Calculating Astronomical Unit from Venus Transit A) Background 1) Parallaxes of the Sun (the horizontal parallaxes) By definition the parallaxes of the Sun is the angle β shown below: By trigonometry, sin β = R / r but the angle β is small and sin β can be approximated by β measured in radians. R is the radius of Earth and r is the distance from the observer to the object. We can get r using the relationship r = R/ β 2) Observation from the Earth Let's consider two observers on the Earth situated in points A and B on the same longitude (meridian), but at very different latitudes. The alignment of AB should be approximately perpendicular to the line of sight to the transit so as to keep the errors as small as possible. Venus is seen as a small disk on the face of the Sun at two different points A' and B'. This is because the lines of sight of A and B towards Venus are not identical. 1
Putting the two observations together using the reference stars it is possible to measure this parallax displacement. 3) A Geometrical Problem Let's consider the plane defined by three points: the Earth's centre O, the Sun's centre C and the Venus centre V. The triangles APV and BPC have the same external angles at P, hence βv + β1 = βs + β2 βv - βs = β2 - β1 = Δβ Where angle Δβ measures the distance between the different positions of Venus's trace on the face of the Sun. Rearranging the last equation gives Δβ = βs (βv/ βs - 1) Now Venus's parallax is βv = AB/(r e - r v ) and the Sun's parallax βs = AB/r e, hence the quotient βv/ βs = r e /(r e - r v ). Substituting this into the equation above gives 2
Δβ = βs (r e /(r e - r v ) - 1) = βs r v /(r e - r v ) In particular, we can get the solar parallax, βs = Δβ (r e / r v - 1) Note that in order to measure Δβ it is necessary to superimpose the two Sun centres at C and then Δβ is the distance between the two traces of Venus observed at same time from A and B. 4) Kepler's Third Law Let's take r e as the Earth-Sun distance and r v the Venus-Sun distance. We can calculate the ratio (r v / r e ) 3 by using Kepler's Third Law as we know that the periods of revolution of Venus and of the Earth are 224.7 days and 365.25 days respectively. (r e / r v ) 3 = (365.25/224.7) 2 therefore r e / r v = 1.38248 5) Final formulae for the Earth-Sun distance. Using this result in the parallax formula from section 3, we get βs = Δβ((r e / r v ) - 1)= Δβ (1.38248-1) therefore βs = 0.38248 Δβ And finally using the parallax formula from section (1), the distance from the Earth to the Sun r e is r e = AB/βs 3
So we need to find AB from the location of the observers and to measure Δβ from observational data of the transit. B) Observational data needed 1) Distance between observers at points A and B The distance AB can be deduced from the latitude of the two points of observation. In the diagram, ϕ1 and ϕ2 are the latitudes of A and B, and R is the radius of the Earth. In the right angled triangle that divides the isosceles triangle RAB sin ((ϕ1 + ϕ2)/2) = AB/2)/R. Then the distance AB is AB = 2 R sin ((ϕ1 + ϕ2)/2) Be careful. If both cities are in the same hemisphere, the angle is (ϕ1 - ϕ2)/2 and also the geometrical situation changes if both cities are on different longitudes. 2) Distance Δβ between two observed paths of Venus In order to calculate Δβ we need the data obtained by two observers at the points A and B on the same longitude (meridian). In any case it is necessary to have a "photograph" of the paths of Venus visible from each location or the times that Venus crossed the Sun's disk. 4
Calculation of Δβ by direct measurement Measure the diameter of the Sun D and the distance between the two paths Δβ, that is to say A'B', on a photograph. The angular diameter of the Sun, seen from the Earth is 30' (minutes of arc or 30 / 60 ). By means of simple proportion, the distance between the observations of Venus is linked to the Sun's diameter by Δβ /30' = A'B'/D therefore Δβ = (30') (A'B'/D ) but the formula requires the Sun's angular diameter to be expressed in radians. Therefore Δβ = (30 π/10800) (A'B'/D) Δβ = (π/360) (A'B'/D) C) Observations that you can make in 2012 1) Single observations that are quick and easy to make Make plans to record an image of the transit when Venus is on the mid-line of the face of the Sun (point A'). You then need to share your results with another observer who is located on the same longitude (meridian) and who will observe at more or less the same time (point B'). You have to measure the distances DA and Δβ (from the centre of the Sun CA and CB) as shown in the diagram. Obviously the diagrams must be adjusted to be the same size to allow DA and Δβ to be compared. To obtain the highest accuracy you will need to take a photograph or make a still video image. 5
You can try to make a pencil sketch of a projected image but the problem will be that the image will drift across your screen as the Earth turns. This will make it difficult but not impossible to be precise. We suggest you try a combination of methods in case one of them lets you down! To calculate Δβ you need to measure the diameter of the image D, and D A and D B to the same scale. Then taking the angular diameter of the Sun, seen from the Earth as 30' (minutes of arc) then and the distance AB is Δβ = (π/360)((d B - D A )/D) =... radians βs = 0.38248 Δβ =... radians AB = 2. 6378. sin ((ϕ1 - ϕ2) / 2) =... km where ϕ1 and ϕ2 are the latitudes of the observers and the radius of the Earth, R = 6378 km. If the observers are in opposite hemispheres, the angle is (ϕ1 + ϕ2) / 2. and finally the Earth-Sun distance r e = AB/βσ =... km If you want, you can find your own value for the angular diameter of the Sun but you will need to know the focal lengths of your telescope's eyepiece and objective lens and you will have to refer to an optics text book for the necessary formula. 2) Longer observations that are quite easy to make 6
If you are able to make observations throughout the transit, you will be able to plot the path of Venus and note the times of first and second contacts. If you can observe for at least half an hour you can reconstruct the whole transit as follows. Record the exact starting and finishing times of your observations. On a scale drawing, mark a line to represent the path of Venus during your observations. Now extend this straight line until it touches the limbs (edges) of the Sun's image. This is shown in the diagram. By simple ratios you can find the time for the total transit (for instance t A ). You then need to share your results with an observer who is located at a different latitude. You will then have their transit time t B and so will be able to calculate the Earth-Sun distance, also known as the astronomical unit (1 AU). The great advantage of this method is that you do not need to be on the same longitude (meridian) as the other observer. [Source: http://skolor.nacka.se/samskolan/eaae/summerschools/tov2.html] 7