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 CONTENTS Introduction Fermi's Piano Tuner Problem How Old is Old? If the Terrestrial Poles were to Melt... Sunlight Exerts Pressure Falling Eastward What if an Asteroid Hit the Earth Using a Jeep to Estimate the Energy in Gasoline How do Police Radars really work? How "Fast" is the Speed of Light? How Long is a Light Year? How Big is a Trillion? "Seeing" the Earth, Moon, and Sun to Scale Of Stars and Drops of Water If I Were to Build a Model of the Cosmos... A Number Trick Designing a High Altitude Balloon Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water Calendar Calculations Telling Time by the Stars - Sidereal Time Fields, an Heuristic Approach The Irrationality of The Irrationality of The Number (i)i Estimating the Temperature of a Flat Plate in Low Earth Orbit Proving that (p)1/n is Irrational when p is a Prime and n>1 The Transcendentality of Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions Maxwell's Equations: The Vector and Scalar Potentials A Possible Scalar Term Describing Energy Density in the Gravitational Field A Proposed Relativistic, Thermodynamic Four-Vector Motivational Argument for the Expression-eix=cosx+isinx Another Motivational Argument for the Expression-eix=cosx+isinx Calculating the Energy from Sunlight over a 12 hour period Calculating the Energy from Sunlight over actual full day Perfect Numbers-A Case Study Gravitation Inside a Uniform Hollow Sphere Further note on Gravitation Inside a Uniform Hollow Sphere Pythagorean Triples Black Holes and Point Set Topology Additional Notes on Black Holes and Point Set Topology Field Equations and Equations of Motion (General Relativity) The observer in modern physics A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File On Expansion of the Universe - PDF File

## Telling Time by the Stars - Sidereal Time

Problem:
Let the vernal equinox occur at noon solar time on March 21 of a certain year. Estimate the sidereal time at 3:00 pm solar time on November 29 of the same year.

Solution:
Fix the earth-sun line in [inertial] space. Let the earth rotate on its axis once a day, and let the celestial sphere rotate about the celestial poles once a year. As viewed from the north celestial pole, the earth will rotate counter clockwise, and the celestial sphere, clockwise. The rotation rate of the earth is one rotation every 24 solar hours (length of the solar day). The rotation rate of the celestial sphere is one rotation every tropical year (365.2422 days). The relative rotation rate of the earth and celestial sphere is

360o/day + 360o/(365.2422 days)

= 360o/day + 0.9856o/day

= 360.9856o/day

Thus, relative to the earth, the celestial sphere completes one rotation (sidereal day) in something less than a solar day. In fact, the sidereal day is just

(360/360.9856) x 24 hours = 23.9345 hours

= 23 hr 56 min 4.2 sec

i.e., approximately 3 min 56 sec shorter than the solar day. Another way to show the same thing is to form the ratio of the length of the sidereal day to that of the solar day:

(24 hr/solar day)/(23.9345 hr/sidereal day) = 1.0027 sidereal day/solar day

The tropical year is then

(365.2422 solar days) x 1.0027 = 366.2284 sidereal days

Now we may solve the problem of estimating the sidereal time on 3:00 pm, November 29. From noon on March 21 to noon on November 29 is 253 solar days. From noon to 3:00 pm on November 29 is an additional 3/24 = 0.125 solar days. Hence, the total elapsed time from the vernal equinox to 3:00 pm on november 29 is

253.125 solar days

Now 253 solar days = 253 x 1.0027 = 253.683 sidereal days. Thus, solar noon on November 29 is

(0.683 sidereal days) x (23.9345 hr/sidereal day) = 16.347 hr

= 16 hr 21 min.

(i.e., a sidereal clock at noon solar time on November 29 reads 4 hr 21 min ahead of a solar clock).

Also, 0.125 solar days = 0.125 x 1.0027 = 0.125 sidereal days (to within the accuracy of the calculation).

(0.125 sidereal days) x (23.9345 hr/sidereal day) = 2.992 hr

= 3 hr 00 min (to within rounding accuracy)

The sidereal time at 3:00 pm, November 29 is, therefore

(16 hr 21 min) + (3 hr 00 min) = 19 hr 21 min

An alternative approach would be to convert 253.125 solar days into sidereal days in a single step:

(253.125 solar days) x 1.0027 = 253.808 sidereal days

If all clocks start at noon on the vernal equinox, then the sidereal clock, at 3:00 pm solar, November 29, is reading 0.81 day past sidereal noon or

(0.808 sidereal day) x (23.9345 hr/sidereal day) = 19.339 hr

= 19 hr 20 min

Accuracy in the last digit is due to rounding error.