these loci changes with the declination of the Sun over the course of the year. The shadow lengths and the X-positions, along with the hour angle of the Sun obtained from the times of the observations and the time of minimum shadow length, make it possible to derive the declination of the Sun. In this case it is best to use observations obtained when the Sun is roughly 1 h or more from the meridian.
(a) The shadow length of a 632 mm gnomon on the day of the summer solstice, as measured at College Station, Texas. For illustrative purposes a hyperbola is fit to the data. (b) The shadow length of a 550 mm gnomon on the day of the winter solstice, as measured at the same location. A fourth order polynomial is fit to the data.
Geometry of the lunar eclipse (not to scale). At point G the Moon is halfway into the shadow of the Earth. At H the Moon is halfway out of the shadow. Simple arguments and measurements originating with Aristarchus allow us to estimate the distance to the Moon in terms of the Earth’s radius.
The angular motion of the Moon as a function of hour angle for an observer at College Station.
The cross staff. The cardboard cross piece slides up and down the yardstick. By using simple geometry we can use this device to determine the angular separation of objects in the sky.
The topocentric and geocentric right ascension and declination of the Moon on May 15, 2011. The topocentric values are calculated for College Station. The zenith angle of the Moon is greater for an observer situated on the surface of the Earth compared to a hypothetical observer at the center of the Earth. In other words, by observing on the surface of the Earth, the Moon appears to be lower in the sky compared to what would be seen by an observer at the center of the Earth. This elevation angle offset converts to varying shifts in right ascension and declination over the course of the night.
The angular motion (dashed curve) of an Earth-orbiting object as viewed by a hypothetical observer at the Earth’s center, as a function of its distance in . The mean angular motion (solid curve) of an Earth-orbiting object as viewed from College Station. The average is taken over 8 h centered on the meridian transit. On October 21, 2010 we measured the Moon to move 0.40±0.03°/h from six observations over 9.1 h. The implied value of the Moon’s distance is between 57 and on that occasion. On May 15, 2011 we measured the Moon to move 0.508±0.094°/h from three sets of observations taken over 3.9 h. The implied distance to the Moon is roughly 47 to .
Asteroid 1996 HW1 is circled in this unfiltered 90 s image obtained by DeBenedictis with a 15 cm Takahashi refractor at the Etscorn Observatory, Socorro, New Mexico, on July 24, 2008 at 08:17:27.8 UT. North is up, east is to the left.
Asteroid 1996 HW1 is circled in this unfiltered 90 s image obtained by Tabak and Pasricha with a 25 cm Meade reflector at Besant Hill School in Ojai, California, on July 24, 2008 at 08:17:28 UT, with an uncertainty of no more than s. North is up, east to the left. This image was taken at the same time as Fig. 10, but the asteroid is roughly 5 arc sec to the left (east) as observed at the other end of a baseline of 1130 km.
Gnomon data. CDT and CST represent Central Daylight Time and Central Standard Time. The height of the Gnomon was 632 mm on June 21, 2010 and 550 mm on December 21, 2010. All lengths are in millimeters.
Angular separations (May 15, 2011).
Derived and true topocentric positions of Moon (May 15, 2011). Also, the angular distance between the derived position of the Moon based on observations with the cross staff versus the true topocentric position of the Moon. For the fourth and fifth determinations the uncertainties in the right ascension and declination derive from the assumption that the angular separations of the Moon versus Vir and the Moon versus Saturn were accurate to .
Steeger orbit determination for asteroid 1996 HW1. The mean anomaly increases by per day according to JPL Horizons, so it is at the epoch of the Steeger solution.
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