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Distance to the Moon
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Introduction :

 

                                 The method of parallax is a technique used to measure the distance to an object by taking two observations of it from two separate locations located in space. Here in this experiment we shall use this technique to find the distance to the moon by observing it from two locations on the same line of longitude. Observations shall be taken at the time of meridian transit. The observations shall be used to calculate the horizontal parallax of the Moon, which will give us the distance to the moon.

 

 

Theory:

                                         

                                          Fig 1:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»O«/mi»«msub»«mi mathvariant=¨bold-italic¨»P«/mi»«mn»2«/mn»«/msub»«mi mathvariant=¨bold-italic¨»Q«/mi»«/math» is the parallax of O with respect to the base line P1P2.

Let us first understand what parallax is. Consider a certain object O which is being observed from two different locations, P1 and P2 (See Fig 1). The parallactic angle is defined as «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»p«/mi»«mo»=«/mo»«mo»§#8736;«/mo»«msub»«mi mathvariant=¨bold-italic¨»P«/mi»«mn»1«/mn»«/msub»«mi mathvariant=¨bold-italic¨»O«/mi»«msub»«mi mathvariant=¨bold-italic¨»P«/mi»«mn»2«/mn»«/msub»«mo».«/mo»«/math» Note that this is equal to angle«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»O«/mi»«msub»«mi mathvariant=¨bold-italic¨»P«/mi»«mn»2«/mn»«/msub»«mi mathvariant=¨bold-italic¨»Q«/mi»«/math» Thus, if we are observing O with respect to a distant background, then the parallactic angle is the change in the apparent position of the object w.r.t. the fixed background. We say that p is the parallactic angle of O between P1 and P2. The line P1P2 is called the baseline.

We define the geocentric parallaxp, to be the parallactic angle for any object, between the actual observer, and a hypothetical observer located at the center of the earth. (See Fig 2). We define the horizontal parallax ,P, to be the geocentric parallax when the object appears on the horizon for the observer.

                                                                    

                          Fig 2: The angle p is the parallax of the moon. The special case of angle P is called the horizontal parallax.
Both are with respect to the baseline OC

First we shall derive a relation between P and p,

We notice that

                                                                       «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mfrac»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»p«/mi»«/mrow»«msub»«mi mathvariant=¨bold-italic¨»R«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«mrow»«mn»180«/mn»«mo»§#176;«/mo»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»0«/mn»«/msub»«/mrow»«/mfenced»«/mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8658;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»p«/mi»«mo»=«/mo»«mfrac»«msub»«mi mathvariant=¨bold-italic¨»R«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mi mathvariant=¨bold-italic¨»d«/mi»«/mfrac»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»0«/mn»«/msub»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»

Where RE is the radius of the earth, d is the distance of the object from the centre of the earth, z0 is the zenith distance for the real observer, and z is the zenith distance for the hypothetical observer at the centre of the earth who has the same zenith as the actual observer ( the geocentric zenith distance). Also, we see that sin P = R/d . Thus 

                                                                     «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨»sin«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»p«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mo»§nbsp;«/mo»«mo»§#215;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mo»(«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»0«/mn»«/msub»«mo»)«/mo»«/math»

Also, since both P and p are small angles, 

                                                                                            «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»p«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold¨»sin«/mi»«mo»(«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»0«/mn»«/msub»«mo»)«/mo»«/math»

Note that sin P gives the distance to the object measured in Earth Radii (RE).

Let us now get back to using these newly derived equations for measuring the distance to the moon. Consider two observatories O1 and O2 located on the same longitude on the surface of the earth. Both of them shall be used to measure the zenithal distance of the moon during meridinal transit (The moons azimuthal distance is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo»§#176;«/mo»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»r«/mi»«mo»§nbsp;«/mo»«mn»180«/mn»«mo»§#176;«/mo»«/math»). Let the observatories have latitude «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»1«/mn»«/msub»«mi mathvariant=¨bold-italic¨»§#925;«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»and«/mi»«mo»§nbsp;«/mo»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»2«/mn»«/msub»«mi mathvariant=¨bold-italic¨»S«/mi»«/math»

                                                 

                                  Fig 3: The situation being considered to find the distance to the moon
 
 

Let z10 and z20 be the observed zenith distance from the two observatories; and z1 and z2 be the corresponding geocentric zenith distance. Let p1 and p2 be the two parallactic angles for the two observatories.

 

NOTE: We take all angles as positive counterclockwise and negative clockwise.Furthermore latitude in northern hemisphere is taken positive and in southern hemisphere is negative. Here in Fig 3, z1 and«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»1«/mn»«/msub»«/math» are positive and z2 and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»2«/mn»«/msub»«/math»are negative.

Now, we see that

                                                                         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»2«/mn»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mo»§nbsp;«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»10«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»1«/mn»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»a«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»20«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»2«/mn»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold¨»sin«/mi»«mo»(«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»10«/mn»«/msub»«mo»)«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»a«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold¨»sin«/mi»«mo»(«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»20«/mn»«/msub»«mo»)«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»b«/mi»«/mtd»«/mtr»«/mtable»«/math»

Using Eqn. 4

                                                                     «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»P«/mi»«mo»=«/mo»«mfenced»«mrow»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»2«/mn»«/msub»«/mrow»«/mfenced»«mo»/«/mo»«mfenced»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»10«/mn»«/msub»«/mfenced»«mo»-«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»20«/mn»«/msub»«/mfenced»«/mrow»«/mfenced»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mn»5«/mn»«/math»

We now use Eqn. 3 to eliminate p1 - p2

                                                                      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»p«/mi»«mn»2«/mn»«/msub»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfenced»«mrow»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»10«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»20«/mn»«/msub»«/mrow»«/mfenced»«mo»-«/mo»«mfenced»«mrow»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»1«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»2«/mn»«/msub»«/mrow»«/mfenced»«/math»

We now use Eqn 2 to eliminate z1 - z2, and use this in Eqn. 5

                                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»P«/mi»«mo»=«/mo»«mfrac»«mfenced»«mrow»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»10«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»20«/mn»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi mathvariant=¨bold-italic¨»§#934;«/mi»«mn»2«/mn»«/msub»«/mrow»«/mfenced»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»10«/mn»«/msub»«/mfenced»«mo»-«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«msub»«mi mathvariant=¨bold-italic¨»z«/mi»«mn»20«/mn»«/msub»«/mfenced»«/mrow»«/mfrac»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Eqn«/mi»«mo».«/mo»«mo»§nbsp;«/mo»«mn»6«/mn»«/math»

Here, we can measure all the values in the RHS. Thus we can calculate P. Then, sin P will give us the distance to the moon in units of the earths Radii.

 

 

 

 

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