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Journal of Physical Science and Application 6 (6) (2016) 57-62
doi: 10.17265/2159-5348/2016.06.008
DAVID
PUBLISHING
Measurement of Mercury Distance, by Observing Transit
of Mercury
Samaneh Shamyati1, 2, 3
1. A member of the Astronomy Society of IRAN, 15875-3671, Iran
2. A member of Astronomy Center of Rey, Shahre Rey, Tehran, 1876943116, Iran
3. Student of physics Payame noor University (PNU), Tehran, 19395-3697, Iran
Abstract: We will measure the distance between the Sun and the Moon by considering the Parallax and also by considering the
Mercury Sun distance. The goal is to utilize the observed data. The way that we have used in this paper is based on two observational
data in Iran (Mashhad 36°31′ N, 59°48′ E) and Canada (Yukon Territory 69°27′ N, 33°2′ E). It is required to mention that in Mercury
transition 2016, I had personally collected the first place, Mashhad. In measuring the distance between Mercury and the Sun we have
utilized the trigonometry terms and the related formulas. Using all the collected data and studying on the contacts of the mercury, has
helped us to reach the best results.
Key words: Distance, Sun, Earth, Mercury, IRAN, Solar, Sun-Mercury, parallax.
Nomenclature
R Earth = 150015880 =1000105867 Au
R Mercury = 65724412 =0.438163746
0.43 Au
Most distance of the sun: 70 × 106 = 0.46 Au
The average distance from sun: 58 × 106 = 0.38 Au
The shortest distance to the sun: 46 × 106 = 0.30 Au
Today the way is used for calculating the earth to
Sun distance. In this paper both the Earth-Sun distance
and Mercury-Sun distance have been used. The
importance of this paper is due to the fact that we
could use the observations to calculate the
Mercury-Sun distance, using the Mercury transition.
We could also confirm the previous measurements of
the distance. The used formulas in this paper are all
based on the results of working on this phenomenon
and measuring the Sun-Earth distance. However the
Mercury-Sun distance had been required.
2. Calculation for the Distance of the Sun
and Earth [4]:
1. Introduction
The start of this subject could go back to the year
1761, at which Edmond Halley had suggested a
competition in the field of Mercury transition and Jin
Nicolas Delicias collected some data then.
Corresponding author: Samaneh Shamyati, bachelor,
research fields: astronomy, astrophysics (sun, aurora borealis)
and cosmology.
The point O indicated the center of the sun, C is the
center of earth and M is the center of Mercury.
The observer in the point A, observes the Mercury’s
position A’ on the surface of the sun. Likewise, the
observer in the point B, observers the Mercury’s
position B’ on the surface of the sun. Considering the
theorem that: The measure of an exterior angle of a
triangle is equal to the sum of the measures of the two
interior angles that are not adjacent to it, for the two
58
Measurement of Mercury Distance, by Observing Transit of Mercury
triangles BPM and APO:
βB αM ρ1 , αs βA ρ2
ρ2 ρ1
βB αM αs βA
The centers of Mercury, the Sun and Earth are
placed on one line. Therefore:
αM αs βA βB
where, ∆
have:
is the Parallax, by simplification we will
Δβ
αs
αM
αs
1
d
α
2
D
2
If α be a tiny angle therefore:
1
α
tan α
tan
2
2
By replacing them in the above equation, we have:
d
tanα
D
Considering the above expression and knowing that
Re and RM are alternatively the Sun-earth distance and
the Sun-Mercury distance, and that αs and αM are both
tiny angles, we can use the below trigonometry
relations in the two triangles ABO and ABM.
AB
AB
, αs
αM
Re RM
Re
tan
Considering the above relations, we can calculate
αs and αM :
Re
αM
αs
Re RM
By replacing it in the relation ∆ , we will have:
∆β
αs
Re
Re RM
1
RM
αs
Re RM
Therefore:
Re
αs ∆β
1
ΙΙ
RM
According to Fig. 2, αs would be the parallax of the
Re
RM
3
Te
TM
2
,
Re 3
365.25 2
Re
2.583
RM
RM
87.969
By replacing it in the expression (ІІ), we have:
1.583 ∆
in relation
Using the first relation, we can state
(І) like this:
AB 0.631 AB
Re
αs
∆β
As a result, to have Re we need the distance
between the two observers (Iran-Canada) and ∆ of
the observing data. Based on the observing data we
have Table 1.
First by using trigonometry, we calculate the
distance between Iran and Canada:
cos C I
cos 90 φc cos 90 φI
sin 90 φc sin 90 φI cos Δλ
IC Cos 1 θ
IC: (Distance of Iran to Canada)
cos C I
cos 90 φc cos 90
sin 90 φc sin 90 φI cos Δλ
cos 21° cos 31°
sin 21° sin 31° cos 74
0.938 0.85
0.35 0.51 0.27
0.79 0.048 0.838
IC
CI
CI
33.07
cos 1 0.838
cos 1 xθ
Re . θ
θ: Rad
IC
re . θ
9481003.65 αs
αM
αM
IC
φI
6371 33.07
9481003.65
150000000
9481003.65
150000000 46000000
0.0632
π
2
αs
AB
Re
0.091
AB
Re RM
Sun from the views of observers A and B. The ratio
Re
RM
can be calculated via Kepler’s third law (the
Mercury orbital motion (a Mercury year: 87.979 days),
an earth year lasts 365.25 days.
Fig. 1
Calculation for the distance of the Sun and Earth.
Measurement of Mercury Distance, by Observing Transit of Mercury
Fig. 2
d/2
Table 1
Iran.
59
Calculation for the Angle of α.
D
Observation form (the transit of Mercury), Ahmad Nezam&Samaneh shamyati, Ferdowsi University ,Mashhad,
60
Me
easurement of
o Mercury Diistance, by Observing
O
Tra
ansit of Mercu
ury
Fig. 3
photoo by Ahmad Neezam, Observaatory of Ferdow
wsi University
y, Mashhad, Iraan.
Fig. 4
Transsit of Mercuryy :2016 May 099, F. Espenak, www.Eclipsew
w
wise.com.[1].
Δβ
αs
αM
αs
1
0.0632
0.091
0.0632
1
0.027
Re
AB
αs
Re
94881003.65
0
0.0632
1500015880 mK
∆
:β
BMP B
∆
:α
APO S
αM αS
Δβ
αM
αM
P1
βA
P1
βA
βB
α1
αM 1
βB
αM
αS
βA
Δβ
α1
αM
I
61
Measurement of Mercury Distance, by Observing Transit of Mercury
α
d
RM 2
TM 2
87.969
D
RO
EO
365.25
AB
∆
:α
ABO 1 RO
AB
∆
:α
ABM M RO RM
1
I
Table 2
Δβ
αM 1
1
RM
2
RO
0.24
II
α1
αM
RO
RM
αM 0.24
Δβ
RO
Δβ
RO RM
RO
RM
RS
RM
RO
0.24
αM .
RM
RO
Canada (PEI-Yukon Territory).[5].
II
4.17Δβ
0.24
AB
Δβ
αM
RO
Δβ
0.24
RM
4.17Δβ
Δβ
.
αM
RM
62
Measurement of Mercury Distance, by Observing Transit of Mercury
Table 3
Geocentric phases of the 2016 transit of Mercury. [2, 3].
3. Results and Discussion
Now we should replace the numbers in the
equations. Therefore, we would have the distance
between the Sun and Earth. Then according to Fig. 1
and calculation via Spherical trigonometry, we can
measure the distance between the Sun and Mercury.
RM
9481003.65
150000000 0.24
0.027
150000000 84275588
65724412 0.4 Au
Ehsan Mehrjoo (Astronomy & Astrophysics
Olympiad Teacher); Fatemeh Hematian (language
translator);
Shahla
Shoja-Amineh
(amateur
astronomer); Ahmed Nezam (amateur astronomer).
References
[1]
[2]
Acknowledgement
[3]
Special thanks to: Dr. Mehdi Daneshyar (the
chairman of Shahrerey Astronomy Center); Dr. Atila
pro (head of the Middle East IOTA); Dr. Ebadeh
Moradi (a professor in Farhangiyan University);
[4]
[5]
Explanatory
Supplement
to
the
Astronomical
Ephemerise and the American Ephemeris and Nautical
Almanac, 1974, Her Majesty’ Nautical Almanac Office,
London.
Espenak, F. 2015. Six Millennium Canon of Transits:
-2999 to +3000. Portal, AZ: Astropixels Publishing.
Meeus, J. 1989. Transits. Richmond, VA: Willmann-Bell,
Inc.
Hashemi-Niari, F. 2016. “Calculate the Distance from
Earth to the Sun by Observing Transit.” Astronomy
Magazine 5: 23.
Espenak, F. 2016. 2016 Transit of Mercury. Observer's
Handbook 2016, Royal Astronomical Society of Canada.