Introduction:  The  aim  of  this  paper  is  to  find  out  the  methods   to  calculate  the  distance  between  the  Earth  and   the   star.   The   whole   paper   is   focused   on   the   finding   the   distance   between   the   Earth   and   the   sun.   I   decided   to   focus   on   finding   the   distance   of   sun   and   Earth   is   because   the   universe   is   expanding   and   this   will   make   the   calculation   change   through   time.   In   the   solar   system   the   gravity   is   much   larger   than   the   force   of   the   universe   expanding,   so   the   distance   will   not   expand.   (“Newton   Science   Magazine”)   In   this   paper   I   have   created   three   ways   to   find   the   distance   in   the   solar   system.   The   first   one   is   based   on   if   I   know   the   size   of   the   image   of   the   object   and   the   real   length   I   can   find   out   the   distance.   In   this   method   I   have   designed   an   experiment   to   prove   the   hypothesis   and   also   to   find   the   equation.   The   second   method   I   create   is   a   combination   between   the   Kepler’s   Third   Law   and   the   trigonometry.   I   made   the   hypothesis   of   the   position   of   the   planets   and   the   sun   and   at   last   I   used   the   calculation   to   prove   the   method   works.  The  last  method  I  create  is  based  on  the   Centrifugal   force   and   gravity   force.   The   three   methods   all   contain   some   error   and   because   there  are  some  limitations  of  my  methods.

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Method 1:

Personal Project  method  1:    Create  by  Adam  Grade  10   1.        Method1   In  order  to  find  out  the  distance  of  the  star  the  first  thing  I  am  going  to  do  is  to   find   out   the   pattern   between   how   many   percent   you   see   the   object   and   the   distance.   It’s   a   comment   knowledge   that   the   object   decreases   when   you   step   farther   away   from   the   object.   The   most   important   thing   is   how   does   the   object   decrease?   In  order  to  find  out  this  idea  of  how  object  decreases  I  need  a  real  experiment.   Designing  the  experiment  form:   Title:  The  pattern  between  how  many  percent  you  see  the  object  and  the  distance.                                                                       By:  Adam  Hsieh                                  Experiment  number:  01   1.The  question:   How  big  the  object  you  see  when  you  stand  X  meters  away  from  the  object?

2.What are  you  going  to  find?   I  am  going  to  find  the  equation  between  the  distance  and  how  big  the  object  you  see.           3.Important  things  about  the  experiment:              1.  When  I  do  the  experiment,  need  to  close  one  eye.  (If  you  use  two  eyes  the  result   will  be  in  error,  this  is  because  there  is  a  distance  between  left  eye  and  right  eye.)              2.  When  you  measure  you  need  to  keep  the  same  distance  from  the  ruler.              3.  The  object  must  be  90  degrees  with  the  table  of  floor.       4.What  do  you  need  for  the  experiment?        1.one  30  cm  long  ruler                                                                                    6.a  protractor      2.some  objects  (different  length)                                            7.  a  table        3.a  3  meters  long  ruler                                                                                    8.  12  pens  or  pencils        4.some  tapes          5.some  books   5.The  prediction  for  the  experiment:            The  size  of  the  object  you  see  decrease  when  you  step  farther  from  the  object.

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6.Steps of  the  experiment:              Step1:  Use  the  tapes,  protractor  and  the  book  to  make  the  object  stays  90  degrees   with  the  flat  table  or  floor.              Step2:  Use  the  3  meters  long  ruler  to  make  sure  you  stand  in  1,2,3…8,9,10  meters   away  from  the  object,  and  then  put  a  pencil  or  pen  on  the  place  you  need  to  stand.              Step3:  Use  the  30  cm  long  ruler  to  find  the  distance  between  you  and  the  measure   point.  (where  the  ruler  is)              Step4:  Make  sure  the  ruler  is  parallel  with  you.            Step5:  Close  an  eye,  and  each  time  you  measure  try  to  keep  the  same  eye  closed.              Step6:  Measure  the  size  of  the  object  you  see  with  the  ruler.              Step7:  Repeat  from  Step  2  to  Step  6  until  you  change  another  object.                  Step8:  When  you  change  another  object  do  the  Step  1  to  Step  6  again.            (The  experiment  below  all  close  the  left  eye)   7.The  data:     Table:  01                  Object  1:  The  flag                    The  distance  between  you  and  the  measure  point:  30  cm              The  original  length  of  the  object:  60.50cm                       Distance  from  the  object:  (meters)    The  size  you  see  the  object:  (cm)                                                                        1   21.20                                                                        2   9.95                                                                        3   5.85                                                                        4   4.55                                                                        5   3.45                                                                        6   2.80                                                                        7   2.45                                                                        8   2.15                                                                        9   1.90                                                                      10   1.45     Picture  of  the  object  or  process  of  the  experiment:

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Object 2:    music stand                              The  distance  between  you  and  the  measure  point:  30  cm              The  original  length  of  the  object:    126.63cm                         Distance  from  the  object:  (meters)    The  size  you  see  the  object:  (cm)   1   37.50   2   17.00   3   12.80

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4 5   6   7   8   9   10   Picture  of  the  object  or  process  of  the  experiment:

9.50 6.95   5.85   5.05   4.89   3.65   2.95

Object  3:  music  stand                    The  distance  between  you  and  the  measure  point:  30      cm              The  original  length  of  the  object:  142.4  cm                             Distance  from  the  object:  (meters)    The  size  you  see  the  object:  (cm)   1   40.18   2   18.75   3   13.20   4   9.55   5   8.30   6   6.35   7   5.85   8   4.80   9   4.05

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10 Picture  of  the  object  or  process  of  the  experiment:

3.55

8.Does  the  data  support  the  prediction:    The  data  does  support  the  prediction,  so  we  can  use  these  data  to  find  the  equation.

2.The  Graph  of  each  data:

Graph01-­‐1 Object  1  The  flag

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Graph 01-­‐2        Object  2  The  120  cm  music  stand

Graph01-­‐3    Object  3:  The  140  cm  music  stand

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Relationship between  each  graph:    The  graph  of  three  objects:   Graph:  01-­‐4

The  Graph  plus  the  average  (the  orange  line):        Graph  :  01-­‐5

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What I  find  out  from  the  object:    1.  The  equation  of  the  distance  and  how  big  the  object  you  see  is  in

this equation:

1. Equation of  object1:   F (x) = 24.63 ℯ^(-0.32 x)

e=2.7182818285 x

⇒ f (x) = 24.63 × ⎡⎣(2.7182818285)−0.32 ⎤⎦

⇒ f (x) = 24.63 × (0.7261490371)x   2. Equation  of  object2:   F (x) = 47.48 ℯ^(-0.31 x)

e=2.7182818285

⇒ f (x) = 47.48 × ( 2.7182818285 −0.31 ) ⇒ f (x) = 47.48 × ( 0.7334469562 )

x

x

3. Equation of object3: F (x) = 51.07 ℯ^(-0.31 x)

e=2.7182818285

⇒ f (x) = 51.07 × ⎡⎣( 2.7182818285 )

f (x) = L × ( 0.7334469562 )

x

−0.31 x

⎤ ⎦

⇒ f (x) = 51.07 × ( 0.7334469562 ) 4. Put the equation into order: L= the length of the object X= the distance between you and the object F (x)= the size of the object you see x

f (x) = L × ( 0.7334469562 )

x

2. The  problem  of  the  equation:        1.  When  x=0,  y  should  be  the  length  of  the  object,  but  in  the  three  equation   when  x=0,  y  is  not  the  length  of  the  object.

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2. The  equation  tells  us  that  if  you  stand  1  meter  away  from  the  object  the  size   you  see  the  object  will  always  be  0.7334469562  of  the  object.  I  found  out  this  is   not  true  with  the  reality,  in  fact  the  length  of  the  object  you  see  decrease  slower   and  slower  and  will  not  always  decrease  in  the  same  speed.      3.  The  data  shows  there  must  be  something  wrong  in  this  situation.

3. Find  out  the  mistake:        1.  If  the  equation  is  wrong  shows  that  maybe  this  is  not  an  exponential

equation.      2.   Maybe   there   is   no   pattern   between   the   distance   and   the   size   you   see   the   object,   but   the   pattern   between   the   percentages   of   the   object   you   see   and   the   distance  between  you  and  the  object.

4.Make an  improvement:

I  think  number  2  is  more  correct  than  number  1,  so  I  am  going  to  make  a  new   table.

5. Table 02:                                                                                                    Object1

The distance between you and the object (cm)

The percentage of the object you see (%)

0 1 2 3 4 5 6 7 8 9 10

100 35.04132231 16.44628099 9.669421488 7.520661157 5.702479339 4.628099174 4.049586777 3.553719008 3.140495868 2.396694215

Percentage= the length of the object you see/ the original object length

Object2                 The distance between you and the object The percentage of the object you see (m) Percentage= the length of the object you see/ the (cm) original object length

0 1 2 3 4 5 6 7 8 9 10

100 29.61383558 13.4249388 10.10818921 7.502171681 5.488430862 4.619758351 3.987996525 3.86164416 2.88241333 2.329621733

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Object3                 The distance between you and the object The percentage of the object you see (%) Percentage= the length of the object you see/ the (cm) original object length

0 1 2 3 4 5 6 7 8 9 10

100 28.21629213 13.16713485 9.269662921 6.706460674 5.828651685 4.459269663 4.108146067 3.370786517 2.844101124 2.492977528

6. Graph of Table 02: Graph 2: Object1 Graph: 02-1

Object 2 Graph:02-2

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Object 3 Graph:02-3

7. What I found out of Table 02: 1. When I look at the three graph of the table 2, the graph is not correct. 2. The lines in these three graph aren’t going to y=100, when the x=0. 3. This tells me an important massage: This graph is not exponential line!

8. Make prediction: 1. When you walk away from the object each meter the object will decrease in a different speed. 2. I think I need to create a new table for the decrease of percentage from each object.

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9. Table 3: The multiplying power between N meters and N-1 meters Object1 N meters (distance) The multiplying power (The percentage of object you see in N meters/ the percentage of the object you see in N-1 meters)

1 2 3 4 5 6 7 8 9 10

N meters (distance)

0.3504132231 0.4693396226 0.5879396986 0.7777777777 0.7582417583 0.811594203 0.8749999999 0.8775510203 0.8837209304 0.7631578947 Object2 The multiplying power (The percentage of object you see in N meters/ the percentage of the object you see in N-1 meters)

1 2 3 4 5 6 7 8 9 10

N meters (distance)

0.2961383558 0.4533333335 0.7529411762 0.7421875002 0.7315789475 0.8417266186 0.8632478632 0.9683168317 0.7464212679 0.8082191783 Object3 The multiplying power (The percentage of object you see in N meters/ the percentage of the object you see in N-1 meters)

1 2 3 4

Â

0.2821629213 0.4666500754 0.703999999 0.7234848485

13 Â

5 6 7 8 9 10

0.8691099476 0.765060241 0.9212598424 0.8205128206 0.8437500001 0.8765432097

10. Dots of the Table 3: Object 1: Graph: 3-1

Object 2 Graph: 3-2

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Object 3 Graph:3-3

Put all three in one picture: Graph:3-4

11.What I found out from the picture: 1.The picture shows the multiplying power goes up, but the lines go a little down when x is 10. 2. Finding the multiplying power of each dot is even harder, because if there is a little error during the measurement and the data will change a lot. 3.So I decided to find the average of table 2.

12. The average of the Table 02: Â

15 Â

N meters (distance)

Table:2-2 The multiplying power (The percentage of object you see in N meters/ the percentage of the object you see in N-1 meters)

1 2 3 4 5 6 7 8 9 10 13. The Graph of the average in 12. Graph:2-4

0.3095715 0.463107677 0.681626958 0.747816709 0.786310218 0.806127021 0.886502569 0.888793558 0.824630733 0.815973428

14. Find the equation of average graph: 1. The data shows that the distance 7 and 8 is not correct. 2. The graph shows that the line goes up very fast first, and then quickly slows down, but the graph keeps going up. 3. Each dot in the data should not be bigger than 1, and smaller than 0. 4. Even though the graph still goes up very slowly, I will just consider the line in the back is a linear line. 5. In this graph I will use a quadratic line plus a linear line to show the equation. 6.Finding the equation: 6-1 Quadratic line: y = ax 2 + bx + c          (1,  0.3095715), (5, 0.786310218), (10, 0.815973428) Put in the equation:

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0.3095715=a+b+c ---------1 0.786310218= 25a+5b+c --------2 0.815973428= 100a+10b+c ---------3

3 − 1 ⇒ 99a + 9b = 0.506401928 − − − −4 2 − 1 ⇒ 24a + 4b = 0.476738718 − − − −5 5 ÷ 4 ⇒ 6a + b = 0.1191846795 − − − −6 4 ÷ 9 ⇒ 11a + b = 0.05626688089 − − − −7 7 − 6 ⇒ 5a = −0.06291779861− − − −8 8 ÷ 5 ⇒ a = −0.01258355972 a = −0.01258355972 ⇔ 6a = −0.07550135833

⇒ b = 0.1946860378 ⇒ 0.3095715 = −0.01258355972 + 0.1946860378 + c ⇒ c = 0.1274690219                                            Put  in  together:   y = −0.01258355972x 2 + 0.1946860378x + 0.1274690219              6-­‐2  The  linear  line                        The  linear  line  comes  after  the  vertex  of  the  quadratic.                    Find  the  vertex  of  the  equation:   −b                             Vertex = , y = ax 2 + bx + c   2a The  vertex  of  the  equation  is:   −0.1946860378             =7.735729878.   −0.02516711944 Put  the  vertex  X  in  the  quadratic  equation:          Y=3.14741787   This   is   not   correct   because   the   size   of   the   object   won’t   be   bigger   as   you   stand   farther.  I  think  when  y  is  almost  1  the  line  will  be  almost  parallel  with  X-­‐axis.   Put  y=1  in  the  equation   0=   −0.01258355972x 2 + 0.1946860378x + 0.1274690219 -­‐1   0=   −0.01258355972x 2 + 0.1946860378x − 0.8725309781   x=

x=

−0.1946860378 ±

( 0.1946860378 )2 − 4(−0.01258355972)(−0.8725309781) 2(−0.01258355972)

−0.1946860378 ± −6.015529368 *10 −3 2(−0.01258355972)

X  is  undefined,  because  the  number  in  the  square  root  cannot  be  less  than  0.

13.The possible equation of the model: 1.Model 1: L=The Length of the object F (x)= the size of the object you see (the measure point is 30 cm from your eyes) X= the distance between you and the object B= the multiplying power The equation:

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f (x) = L × ( b )

x

b ∝ x,1 > b > 0

When x → ∞ ⇔ b → 1 2.Model 2:

L= The length of the object F (x)= the size of the object you see (the measure point is 30 cm from your eyes) X= the distance between you and the object A = the multiplying power between x and x-1 meters

f (x) = L × ( a1 × a2 × a3 × a4 × ......× ax−2 × ax−1 × ax ) ⇒ f (x) = L × ax !(a1 × a2 × a3 × ......× ax−1 × ax )

a1 = 0.3095715,a∞ → 1

14. Which  model  is  more  correct:   In my opinion the second model is better than the first one, because the equation is not exponential line. There is still a problem about my equation: I still need to find the equation for the second model, the equation for the multiplying power A is still not found. 15.Think over again what is the purpose of the equation: I am going to use this equation to find the distance from the moon, and I need to know both the size of the moon and the size I see the moon, and then using the equation, the equation for how many percent of the object will you see if you stand in X meter away from the object. Here is how I am going to use the equation: The things I need to know for finding the distance: 1.The length of the object (L) 2.The size of the object you see [f (x)] < the measure point is 30 cm away from your eyes> 3.Pretend I already find out the equation for ax ! L= The length of the object F (x)= the size of the object you see (the measure point is 30 cm from your eyes) X= the distance between you and the object

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f (x) = L × ax ! Put  the  number  f  (x)  and  L  in  the  equation,  and  then  you  will  get  the

a!

number of   x .   Put  the  number  in  the  equation  for   ax !  and  then  you  will  find  the   distance  (x).   This  is  how  it  works.

<Wait a minute! If I know the length of the object, and the size I see the object, and the distance of the measure point, I already can find the distance of the object! > This can be simply solved! 16. The simplest way to find the distance:

The picture is from: http://www.google.as/imgres?q=相似三角形 &hl=en&gbv=2&biw=1280&bih=646&tbm=isch&tbnid=9fNnITOzjC8sTM:&imgrefurl=http://gsyx.c ersp.com/article/browse/3603922.jspx&docid=WCILr1FRp7ikZM&w=186&h=241&ei=VgVdTszTP MTXrQeYyY3GDw&zoom=1&iact=hc&vpx=418&vpy=264&dur=1279&hovh=192&hovw=148&tx= 110&ty=138&page=4&tbnh=149&tbnw=115&start=46&ndsp=15&ved=1t:429,r:6,s:46

In the picture point A is the point where the eyes of the people are, and line AD is the measure point (30cm), line DE is   the   size   of   the   object,   and   line   CB  is  the  length  of  the  object.  We  are  trying  to  find  line   AB ,  which   is  the  distance  between  you  and  the  object. The equation: If we want to find the distance we can use the simple equation:

AD : AB = DE : BC

And  this  is  the  final  equation  for  solving  the  distance.

It’s time  to  try  on  the  moon!     17.  The  new  experiment02:

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Title: Finding the distance from Earth and moon. Experiment number: 02 Date: 1.The question: How far is the moon from earth? 2.What  are  you  going  to  find?   The distance between moon and Earth. 3.Important  things  about  the  experiment:   1.Try to limit the error of the observation. 2.The real size of the moon 3.The distance of the measure point 4. Please make the observation during the full moon 5. When you make observation please close one eye 4.What  do  you  need  for  the  experiment?   1. A ruler 3. A flat table (which is 90 degrees with the floor) 2. Some tape 5.The  prediction  for  the  experiment:   The moon is pretty far from the Earth. 6.Steps  of  the  experiment:   Step 1. Taped the ruler on the leg of the table, and make sure the ruler is 90 degrees with the Earth surface. Step 2. Close one eye, and when you measure keep your eye 30 cm from the ruler. Step 3. Start measuring, and measure few times to limit the error. Step 4. Use the equation that I found in the first experiment to find the distance of the moon. 7.The  data:   The distance of the measure point (cm) 10

The size of the moon you see (cm)

The distance of the

NA

NA

20

4

moon < 10 km>

20 30 40 50

NA NA NA NA

NA NA NA NA

I am not planning to use this way to find the distance, because this will cause too much error and this is not a smart way to solve this problem, so I going to use different method to find out the distance.

Short  Conclusion:

Because the  “Error”  is  too  huge  and  this  make  the  method  can  only  be   used   for   finding   the   close   distance   objects,   not   for   finding   the   distance  between  the  Earth  and  other  planets.

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Method 2  Cited  from  website                                                      (Alternative  for  method  1)

It’s  not  a  smart  idea  to  measure  the  distance  by  just  looking  how  big  the  object   is,   because   when   you   are   very   far   away   from   the   object   each   tinny   measurement   error  will  cause  the  huge  distance  error.  For  example  if  the  object  is  1  light  year   far   and   you   see   the   star   is   0.001   cm,   and   even   the   error   is   0.000001   will   still   cause  a  gigantic  error  for  distance.          Today   no   one   is   finding   the   distance   of   the   moon   by   just   looking   the   size   of   moon.  People  are  using  radar and  the  radio  wave  to  find  out  the  distance  of  the   near   planet.”   The   speed   of   the   radio   wave   is   same   as   the   light   speed,   which   is   about   299792.485km/sec.”   (Number   from   Newton   science   magazine)   Today   people  find  the  distance  between  the  near  planets  using  the  time  of  the  reflection   of  the  radio  wave.

The  equation  of  the  wave:

v=λ× f

V=  the  speed  of  the  wave   F=  frequency  (Hz)-­‐-­‐-­‐-­‐times/second   λ =  Wave  length       In  the  Vacuum  (space)  the  speed  of  the  radio  wave  will  always  be  the  light  speed.

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If we  want  to  know  a  planet  near  us  we  can  find  out  the  distance  by  finding  the   time  of  the  radio  wave  comes  back.   For  example:  If  there  is  a  planet  near  earth  and  it  takes  10  seconds  for  the  radio   wave  to  come  back  to  earth.  And  then  we  can  easily  calculate  the  distance  of  the   planet.     The  distance  of  the  planet=  10*  299792.485/2=  1498962.425km

The equation:     The  distance  of  the  planet  (km)=  time  of  the  radio  wave  to   reflect  back  (seconds)*299792.485/2     This  is  the  best  way  to  find  the  distance  between  some  close  planets.

Radio wave  is  a  special  long  wave,  which  cannot  be  seen  by  human  eyes.  Radio   wave   can   travel   without   any   media,   and   they   also   travel   light   speed.”   Radio waves are a type of electromagnetic radiation with wavelengths in the

electromagnetic spectrum longer than infrared light.” (“Radio wave” Wikipedia) Radio wave is a good way to find the distance of the planets in our solar system, but there is still a problem. Sun does not reflect radio wave, so we can’t use this method to find the distance.

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Method 3:   Aim: Create a method to find the distance between the Earth and sun Searching the way of finding the distance. There is a way using the Kepler's laws of planetary motion. There are three laws: “Law 1: The orbit of every planet is an ellipse with the Sun at one of the

two foci. Law 2: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. Law 3: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.” ( “Kepler's laws of planetary motion“Wikipedia.) We can  use  the  third  law  of  the  Kepler’s  law  of  planetary  motion  to  find  the   distance  between  sun  and  earth.   The  equation:

In  here  “a”  is  the  semi-­‐major  axis  of  the  planet  orbit.   is  the  planetary  orbital   period,  and  K  is  any  number.     Any  planet  in  our  solar  system  the  number  K  will  always  be  the  same,  so  we  can   use  this  data  to  find  the  farthest  distance  between  the  Earth  and  the  sun.     Start  to  find  the  distance  between  the  Earth  and  the  sun:     1. Important  things  to  know:                  According   to   the   first   law   of   the   Kepler’s   law   of   planetary   motion,   the   orbit  of  the  planet  is  not  circle  is  ellipse,  so  the  distance  between  the  sun  and   the  Earth  is  not  always  the  same.  Using  the  third  law  we  can  find  the  farthest   distance   between   the   sun   and   the   moon,   and   not   even   this   we   can   use   this   method  to  find  the  distance  between  any  planets  in  the  solar  system.     2. Things  need  to  know  before  finding  the  distance:   1.The  planetary  orbital  period  of  the  Earth  (365.2  days)  -­‐-­‐-­‐-­‐common   knowledge.     2.  The  planetary  orbital  period  of  the  Venus  or  other  planet.        Way  to  find  the  orbital  period  of  other  planet:          In  order  to  find  the  orbital  period  of  the  Venus  or  other  planet,  you  have  to   make   observation   using   a   telescope.   You   can   find   the   orbit   period   of   other   planet  by  watching  how  many  days  between  the  planet  cross  the  sun  twice.

τ

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3.The distance  between  Earth  and  other  planet.  (This  can  easy  be  found  by   using  method  2,  the  radio  wave.)     3. Start  using  the  equation  to  find  the  distance  between  the  sun  and  Earth:   The  orbital  period  of  Earth  (365.256363004 days)-----(data from “Earth” Wikipedia.)

The orbital period of Venus (224.700 69 days)------(data from “Venus” Wikipedia)

The maximum distance between Earth and Venus ( 261× 10 6 km)-­‐-­‐-­‐-­‐-­‐(data from  “Venus  fact  sheet”  Author/Curator: Dr. David R. Williams, dave.williams@nasa.gov NSSDC, Mail Code 690.1 NASA Goddard Space Flight Center Greenbelt, MD 20771 +1-301-286-1258 ”

The minimum distance between Earth and Venus ( 38.2 × 10 6 km)-­‐-­‐-­‐-­‐-­‐(data

from “Venus  fact  sheet”  Author/Curator: Dr. David R. Williams, dave.williams@nasa.gov NSSDC, Mail Code 690.1 NASA Goddard Space Flight Center Greenbelt, MD 20771 +1-301-286-1258 ”

Use the calculator, start calculate: = The maximum distance between the Earth and the sun =  The  maximum  distance  between  the  Venus  and  the  sun      B=  the  distance  between  Earth  and  Venus

= The  orbital  period  of  Earth    =  The  orbital  period  of  Venus

Put in  numbers:

And

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( ae ) = 0.7233321085 ( ae ) + b

38.2 × 10 6 ≤ B ≤ 261× 10 6 6 but  when  ae=av+b  ,  B  =   38.2 × 10   As  you  see  the  picture  below  when  Venus  is  on  the  farthest  point  from  Earth   Venus  is  on  the  opposite  side  of  the  Earth,  and   ,  so  when  B  is  the   smallest  value  then  Venus  is  on  the  closest  point  of  Earth.  (38.2  x  106)

Picture from:   http://www.google.co.id/imgres?q=orbit+of+venus+and+earth&um=1&hl=id&biw=1280&bih=646&tbm=i sch&tbnid=RFI4sMF_BF-­‐4wM:&imgrefurl=http://sci.esa.int/science-­‐ e/www/object/index.cfm%3Ffobjectid%3D38251&docid=-­‐ odVD30NVHCH9M&w=410&h=410&ei=GUlgToSNMMKtrAfmpun7Dw&zoom=1&iact=hc&vpx=850&vpy=9 0&dur=539&hovh=224&hovw=224&tx=135&ty=124&page=1&tbnh=149&tbnw=149&start=0&ndsp=16&v ed=1t:429,r:4,s:0

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0.2766678915(ae ) = 38.2 × 10 6 38.2 × 10 6 ⇒ ( ae ) = 0.2766678915 ⇒ ( ae ) = 138071677.9

so the  distance  between  the  sun  and  the  Earth  is  138071677.9km.   “The   real   distance   between   the   sun   and   the   Earth   is   149597870.7   km.”-­‐-­‐-­‐-­‐-­‐data  from  Newton  science  magazine  number  33  page  68.     My  calculate  error  is  1-­‐(138071677.9/  149597870.7  )=7.704783996%     Ways  to  improve  the  error:   The   biggest   error   of   this   question   is   the   shortest   distance   between   the   Earth   and   Venus  is  not  the  distance  we  want.

Picture from:   http://www.google.co.id/imgres?q=orbit+of+venus+and+earth&um=1&hl=id&biw=1280&bih=646&tbm=i sch&tbnid=bUQvnUbbAzQGNM:&imgrefurl=http://www.harmony.demon.co.uk/spangledskies/venustrans. html&docid=H2DmYJfIAVBumM&w=330&h=260&ei=GUlgToSNMMKtrAfmpun7Dw&zoom=1&iact=hc&vpx =988&vpy=103&dur=575&hovh=199&hovw=253&tx=204&ty=110&page=1&tbnh=144&tbnw=183&start= 0&ndsp=16&ved=1t:429,r:5,s:0

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The orbit  of  the  planet  is  not  in  a  same  flat  line,  and  each  orbit  has  different  angle.

The picture  is  draw  by  myself.   Point  A  is  sun.   Point  B  is  Venus.   Point  C  is  Earth.   Line   BC  is  the  shortest  distance  between  the  Earth  and  Venus  ( 38.2 × 10 6 )   And  we  used  the  third  law  of  Kepler’s  law  to  found  out:

AB = 0.7233321085AC And   BD ⊥ AC   In  order  to  find  the  distance   AC ,  we  need  to  find  the  angle  of ∠BAD .   ∠BAD  is  the  angle  of  the  Inclination to Ecliptic. Ways to find the other planet’s Inclination to Ecliptic is extremely difficult. There is an equation that is use for finding the Inclination, and it is so hard that I can’t understand, so I am not going to talk about it. The equation:

<From  “Inclination”  Wikipedia>   ∠BAD =  3.394 71° <data from “Venus” Wikipedia> Time to calculate:

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“The real  distance  between  the  sun  and  the  Earth  is  149597870.7   km.”-­‐-­‐-­‐-­‐-­‐data  from  Newton  science  magazine  number  33  page  68.

My calculate  error:  1-­‐(135837659.1/149597870.7)=1-­‐ 0.908018667=9.198133301%     Conclusion  for  method  3:      The  second  time  my  calculate  error  is  even  bigger!        I   must   start   to   think   why   the   calculation   went   wrong.  My  calculation

increased the   error   and   it   should   decrease   the   error   because   I   add   the   factory   of   Inclination to Ecliptic.  The  best  way  to  solve  this  question  is  to  ask  the  expert  and   set   up   the   interview.   I   set   up   the   interview   with   the   math   teacher.   In   the   Interview   he   gave   me   the   idea   of   the   distance   that   I   used   is   not   the   closest   distance   between   the   sun   and   the   moon.   This   made   me   think   of   is   there   any   other  factories  that  I  still  need  to  add.     Even   though   I   add   the   factor   of   inclination,   but   it   is   not   enough.   I   found   out   I   still   need   5   more   factories.   Those   five   factories   are   semi-­‐major   axis,   orbital   eccentricity,   longitude   of   the   ascending   node,   argument   of   periapsis,   and   mean   anomaly.  I  think  I  need  to  add  them  one  by  one.

Correcting the  equation:

1. The first  factor  I  want  to  add  is  the  argument  of  periapsis.

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In order  to  add  new  factories  I  need  some  pictures  to  help  me  understand   the  orbit.

Picture  from:  http://en.wikipedia.org/wiki/Longitude_of_the_ascending_node     Then  after  I  saw  this  picture  I  need  to  draw  down  my  own  picture  to  make  this   less  complex.

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In this  picture  dot  S  represent  the  sun,  the  dot  V  represent  the  Venus,   and  the  dot  E  represent  the  Earth.  If  I  want  to  find  the  distance  I  need   to  draw  another  picture  in  the  center  part  of  the  picture  above.     2.Drawing  the  new  picture:

This picture  shows  the  3D  of  the  distance  between  the  sun,  Venus   and  also  Earth.     3.Things  need  to  know  before  calculation:         VE =The  shortest  distance  between  the  Earth  and  Venus  ( 38.2 × 10 6 km)                     SE =The  maximum  distance  between  the  sun  and  Earth             SV =  The  maximum  distance  between  the  sun  and  Venus             ∠1 =  The  Inclination of Venus to Ecliptic (3.394 71° <data from “Venus” Wikipedia>)

∠2 = The angle of argument of  periapsis.  (54.852   29o <data  from  “Venus”   Wikipedia>)          We  used  the  Kepler’s  Third  Law  we  find  out  that:  0.7233321085 SE =   SV

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4.Time to  calculate:

From  the  equation  1,2,3  we  can  find  the  line  SE  in  the  following  equation:

5.  The  distance  between  the  Earth  and  the  sun  is   1.36438097320209 × 10 8 km ,  136438097.320209km

The real  distance  between  the  Earth  and  the  sun  is  149597870.7  km   (Newton  science  magazine).     My  calculate  error:  1-­‐(my  calculation/  the  real  distance)

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My calculate  error:  1-­‐(136438097.320209/149597870.7)=1-­‐ 0.91203234833348=  8.796765166652%     The  error  had  decreased,  but  I  haven’t  used  the  angle  of  argument  of   periapsis,  it  means  that  the  equation  is  same  as  the  one  I  did  with  the   inclination.   The   error   decreases   because   I   used   11   digits   in   the   first   calculation,   and   I   used   14   digits   in   the   second   calculation.   In   this   method  I  think  it  is  hard  to  reduce  the  error  because  the  planets  are   moving  all  the  time.  Each  angle,  and  the  measurement  between  two   planets  and  also  the  orbit  period  of  two  planets  might  contain  error   that  can’t  be  reduced.  The  error  in  this  equation  is  less  than  10%  and   this  is  acceptable.     6.  The  write  the  method  in  another  form.          This  method  is  created  by  myself,  and  is  used  to  find  the  distance   between  the  sun  and  the  planets.              DA=  the  distance  between  the  Planet  A  and  Sun          DB=  the  distance  between  the  Planet  B  and  Sun   (Planet  A,  and  Planet  B  are  in  the  same  solar  system.)                              (DB>DA)      =  The  orbital  period  of  Planet  A      =  The  orbital  period  of  planet  B          =  The  angle  of  inclination  between  planet  A  and  planet  B          DAB=  the  shortest  distance  between  planet  A  and  planet  B      (Using  the  radio  wave  in  method  2,  or  using  the  method  1  can  find   this  distance.)

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34

2

2

AD + BD = DAB ⇒ [(sin ∠θ ) × DA ]2 + {DB − [DA × (cos ∠θ )]}2 = (DAB )2 ⇒ DAB = [(sin ∠θ ) × (DA )] + {DB − [DA × (cos ∠θ ]} 2

⇒ DAB = [(sin ∠θ ) × ( 3

2

(DB )3 × (τ A )2 2 (DB )3 × (τ A )2 3 )] + {D − [ × (cos ∠θ ]}2 B (τ B )2 (τ B )2

⇒ DAB = [(sin ∠θ ) × DB × (

τ A 23 2 τ 2 ) ] + {DB − [(cos ∠θ ) × (DB ) × ( A ) 3 ]}2 τB τB

⇒ DAB = [(sin ∠θ ) × DB × (

τ A 23 2 τ 2 ) ] + {DB × [1− (cos ∠θ ) × ( A ) 3 ]}2 τB τB

⇒ DAB = (DB )2 × {[(sin ∠θ ) × (

τA τ ) ] + [1− (cos ∠θ ) × ( A ) ] } τB τB 2 3 2

2 3 2

⇒ DAB = DB × {[(sin ∠θ ) × (

τ A 23 2 τ 2 ) ] + {[1− (cos ∠θ ) × ( A ) 3 ]}2 τB τB

⇒ DB = DAB ÷ {[(sin ∠θ ) × (

τ A 23 2 τ 2 ) ] + {[1− (cos ∠θ ) × ( A ) 3 ]}2 τB τB

Final  Equation  for  method  3:

τ A 23 2 τ A 23 2 DB = DAB ÷ {[(sin ∠θ ) × ( ) ] + {[1− (cos∠θ ) × ( ) ]} τB τB If we  want  to  find  the  distance  between  Planet  B  and  the  star  which  is   in  the  same  solar  system,  all  we  need  to  is  put  in  the  following  data:          1. τ A =  The  orbital  period  of  Planet  A        2.    =  The  orbital  period  of  planet  B          3.   =  The  angle  of  inclination  between  planet  A  and  planet  B     4. DAB=  the  shortest  distance  between  planet  A  and  planet  B

Conclusion for  method  3:              This  method  contains  less  error,  but  in  order  to  use  this  method  we   need  to  find  out  four  factors  that  is  a  lot.  In  this  method  I  found  out   there  are  six  factors  need  to  be  add  in  order  to  know  the  orbit  of  the   planet,   however   in   my   method   I   only   need   one.   I   have   tried   to   limit   the   error   but   it   didn’t   work,   so   because   the   error   is   only   about   8

35

percent so   I   think   it   is   acceptable.   The   success   of   the   method   had   support  my  hypothesis  that  is  the  position  of  the  planets.

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Method 4:

1.Aim:  To  find  the  distance  between  the  sun  and  Earth          2.  Hypothesis:  Earth  is  orbiting  the  sun  in  the  stable  orbit,  so  the  force  of  gravity   between  the  sun  and  Earth  should  be  equal  with  the  centrifugal  force.        3.  Testing  the  hypothesis:          In  order  to  test  the  hypothesis  I  need  to  know:             (1).  Equations  of  the  two  forces:     mass × (speed)2 1-­‐1. The  Centrifugal  force  =       radius G × mass1 × mass2 1-­‐2. The  Gravity  force=   ,  G=6.67  x  10-­‐11-­‐-­‐-­‐-­‐-­‐(From   radius 2 “Gravity”  Newton  Science  Magazine.)                        (2).  The  data  which  are  needed  to  find  the  distance:   2-­‐1.            The  mass  of  Earth  -­‐-­‐-­‐-­‐-­‐5.9742  x  1024  kg  (“Earth  Fact  sheet”)     2-­‐2.            The  mass  of  the  sun-­‐-­‐-­‐-­‐1.9891  x  1030  kg  (“Sun  Fact  sheet”)     2-­‐3.            The  average  orbital  speed  of  Earth-­‐-­‐-­‐-­‐29.783  km/s,  107,218km/h   (“Earth  Fact  sheet)     (3.)  What  are  we  going  to  find:                    In  these  two  equations,  I  am  going  to  find  out  the  distance  between  the   Earth  and  sun  that  is  radius  (meters)  in  the  equation  above.     (4.)  Start  the  calculation:      1.   Earth   is   orbiting   the   sun   in   the   stable   orbit,   and   that   means   the   Gravity   force  and  the  Centrifugal  force  should  be  the  same.     Make  the  equation:     mass(Earth) × (speed)2 G × mass(Sun) × mass(Earth) ,   = radius radius 2 Put  in  numbers:   (5.9742 × 10 24 ) × (107218)2 6.67 × 10 −11 × 1.9891× 10 30 × 5.9742 × 10 24   = radius radius 2 6.67 × 10 −11 × 1.9891× 10 30 × 5.9742 × 10 24   ⇒ (5.9742 × 10 24 ) × (107218)2 = radius ⇒ (5.9742 × 10 24 ) × (107218)2 × radius = 6.67 × 10 −11 × 1.9891× 10 30 × 5.9742 × 10 24   ⇒ (5.9742) × (107218)2 × radius = 6.67 × 1.9891× 5.9742 × 1019

⇒ 68677608096.2808 × radius = 7.92614857374 × 10 20 ⇒ 6.86776080962808 × radius × 1010 = 7.92614857374 × 10 20

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⇒ 6.86776080962808 × radius = 7.92614857374 × 1010 ⇒ radius = 1.15410958439731× 1010 M     11541095843.9731m=115410958.439731km     The  distance  between  the  sun  and  Earth  in  my  calculation  is   115410958.439731km     The  real  distance  between  the  Earth  and  the  sun  is  149597870.7  km   (Newton  science  magazine).   My  calculate  error:  1-­‐(my  calculation/  the  real  distance)      My  calculate  error:  1-­‐(115410958.439731/  149597870.7)  =1-­‐ 0.77147460655488=0.22852539344512=22.85%

4.Rewrite  into  the  equation:     MA:  Mass  of  planet  A   MS:  Mass  of  star  (in  the  same  solar  system)   VA:    The  average  orbital  speed  of  planet  A   R:      the  distance  between  the  star  and  the  planet  A  (planet  A  and  the  star  is   in  the  same  solar  system)<in  meters>   G=6.67  x  10-­‐11   M A × (VA )2 G × M A × M S = R R2   G × MA × MS 2   ⇒ M A × (VA ) = R G × MA × MS   ⇒R= M A × (VA )2 G × MS   ⇒R= (VA )2

6.67 × 10 −11 × M S (VA )2 6.67 × M S   ⇒R= (VA )2 × 1011   In  this  equation,  if  we  know  the  mass  of  the  star  and  the  average  orbital   speed  of  planet  A  we  can  find  out  the  distance  between  the  planet  A  and  the   star.  (The  planet  A  and  the  star  must  be  in  the  same  solar  system.)   ⇒R=

5.  Conclusion:              The  calculation  supports  my  hypothesis,  we  can  find  out  the  distance  by  using   this   method,   but   there   is   about   22%   of   error   in   the   calculation.   The   error   is   a

38

huge and   this   is   caused   because   there   are   still   many   other   small   factors   that   might   affect   the   results.   For   example,   when   Earth   orbits   the   sun,   the   Earth   will   also  be  affected  by  the  gravity  of  other  planets  and  also  moon.  The  Earth  also  has   its  own  Rotation  and  this  will  also  create  a  Centrifugal  force.  These  factors  will   make  the  calculation  more  complex,  and  right  now  I  haven’t  have  the  ability  to   add  these  factors,  so  this  method  is  not  a  great  way  to  find  the  distance,  but  the   benefits   of   this   method   is   we   only   need   to   know   the   mass   of   the   star   and   the   average   orbital   speed   of   the   planet,   and   then   we   can   find   the   distance.   There   are   only  2  data  need  to  find  the  distance  and  the  equation  is  simple,  which  is  easier   for  Amateur  stargazers.

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Results:

Comparing each  method  I  create:   Method  1.     Error  of  the  method  (%)   Factors  need  to  find  the  star   The  purpose  of  the  method   Hypothesis:

Experiment method:

Does the  experiment  support  the   hypothesis     Method  3     Error  of  the  method  (%)   Factors  need  to  find  the  star   The  purpose  of  the  method   Hypothesis:

Experiment method:   Does  the  experiment  support  the   hypothesis

Extremely high   3   Find  the  distance  between  planets  and   Earth  (fail,  can  only  be  used  for  short   distance  objects)   If  we  find  out  the  size  of  the  image  and   the  real  length  of  the  object  we  can  find   out  the  distance  by  finding  out  the   percentage  decreased  between  the  size   of  the  image  you  see  and  the  real   object.   Find  an  object  and  then  measure  the   size  of  the  image  you  see  by  using  the   ruler,  which  is  fixed  distance  from  your   eyes.  Repeat  this  step  and  change  the   distance  between  you  and  the  object.     When  the  data  is  enough,  change  the   object  and  measure  again.   The  experiment  support  the   hypothesis  which  make  this  method   can  be  used

8% 4   Find  the  distance  between  sun  and   Earth  (success)   The  position  of  the  planets

Solve and  prove  by  using  the   trigonometry   The  experiment  support  the   hypothesis  which  make  this  method   40

can be  used       Method  4     Error  of  the  method  (%)   Factors  need  to  find  the  star   The  purpose  of  the  method   Hypothesis:

Experiment method:   Does  the  experiment  support  the   hypothesis

22% 2   Find  the  distance  between  sun  and   Earth  (success)   If  the  Earth  is  orbiting  the  sun  in  the   stable  orbit,  this  means  that  the  gravity   force  between  the  Earth  and  sun  and   the  Centrifugal  force  should  be  the   same.   Prove  by  using  math.   The  experiment  support  the   hypothesis  which  make  this  method   can  be  used

The  usefulness  of  my  method:     I  create  a  way  to  measure  and  compare  the  usefulness  of  my  three  methods.   The  usefulness  of  the  method:  The  factors  that  need  for  the  method  /  the  error   (%)  of  the  method.       Number  of  the  method:   The  error  of  the  method   The  factors  that  need  for   (%)   the  method   1   Extremely  high  (depends   3   on  the  distance  change)   3   8   4   4   22   2     Graph:

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The dot  closer  to  the  (0,0)  dot  B  will  be  the  most  useful  way  that  needs  fewer   factors  to  find  the  distance  and  have  less  error.       In  this  graph  the  dot  P1  is  the  third  method,  P2  is  the  fourth  method,  and  the  line   AC  is  the  method  1,  which  the  error  change  when  the  distance  between  the   object  and  you  changes.       In  the  graph  we  know  dot  P1  to  the  (0,0)  is  8.94  units,  and  the  dot  P2  to  the  (0,0)   is  22.09  units.

Conclusion for  the  results  compare:

In  here  we  can  know  that  method  3  is  the  best  method  of  these  three  to  used  for   finding  the  distance  between  the  stars.

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Discussions:

The  distance  of  the  stars  can  find  in  many  different  ways.  The  three  methods   all  have  some  errors  because  there  are  some  limitations.

The  Limitations  of  the  methods:

Method  1:     (1.)          When  we  try  to  measure  an  object  we  see  there  would  be  a  problem   of  parallax.  It  is  because  there  is  a  distance  between  left  eye  and  right   eye  and  when  we  focus  on  the  close-­‐distanced  ruler,  and  tried  to  use  it   measure   the   object   far   behind,   the   measure   will   cause   error.   The   image   of   left   eye   and   the   image   of   the   right   eye   you   see   will   not   be   the   same,  and  this  make  the  measurement  can’t  be  accurate.     (2.)        Everyone’s   eyesight   will   be   different   and   this   will   cause   the   error,   and  the  measurement  will  change  if  we  change  the  observer.     (3.)        The   size   of   the   image   you   see   decreases   slower   and   slower   as   the   distance  increase,  so  this  make  the  method  can  only  be  used  for  close   distance  objects.       Method  3:   (1.)          The   shortest   distance   between   the   Earth   and   Venus   is   not   like   the   picture   I   draw.   There   are   still   five   more   factors   need   to   be   add   for   finding  the  orbit,  but  the  factors  do  not  fit  in  my  equation.       Method  4:   (1.)          The   gravity   force   and   Centrifugal   force   is   not   balanced.   There   are   still   many   other   force   will   affect   the   calculations.   For   example,   the   gravity  force  of  other  planets.     (2.)            The   Earth   has   its   own   Rotation   and   this   will   create   a   centrifugal   force  and  this  will  affect  the  results.     (3.)            The  moon  is  one  of  the  biggest  force  around  Earth  and  this  will  have   a  huge  effect  on  the  calculations     General  limitations:   (1.)            The   space   is   twisted   by   gravity   and   this   will   change   the   distance   between  each  object     (2.)            The   space   is   expanding   and   even   though   the   solar   system   will   not   expand,  but  this  will  affect  the  distance  change  in  stars  and  galaxy.       (3.)        The  path  of  the  light  will  change  because  of  the  twist  of  the  space  and   this  might  affect  the  results.

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Conclusion:

In  this  paper,  I  have  created  three  methods  in  order  to  find  the  distance  to   the  star  (sun).  The  three  methods  are  based  on  three  different  hypotheses.  2  of   the  three  methods  are  successful  and  can  be  used  for  the  real  calculation.  During   the   research,   the   study   of   the   Newton   force,   Einstein’s   General   Relativity,   Kepler’s  Law  and  the  general  ideas  of  the  Universe  are  very  helpful  for  creating   my  paper.   Each   method   has   its   own   limitation   and   this   make   the   error   cannot   be   reduced.  In  this  paper  only  the  method  one  is  tested  by  a  real  experiment  and  the   other  two  methods  that  I  create  are  tested  by  math  calculations.                      The   methods   that   I   created   in   this   paper   can   be   used   for   the   non-­‐ professionals   for   finding   the   distance   to   the   sun.   The   paper   is   very   important   for   the   astronomy   research   and   can   be   one   of   the   sources.   Finding   the   distance   is   one  of  the  most  important  things  in  the  development  of  the  astronomy,  and  the   paper  still  can  be  extended  and  develop  to  a  way  to  find  the  distance  with  high   technology  equipment.  In  the  21st  century,  as  the  global  population  growth  and   the   limitation   of   the   world’s   sources,   developing   the   astronomy   becomes   more   and   more   important.   Finding   alternative   for   the   sources   and   solving   the   global   problems,   developing   higher   technology   are   some   of   the   main   benefits   of   the   astronomy.     This   paper   can   also   inspire   the   reader   to   let   them   reach   their   full   potentials   of   creativity.   At   last,   I   hope   the   paper   can   let   everyone   who   is   filled   with   passion   in   space   can   have   a   chance   to   find   the   distance   of   the   star   at   anytime,  anywhere.

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References:

Primary  Sources:   1.Book:

1-­‐1  Pickering, Ron. Complete Biology for Cambridge IGCSE. 2nd ed. Oxford: Oxford UP, 2010. Print.          1-­‐2. Baines, Fran. DK Online Science Encyclopedia. New York: DK Pub., 2006. Print.

2.Interview:

1.  Date:  2011.  9.  7          People  I  interview:  Math  teacher                      Mr.  Manoharan Karthigasu      Time  of  Interview:  30  minutes        Topic  I  talk:  Asking  questions  of  my  personal  project

2.  The  second  talk  with  the  math  teacher:          Date:  2011.9.14          People  I  interview:  Math  teacher                      Mr.  Manoharan Karthigasu        Time  of  Interview:  47  minutes          Topic  I  talk:  Asking  questions  of  my  personal  project                    3.  E-­‐mail  NASA  and  some  professors  for  questions  (They  never  reply)

Secondary  Sources:     1.Internet:

1-­‐1  "Inclination." Wikipedia, the Free Encyclopedia. Wikimedia Foundation, Inc. Web. 03 Sept. 2011. <http://en.wikipedia.org/wiki/Inclination>. 1-2 Times, Hellenistic. "Venus." Wikipedia, the Free Encyclopedia. Wikimedia Foundation, Inc. Web. 03 Sept. 2011. <http://en.wikipedia.org/wiki/Venus>. 1-3 "Kepler's Laws of Planetary Motion." Wikipedia, the Free Encyclopedia. Wikimedia Foundation, Inc.,. Web. 03 Sept. 2011.<http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion>. 1-4 Bergman, Jennifer. "Venus Statistics." Windows to the Universe. Jennifer.Bergman, 16 Feb. 2001. Web. 03 Sept. 2011. <http://www.windows2universe.org/venus/statistics.html>. 1-5 Williams, Dr. David R. "Venus Fact Sheet." Welcome to the NSSDC! NASA Goddard Space Flight Center Greenbelt, 17 Nov. 2010. Web. 03 Sept. 2011. <http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html>.

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1-6 "Earth." Wikipedia, the Free Encyclopedia. Wikimedia Foundation, Inc., 3 Sept. 2011. Web. 03 Sept. 2011. <http://en.wikipedia.org/wiki/Earth>. 1-­‐7  "Series." Series(mathematics). Wikimedia Foundation, Inc. Web. 30 Aug. 2011. <http://translate.googleusercontent.com/translate_c?hl=zhTW&prev=/search%3Fq%3D%25E6%25B3%25B0%25E5%258B%2592%25E7%25B4%259A%25E 6%2595%25B8%26hl%3Den%26biw%3D1280%26bih%3D646%26prmd%3Divns&rurl=translate.go ogle.com&sl=zhCN&twu=1&u=http://en.wikipedia.org/wiki/Series_(mathematics)&usg=ALkJrhiujFju8SupqznNqvPoSm3es6tsA>. 1-8 Wilson, Currtis. "HAD News--Kepler's Laws, So-Called." HAD News The Newsletter of the Historical Astronomy Division of the American Astronomical Society. LeRoy Doggett Nautical Almanac Office U.S. Naval Observatory Washington, DC 20392, May 1994. Web. 24 Sept. 2011. 1-9 Hyman, Andrew T. "A Simple Cartesian Treatment of I Planetary Motion." Lewis and Clark Collage, Portland, OR 97219, USA, 26 Feb. 1992. Web. 23 Sept. 2011.Printed in UK 1-10 Rubin, Julian. "Heinrich Hertz: The Discovery of Radio Waves." The Orchid Grower: A Juvenile Science Adventure Novel. Julian Rubin, Jan. 2011. Web. 23 Sept. 2011. <http://www.juliantrubin.com/bigten/hertzexperiment.html>. 1-11 Stimac, Tomislav. "Frequency Bands." RADIO WAVES below 22 KHz. Tomislav Stimac. Web. 23 Sept. 2011. <http://www.vlf.it/frequency/bands.html>. 1-­‐12. "Orbital Inclination." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 3 Nov. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Inclination>.   1-­‐13. "Argument of Periapsis." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 13 Sept. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Argument_of_periapsis>. 1-14. "Geology of Venus." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 1 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Geology_of_Venus>. 1-15. Times, Hellenistic. "Venus." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 7 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Venus>. 1-16. "Eccentricity (mathematics)." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 23 Nov. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Eccentricity_(mathematics)>. 1-17. "Orbit." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 4 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Orbit>. 1-18. "Orbital Elements." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 23 Nov. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Orbital_elements>. 1-19. "太阳系行星轨道怎么计算？ - 已解决 - 搜搜问问." 搜搜问问--腾讯旗下最大互动问答社区. SOSO Question, 14 Mar. 2009. Web. 08 Dec. 2011. <http://wenwen.soso.com/z/q122995963.htm>. 1-20. "轨道计算." Hugong. Hudong.com. Web. 9 Dec. 2011. <http://www.hudong.com/wiki/轨道计算 >. 1-21. "Apsis." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 7 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Apsis>. 1-22."List of Gravitationally Rounded Objects of the Solar System." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 6 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_System>.

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1-23. "Ceres (dwarf Planet)." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 7 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Ceres_(dwarf_planet)>. 1-24. "Kuiper Belt." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 6 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Kuiper_belt>. 1-25. "4 Vesta." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 8 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/4_Vesta>. 1-26. "Makemake (dwarf Planet)." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 30 Nov. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Makemake_(dwarf_planet)>. 1-27. "Eris (dwarf Planet)." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 8 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Eris_(dwarf_planet)>. 1-28. "List of Minor Planets." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 21 Nov. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/List_of_minor_planets>. 1-29. "Astronomical Object." Wikipedia, the Free Encyclopedia. Wikipedia Foundation, Inc., 6 Dec. 2011. Web. 08 Dec. 2011. <http://en.wikipedia.org/wiki/Astronomical_object>. 1-30 "Distance." Wikipedia, the Free Encyclopedia. Wikipedia.Inc, 22 Sept. 2011. Web. 01 Oct. 2011. <http://en.wikipedia.org/wiki/Distance>. 1-31."圆周离心力公式_百度知道." 百度知道――全球最大中文互动问答平台. Bai Du.com, 5 Apr. 2006. Web. 27 Dec. 2011. <http://zhidao.baidu.com/question/5737186>. 1-32. "Earth." Wikipedia, the Free Encyclopedia. Wikipedia.INC, 27 Dec. 2011. Web. 27 Dec. 2011. <http://en.wikipedia.org/wiki/Earth>. 1-33 Williams, Dr. David R. "Sun Fact Sheet." Welcome to the NSSDC! NASA Goddard Space Flight Center, 1 Sept. 2004. Web. 27 Dec. 2011. <http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html>. 1-34. Williams, Dr. David R. "Earth Fact Sheet." Welcome to the NSSDC! NASA Goddard Space Flight Center, 17 Nov. 2010. Web. 27 Dec. 2011. <http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html>. 1-35  Stern,  David  P.  “Parallax.”  NASA’s  Polar,  Wind  and  Geotail  Missions.  Dr.  David  P.  Stern,  18   Sept.  2004.  Web.  30  Oct.  2011.  <http://www-­‐istp.gsfc.nasa.gov/stargaze/Sparalax.htm>.     1-­‐36      Stern,  David  P.  “How  Distant  Is  the  Moon?”  NASA’s  Polar,  Wind  and  Geotail  Missions.  Dr.   David  P.  Stern,  15  Oct.  2004.  Web.  30  Oct.  2011.  <http://www-­‐ istp.gsfc.nasa.gov/stargaze/Shipparc.htm>.   1-37    Cordova, Susan. "How to Write a Scientific Paper." NMAS Home Page. By Susan Cordova for the New Mexico Junior Academy of Science. Web. 02 Jan. 2012. <http://www.nmas.org/JAhowto.html>.

2.Report paper: <http://science.ntsec.edu.tw/ezfiles/4/1004/attach/60/2007055.pdf> 陳,  漢洲.  The  Research  of  the  Planetary  Brightness  Outside  the  Solar  System.  Rep.  no.  2007055.   Taipei:  陳漢洲,  2007.  Print.

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3. Online  News  article:          1. almer Science, Jason. "BBC News - Light Speed: Flying into Fantasy." BBC - Homepage. BBC, 23 Sept. 2011. Web. 01 Oct. 2011. <http://www.bbc.co.uk/news/science-environment-15034414>. 2. Shukman, David. "BBC News - Results from Cern Show Particles 'exceeded Speed of Light'" BBC - Homepage. BBC, 23 Sept. 2011. Web. 01 Oct. 2011. <http://www.bbc.co.uk/news/scienceenvironment-15038826>.

5.Magazine:

1.

2.

3.

4.

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Appendices:

The  Interview  with  the  Math  teacher:

1.The first  Interview:    The  first  interview:

Date:  2011.  9.  7          People  I  interview:  Math  teacher                      Mr.  Manoharan Karthigasu      Time  of  Interview:  30  minutes        Topic  I  talk:  Asking  question  of  my  personal  project          What  I  talk:                  1.In   my   calculation   using   the   Kepler's laws of planetary motion there is always an error is there any way to reduce the error? There are some tiny errors in every number that you put in the equation and when you put them all in the equation the error will increase. You are using the distance between Venus and Earth (d1), and the distance between semi-­‐major   axis Earth (d2), and the semi-­‐major   axis of Venus (d3) to find the distance between the Earth and the sun. Using the Kepler's laws of planetary motion you can find the   ratio   of   d2:d3,   and   you   try   to   use  d2-­‐d3  to  find  the  number  of  d3.  The  problem  is  you  use  the  closest  distance   between  Earth  and  Venus  to  be  d2-­‐d3,  but  when  both  Venus  and  Earth  is  on  the   semi-­‐major   axis,   the   distance   between   the   Earth   and   the   Venus   is   not   the   closest   number,   because   of   Earth’s   Inclination   is   different   from   Venus’s   Inclination.   So   it’s  okay  to  have  some  error.

2.   I   am   doing   a   paper   for   my   personal   project   and   my   topic   is   about   finding   distance   of   star   and   as   a   teacher   can   you   give   me   some   advice   of   my   personal   project?                First   of   all   I   want   to   tell   you   that   after   I   see   your   bibliography,   I   think   you   can’t  use  Wikipedia  as  one  of  the  resource,  because  as  a  science  paper  Wikipedia   is  not  admit  as  a  reliable  resources  and  you  should  use  more  papers.  Wikipedia  is   the   easiest   way   to   find   the   information,   but   if   you   really   want   to   use   the   Wikipedia  you  need  to  click  into  the  references  of  Wikipedia.

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2.The second  interview:

The  second  talk  with  the  math  teacher:   Date:  2011.9.14   People  I  interview:  Math  teacher                      Mr.  Manoharan Karthigasu      Time  of  Interview:  47  minutes        Topic  I  talk:  Asking  question  of  my  personal  project   What  I  talk:   M:  Myself   T:  The  teacher     M:  In  my  personal  project  the  first  thing  I  tried  to  do  is  to  find  how  many  percent   the   object   decrease   when   I   stand   in   different   distance   from   the   object.   Then   I   designed   the   experiment   by   measuring   the   size   of   the   image   you   see   in   the   same   measure   point   but   different   distance   from   object.   Then   I   changed   the   length   of   the   object   I   see   into   percentage   by   dividing   the   original   length   of   the   object.   I   tried  this  method  in  three  objects,  so  I  can  have  an  average  number.  Then  I  graph   all   the   data   I   have   by   using   the   computer   program,   but   I   found   out   the   line   of   the   graph   is   not   exponential   line,   because   the   image   we   see   does   not   decrease   in   the   same  speed.  Then  I  have  an  idea  of  finding  the  multiplying  power  between  N  and   N-­‐1  meters,  but  after  I  did  this  I  found  out  is  even  harder  to  find  the  equation  of   the  multiplying  power  because  each  tiny  error  I  have  will  cause  a  huge  error  of   the   multiplying   power   between   each   point.   I   did   my   research   and   I   think   the   graph  is  looking  like  this:     F  (x):  the  size  of  the  image  you  see   L:  the  length  of  the  object   X:  the  distance  between  you  and  the  object     F  (x)=L*a!  (a1*a2*a3*a4*a5*…*ax-­‐2*ax-­‐1*ax-­‐1*ax)     The  problem  is  I  don’t  know  how  to  find  the  equation  of  number  a.     I  have  the  data  and  I  don’t  know  how  to  use  my  data  to  find  the  equation  of  a!       T:  So  when  you  put  “a”  into  the  graph  what  does  the  graph  looks  like?   M:  The  computer  auto  detect  the  graph  is  exponential,  but  in  reality  it  should  not   be  exponential.  What  should  I  use  to  find  the  equation  of  “a”?     T:  What  are  you  trying  to  find,  the  length  of  the  object?   M:  Distance  between  you  and  the  object.   M:   I   am   trying   to   find   out   the   equation   of   the   distance   between   you   and   the   object  and  how  the  image  you  see  decrease  in  percentage,  and  by  knowing  how   big  the  star  is  and  then  I  can  start  my  measurement.  I  can  measure  how  big  the   object  I  see  then  using  the  equation  of  how  many  percent  of  the  object  decrease   in  different  distance,  and  then  I  can  find  the  distance  between  the  object  and  I.  I   can   also   add   a   telescope   when   I   measure   and   then   I   can   divide   the   rate   of   the   telescope  and  by  doing  this  I  can  get  a  better  and  closer  number.   T:   You   may   not   get   a   correct   number   if   you   use   this   way   to   find   the   star,   because   you  are  using  the  eyes.

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T: Another  problem  is  when  you  do  this  by  using  human  eye,  and  when  we  use   cats  or  dogs  to  do  the  measurement,  the  result  will  be  different.  That  means  your   eyes  had  a  large  effect  on  your  result.  If  you  want  to  apply  on  other  animals’  eyes   you  still  need  to  do  some  changes.       M:   Another   problem   I   have   is   how   am   I   going   to   find   how   big   the   object   is   without   knowing   the   distance?   It   is   because   I   need   to   know   the   length   of   the   object  before  knowing  the  distance.     T:   Then   you   need   to   do   the   research   on   how   did   scientist   find   the   size   of   the   object  without  knowing  the  distance.   (26:22)     Then  the  math  teacher  showed  me  the  video  of  the  size  of  the  stars.     T:  See,  they  know  the  size  of  the  start.  They  must  know  a  way  to  find  the  size  of   the  star.     M:  So,  the  first  thing  I  need  to  do  is  to  find  more  information  about  the  optics,  our   eyes.   Then   go   back   to   find   the   equation.   After   find   out   the   equation,   try   to   find   out  the  size  of  the  object  without  knowing  the  distance.     After   that   the   teacher   told   me   about   the   app   for   iphones,   which   is   called   “   The   invisible  space.”  You  can  see  what  the  space  real  looks  like  when  you  can  see  the   lights   like   x-­‐ray   and   other   lights   that   normal   people   can’t   see.   The   teacher   showed  me  that  app  in  his  phone  and  explains  how  he  made  this  app.  This  app   used  math.       M:  IF  the  space  is  expanding  should  the  distance  that  we  calculate  isn’t  right?   T:  Brain  Green  had  talked  about  this  topic  in  his  book.  You  are  asking  that  if  the   space   expands   every   day,   the   distance   we   calculated   yesterday   is   wrong   for   today.  This  is  not  true,  because  when  the  space  is  expanding,  we  also  expand.   (39:06)     M:   The   gravity   will   change   the   path   of   light,   because   of   the   twist   of   the   space.   Does  that  will  change  the  result  of  finding  the  distance?     T:  Yes,  the  gravity  will  twist  the  space  and  that  might  change  the  path  of  light.  It   is  because  the  distances  between  the  stars  are  huge  and  that  will  become  a  big   error.  But  light  always  goes  the  shortest  path,  so  even  though  the  space  and  the   path  had  been  twisted,  the  distance  you  find  is  still  acceptable.

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~Rene Descartes (1596－1650)

The End

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Distance to stars by Adam Hsieh

Distance to stars by Adam Hsieh