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1、21 February 2009Erotosthenes Project - Method 4 (Werner Schmidt Observatory)For those of us who wish to confine the measurement of the Earth diameter to actions within the boundaries of Cape Cod, this method has the potential to give almost an order of magnitude more accuracy than other methods. The
2、 key advantage is that the method is relatively immune to error due to atmospheric refraction.Measuring the parallax offset of stars relative to the Moon is only affected by the slight difference in refraction over a small angle. The major source of error when taking these measurements is the granul
3、arity of the CCD chip used with the Werner Schmidt Observatory (WSO) telescope.Theory of the approach:Short version:Examine the parallax of stars near the Moon due to the combination of the Earth orbital motion and the Earth rotation. Null out the effects of Earth orbital motion. Extrapolate the par
4、allax angular motion of stars due to the rotation of the Earth during transit of Moon as would be seen by an observer on the Earth equator. Measure a base line on the Cape. Determine the difference parallax angle of a star as seen from each end of that base line. Use the base line length and the par
5、allax of a star to calculate the distance of the Moon. Use the distance to the Moon and angular motion at equator to determine velocity at equator. Sum the velocity for 24 hours to determine circumference of Earth. Divide by pi to determine the Earth diameter.Picture 1 Parallax distance to Moon Not
6、to scaleLong version:We know the orbit of the Moon around the Earth to a reasonable degree of accuracy. The semi-major axis of the Moons orbit is 238,866 mi and the eccentricity is only 0.05490.The Moon appears to move in front of the celestial sphere (as it would be seen from the Earth center) at a
7、lmost constant angular velocity and distance relative to the Earth center. This ranges from about 28 to 40 seconds of arc per hour but changes less than second of arc per hour during any 3-hour period. What is more important is that the acceleration in that change is very small. That is to say, if w
8、e average the angular rate of change 3 hours before and 3 hours after the Moon transit time we will come very close to the angular rate of change for the Moon transit time.(VAo + VCo) / 2 = VBo(almost)(1)where: VAo = Orbital velocity before transit by delta time VBo = Orbital velocity at transit tim
9、e VCo = Orbital velocity after transit by delta timeWe will refer to the apparent motion of stars relative to the Moon as parallax angular velocity. The expression (1) above only relates to the parallax angular velocity due to the orbit of the Moon around the center of the Earth.An observer will see
10、 an additional component of parallax angular velocity due to the rotation of the Earth. This component of parallax angular velocity is a maximum when the Moon crosses its transit position over the observer. At that time, the observer is moving perpendicular to the line of sight to the moon. At all o
11、ther times, the angular velocity due to the rotation of the Earth will be less.In order to measure the diameter of the Earth, we measure the parallax angular velocity at three equally spaced times, the middle of which coincides with the transit of the Moon. The exact time spacing of these measuremen
12、ts will depend on star locations, Moon phase, Sun rise, and Sun set. Hopefully the spacing will be close to 3 hours each.Picture 2 Three test points top view Not to scalePicture 2 Three test points side view Not to scaleWe measure a parallax angular velocity by observing the parallax of stars relati
13、ve to the Moon on CCD images. That is to say, capture start and end CCD images of a star and the moon separated by a known time. Using the Moon as the frame of reference, measure the distance between the star in separate images. Scale that distance by the number of degrees per pixel and divide by th
14、e time difference between start and end.AVn = sqrt(Xn2Xn1)*(Xn2Xn1) + (Yn2Yn1)*(Yn2Yn1) * Scale / dt(2)where AVn = angular velocity from a pair of images (n = A, B, or C) X1 = horizontal position of 1st point relative to the Moon X2 = horizontal position of 2nd point relative to the Moon Y1 = vertic
15、al position of 1st point relative to the Moon Y2 = vertical position of 2nd point relative to the Moon Scale = number of degrees per pixel (see calibration below) dt = time between 1st point and 2nd point measurementsIn practice we make use of equation (1) and (2) to find AV by the alternate method:
16、AV = AVB - ( AVA + AVC ) / 2 (3)What is important is that in the above expression (3), the rate due to the orbital motion will cancel out and what we are left with is the observed difference in angular velocity due to the rotation of the earth, AV.If the Moon were in the same plane that the observer is rotating (i.e. the plane of the observer parallel to the equatorial plane), the parallax angular velocity due to the rotation of the
