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Planetary Distances
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Introduction :

 

 

In this experiment we shall measure the distance of various planets from the sun. Two different techniques shall be employed here. One for the internal planets, and the other for the external planets. Both will rely on the measurement of a certain parameter known as the elongation of the planet. This shall be measured by noting the coordinates of the planets on the ecliptic co-ordinate system.

Theory:

 

Far the purpose of this experiment, we shall make a few assumptions. Firstly, we shall take the orbits of all planets to be circular. This assumption is fairly accurate for most planets. The most eccentric orbit is that of Mercury (e=0.2), But even there we shall see that the calculated distance is close to the actual value. Secondly, we assume that the solar system is entirely planar. This is also fairly accurate, with the exception of that of mercury«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mo»§#8764;«/mo»«mn»7«/mn»«mo»§#176;«/mo»«/mrow»«/mfenced»«/math».

Inferior Planets:

                                                                          

                    Fig 1: The Sun-Earth-Inferior Planet system, showing elongation and maximum elongation.

 

Let S denote the Sun, P denote the planet, and E denote the Earth. Let us consider a frame of reference rotating with an angular velocity equal to that of the earths revolutionary motion. In this frame the line SE will be fixed. Now, the planet P, could be anywhere on the inner circle as shown in Fig 1. We define«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/math» to be the Elongation of planet P. Note that in the case of an inner planet the elongation cannot attain all values. It can attain all values ranging from«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo»§#176;«/mo»«/math» until a certain maximum. Note that this maximum will be attained when the line EP will be tangent to the the inner circle as shown in Fig 1. Let us denote this maximum elongation by«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»§#949;«/mi»«/math»

Now we see that

                                                                                                                   «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«mi mathvariant=¨bold-italic¨»§#949;«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/mrow»«mrow»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8658;«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«mo»=«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«mi mathvariant=¨bold-italic¨»§#949;«/mi»«/mfenced»«mo»§#215;«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/mtd»«/mtr»«/mtable»«/math»

To measure the angle Since SE is nothing but 1 Astronomical Unit(A.U.), thus we get the distance of the planet from the sun in Astronomical Units.

 

To measure the angle«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold-italic¨»§#949;«/mi»«/math» we shall make use of a plug-in called "Angle Measure". This can be enabled in the configuration window.

 

Superior Planets:

 

In order to understand how to measure the distance to a superior planet, we must first understand the concept of a Synodic period. This is defined as the time between two successive similar configuration of the Sun, Planet, Earth System. For example, 2 successive similar configurations of a general superior planet with the earth is shown in Fig 2. Let us try to understand this better. Let the synodic period of the planet P be Sy. Let the time period of revolution of the Earth be TE, and that of the Planet P be TP. Since the Earth is interior, the earth will revolve faster than planet P. By the Time the Earth takes one complete revolution the planet P will have moved further ahead. The earth would have to move a little further ahead up to position E2 to recover the same configuration.

                                                                               

Fig 2: The Sun-Earth-Superior Planet system, at two time frames, separated by one synodic time period, i.e., Sy

 

Let the angular velocity of the Earth be«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«/math», and similarly that for the planet P be«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«/math». Now, as the nE>nP the radius vector SE will gain on the radius vector SP at a rate of nE-nP. In one synodic period the radius vector SE shall gain on the vector SP by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«/math» radians.

Therefore,

                                                                                         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»y«/mi»«mo»§#215;«/mo»«mfenced»«mrow»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«/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»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8658;«/mo»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/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»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»y«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8658;«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»-«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»y«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8658;«/mo»«mn»1«/mn»«mo»/«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»y«/mi»«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»=«/mo»«mn»1«/mn»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»-«/mo»«mn»1«/mn»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«/mtd»«/mtr»«/mtable»«/math»

Note that Sy is a quantity that can be measured easily by an observer on earth.

Now, to measure the distance of the planet P to the Sun, we shall start our observations at opposition, i.e. when the elongation is«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»180«/mn»«mo»§#176;«/mo»«/math». This corresponds to situation 1 in Fig 2. Now we take our next observation after some time t, at situation 2. In this time the elongation has increased from«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo»§#176;«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»t«/mi»«mi mathvariant=¨bold-italic¨»o«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«/math» at a angular velocity of nE-nP.

Therefore,

                                                                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi mathvariant=¨bold-italic¨»§#952;«/mi»«mo»=«/mo»«mfenced»«mrow»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»-«/mo»«msub»«mi mathvariant=¨bold-italic¨»n«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«/mrow»«/mfenced»«mo»§#215;«/mo»«mi mathvariant=¨bold-italic¨»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mi mathvariant=¨bold-italic¨»t«/mi»«mo»§#215;«/mo»«mfenced»«mrow»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/msub»«mo»-«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»/«/mo»«msub»«mi mathvariant=¨bold-italic¨»T«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/msub»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨bold-italic¨»§#960;«/mi»«mo»§#215;«/mo»«mi mathvariant=¨bold-italic¨»t«/mi»«mo»/«/mo»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»y«/mi»«/mtd»«/mtr»«/mtable»«/math»

Thus the angle theta can be calculated. Now,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«mi mathvariant=¨bold-italic¨»S«/mi»«/math» can be measured from the sky. Thus we can now also calculate«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»E«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold-italic¨»S«/mi»«/math». Thus, using sine rule,

                                                                         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«mrow»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»E«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold-italic¨»S«/mi»«/mrow»«/mfenced»«/mrow»«mrow»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mfenced»«mrow»«mo»§#8736;«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mi mathvariant=¨bold-italic¨»E«/mi»«mi mathvariant=¨bold-italic¨»S«/mi»«/mrow»«/mfenced»«/mrow»«mrow»«mi mathvariant=¨bold-italic¨»S«/mi»«mi mathvariant=¨bold-italic¨»P«/mi»«/mrow»«/mfrac»«/math»

Thus we can calculate the distance to superior planets.

 

 

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