John Pazmino
 NYSkies Astronomy Inc
 2012 June 12

    After the transit of Venus in 2012 NYSkies was flooded with 
questions about the use of the transit to determine the Earth-Sun 
distance. There was ample discussion of the method in litterature 
issued for the transit but much of it was too technical or too simple. 
I here give a geometric walk-thru, but you do have to understand some 

Knock-off history 
 --------- ----
    Until the transits of the 1760s we did not know ell at all the 
earth-Sun distance for a linear size of the solar system. Cassini in 
the 1670s tried to capture a parallax from Mars at opposition but the 
instruments ere too crude for confident results.
    A better choice would be Venus, who approaches closer to the Earth 
than mars, yielding a larger and more easily measured parallax. 
Several attempts were made in the late 1600s during times when Venus 
was in dark sky as evening or morning star. Horrendous troubles broke 
out, explained below, that upset these efforts. 
    Halley observed in the early 1700s a transit of Mercury and hit on 
the plan to observe the future Venus transits for getting a solid 
parallax. He passed on long before the transits but his mission was 
taken up by England and France. Both had colonies scattered across the 
globe from which to observe the events. 
    The idea was to trace the exact path of Venus across the Sun's 
disc, then work out the shift as a function of separation of the 
observing sites. The effort was immense for its time, the first global 
science project in history. The results were far better than what we 
had until then. 
    The project was repeated for the transits of the 1880s, with much 
improvement in the Earth-Sun distance.Even in the 20th century, 
parallax measurements were attempted on close-passing asteroids, like 
Icarus and Hermes. 
    Since the Space Age, with interplanetary probes, transits of Venus 
are no longer needed. Distances thruout the solar system are measured 
by signal relaying with the spaceprobes. nearby objects, mainly Earth-
threat asteroids, are measured by radar pinging. 
    The transits of 2004 and 2012 were observed for information about 
Venus and to test instruments built for detecting extrasolar planets. 
Both transits were widely observed by NYSkies, other astronomers and 
the public. 
    As a segment of human history, the stories of England and France 
work for the two 1700s Venus transits is full of adventure, danger, 
war, high science, travel, glory. For us in the United States, they 
prodded the colonies to agitate for a native American identity, 
leading in the late 1700s to the War of independence. 

Parallax from transits
    Halley's plan was to trace the exact path of Venus across the Sun 
from as many dispersed places on Earth as possible. From building 
triangles from venus to paris of these observing sites the parallax of 
the planet could be worked out. The value so obtained was converted 
into the distance from Earth to Sun and by extension the distances to 
all planets in the solar system. 
    The diagram is grossly exaggerated. A and B are the locations on 
Earth, V is Venus. The angle at V is really tiny. It turns out that 
for A and B one Earth radius apart. This angle is about the size of a 
US 25c coin, the quarter, seen from two short Manhattan blocks away, 
about 170 meters. 

        A                                              / 
        +------                                ------/ b 
        |     ------                      ------   / 
        |          ------      V     ------
        |                ------+------ 
        |          ------            ------
        |     ------                      ------       / 
        +------                                ------/ a 
        B                                          /   

    The lines of sight from A and B they V  intersect the solar disc 
at a and b. At these points Venus is a round black dot against the Sun 
but at two different places on hum. 
    The slanted lines a and b are the paths of venus across the Sun' 
disc, not shown,. The displacement of the paths from sites A and B go 
into the computation of parallax and then the Earth-Sun distance.  
    Halley figured out that it was necessary only to time the ingress 
and egress of the Venus disc with the Sun, since the angular speed of 
the planet was known from solar system dynamics. The span of time it 
took Venus to cross the Sun was equivalent to the length and position 
of her path across the Sun. 
    As simple as this sounds it was utterly heroic that these 
triangles were built from reports returned from the 1700s and 1800s 
transits. Apart from the problems of long-endurance  travel, crossing 
war zOnes, setting up stations, and weather, there was the imperfect 
geography of the day. A team set up on a place whose exact lat-lon was 
not well established, making the base of the triangle loosely defined. 
    It proved impossible to observe from the ideal stations one Earth 
radius apart face-on to the Sun. We had to scatter stations all over 
the world, even to places only discovered a few years earlier. The 
geometry of the triangles between arbitrary pairs of stations was far 
more difficult than for the A and B stations in the diagram above. The 
actual parallax was then fluffed up into the ideal situation of A and 
 with many cross-checks in the network of station s. 
    We cut thru these difficulties and consider the ideal situation of 
a parallax taken across a full radius of Earth, say from where the 
transit occurred in the local zenith and in the local horizon. This 
value is the horizontal parallax. 

Why use a transit?
    Traditional parallax is measured against the stars, assumed 
permanently fixed on the celestial sphere. Altho by the mid 19th 
century stars were known to have proper motion against the celestial 
sphere, this motion was infinitesimal compared to a sensible parallax. 
    It would be normal to acquire the parallax of Venus at nay time 
she was in dark sky against the stars. In the diagram above, without 
the slanted lines for a path on the Sun, points a and b are the 
positions of Venus relative to surrounding stars. 
    Opportunities for dark-sky Venus parallax happen frequently enough, in 
both morning and evening, in the next couple months. 
    The measured value would be applied to the triangle of Sun-Venus-
Earth with sides fixed by the already known scale of the solar system. 

                            / \ 
                     /          \ 
               /                   \ 
              E                     S 

    S-E is the Sun-Earth distance. S-V is the Sun-Venus distance. E-V, 
the distance from Earth to Venus that is determined from the parallax  
of Venus captured against the stars when Venus stands in a dark sky.  
All three sides are known ratios of each other from the application of 
known planetary motion. 
    Once the linear length of the Earth-Venus arm of this triangle was 
determined, the other sides fall into line with their own linear 
dimensions. In the ideal case we should have obtained the Sun-Earth 
distance in the late 1600s. limited only by instrumental resolution 
for small angular displacement against the stars.                       
    When fixes on Venus were tried, we got discordant parallaxes.  The 
uncertainties were due to the Venus phases and atmosphere. The phases 
made it tricky to fix on the center of the planet's disc. 
    The atmosphere, discovered as such during the 1700s transits, 
distorted the shape of Venus, further confusing the location of her 
geometric center. 
    A transit presents a nitid scene of just the black round dot of 
Venus and the bright disc of the Sun. Also, during a transit Venus is 
closest to Earth, at inferior conjunction, presenting the largest 
parallax she can offer. 

The Sun moves
    During a Venus transit we do NOT observe directly the full 
parallax of Venus. The Sun, against which we see Venus, is himself 
close enough to exhibit his own, unknown, parallax. 
    The orbit radius of Venus is 0.723 that of Earth's. Accounting for 
excentricity and place of each planet relative to its own perihelion, 
the distance of Venus from Earth in the 2012 transit is 0.289 Earth 
orbit radius. It's larger than the mean distance  mainly because Earth 
in June 2012 is nearing her apohelion, being a little father form Sun 
than 1.000 radius. 
    This means that the parallax of Venus is 1/0.289 that of the Sun's 
or 3.460 times greater. The parallax angle is so tiny we can use a 
simple proportion via the small-angle principle. 
   Against the celestial sphere Venus AND SUN are displaced by 
parallax such that Venus ON THE SUN shifts track by ((3.460)-
(1))/(3.460) = 0.711 of its full value  against the celestial sphere. 

        ------------------------ geocentric position of Sun & Venus 
               |    | 
               |    | Sun parallax 
               |   \|/ 
         Venus |   ------------- horizontal position of Sun 
      parallax |              | 
               |              | Venus shift 
               |              | on Sun 
               |              |        
              \|/            \|/ 
              ------------------ horizontal position of Venus 

    By missing the parallax shift of the background against which 
Venus is measured, erroneous values are calculated for the Earth-Sun 
distance. We must inflate the MEASURED displaced on the Sun by 
(3.460)/(2.460) = 1.407 times to remove the Sun's own parallax. 
    From modern computations, like in the circumstances published for 
the 2012 transit, the Venus shift on the Sun is 21.8 arcseconds. This 
is about 1/3 of Venus's angular diameter. This must be inflate by 
1.407 times to get 30.6 arcseconds. 

Earth-Sun distance 
    Now we can build the triangle. The base is 6,400km and apex is 
30.6 arcsecond. By direct trigonometry we have that the long sides of 
this triangle, assumed equally long, is 6400km/tan(30.6 arcsec) = 
43,740,000km (rounded). 
    From the scale of the solar system  this distance is 0.289 of the 
Earth-Sun distance.  we have 46.740,000km/0.289 = 149,235,000km.   
This full distance is the Astronomical unit. Given some rounding here 
this is only a whisker off of the actual value of 149,600,000km. The 
discrepancy of 365,000km is less than the Earth-Moon distance. 
    The short explanation here of how we used the Venus transits to 
find the Earth-Sun distance, and then the size of the whole solar 
system, is an ideal situation. The actual work in the 1700s and 1800s 
took many years after the expeditions returned to their home 
countries. The final accepted size of the solar system was issued near 
the run of the 20th century . It stayed in force until radar and 
spaceprobes improved it in the 1960s. We now know the value to a 
couple kilometers, the diameter of a small town on Earth. 
    It was an awesome thrill of this generation of astronomers to 
witness the two transits of the 21st century. The next pair come in 
the next century, long after all of us present today are passed on. 
With the separation between pairs being over a hundred years, it was 
possible that some astronomers lived and died without ever seeing one. 
    In addition to the sight of a whole effing planet plodding its way 
across the Sun, there are spinoff lessons in history and  global 
affairs, and, yes, maths and geometry.