Geometrical aspects of Venus transit

Bertuola, A. C.; Frajuca, C.; Magalhães, N. S.; Santos Filho, V. dos

doi:10.1590/S1806-11173731941

A. C. Bertuola ¹1 E-mail: acbertuola@gmail.com.

Instituto Federal de São Paulo, São Paulo, SP, Brasil

C. Frajuca

Instituto Federal de São Paulo, São Paulo, SP, Brasil

N. S. Magalhães

Universidade Federal de São Paulo, São Paulo, SP, Brasil

V. dos Santos Filho

H4D Scientific Research Laboratory, 04674-225, São Paulo, SP, Brasil

1 E-mail: acbertuola@gmail.com.

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Figure 1
Mosaic formed by photos in which it is shown the composition of the positions of Venus during the transit. From frames 1 up to 5, one observes the complete composition of the transit. For the others remaining photos, the composition of the transit is incomplete.

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Figure 2
Geometrical treatment of Venus transit of the photo 5 from . In the frame (a), we drew with Photoshop three straight lines, being one the diameter of the Sun and the others enclosing all the spots of Venus during the transit. In the frame (b), we zoom in the right extreme of the diameter, so that we could securely obtain one point in the illuminated region and the other one in the dark region. In the frame (c), we zoom in one of the spots, so that it were entirely contained inside a square drawn with the software. Inside the spot, a circle was drawn in order to outline the effective dark region.

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Figure 3
Initial triangulation proposed from Venus transit. In the scheme, we show the basic triangulation, in which is drawn the entire planet and its spot on the Sun.

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Figure 4
Triangulation proposed from Venus transit in the case of the first model. In the scheme, we have a representation of the static model, in which it is not considered the Earth revolution movement.

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Figure 5
Triangulation proposed from Venus transit in the case of the second model. In the scheme, we have a representation of the translational model, in which we consider the Earth revolution movement.

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Figure 6
Scheme for visualization of the correction in the translational model due to the rotation of the Earth. In the calculation, we first consider a translation of the center of the Earth from C up to

C^{'}

, characterizing the translation

\bar{T U}

. In the end of the translation, one realizes a rotation of 100.05° and the observer in the Earth is dislocated from U to V, characterizing the effective distance

\bar{T V}

. So, the variation in the positions of the observer corresponds to

\bar{U V} = Δ l

. In both circles, the parameter

δ

is the angle of observation in relation to the horizon.

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Table 1
The quantity

d_{E S}

is the Earth-Sun distance (in km),

D_{S}

is the Sun diameter,

T_{E}

is the Earth revolution period (in days),

T_{V}

is Venus orbital period (in days), t is the duration of transit (in hours) and

R_{E}

is the radius of the Earth (in km).

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Table 2
Ratios obtained from the photo 5. Each numerical matrix element represents the ratio of the diameter or length in the column by the diameter in the row. We considered internal values for

D^{'}

and

L^{'}

, respectively indicated by the first and third columns; and external values for both, respectively indicated in the second and fourth columns.

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Table 3
Average values of the ratios and their respective errors:

Δ

(standard deviation) and

Δ (%)

(percentage error). We used the percentage error to study the quality of the images.

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Table 4
Numerical results calculated by means of the indirect geometrical method for the static and translational cases. Note that the best results for the Earth-Venus distance and Venus diameter were obtained from the translational method. The first line of the numerical results refers to the complete transit and the last one corresponds to the incomplete one.

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Table 5
Final results for the Earth-Venus distance and Venus diameter by means of the translational model with correction.

(1)

(2)

(3)

(4)

(5)

(6)

$d_{E S} (km)$	$\frac{d_{E S}}{D_{S}}$	$T_{E} (day)$	$T_{V} (day)$	$t (h)$	$R_{E} (km)$
$1.521 \times 1 0^{8}$	109.3	365.26	224.7	6.67	6378

–	$D^{'}$	$D^{'}$	$L^{'}$	$L^{'}$
$D_{S}$	0.03191	0.02808	0.8596	0.8239
$D_{S}$	0.03197	0.02813	0.8587	0.8254

Photo	$α$	$Δ$	$Δ (%)$	$β$	$Δ$	$Δ (%)$
1	0.0284	0.0028	9.7	0.8108	0.025	3.1
2	0.0297	0.0024	8.0	0.8250	0.031	3.8
3	0.0311	0.0009	3.0	0.8039	0.028	3.5
4	0.0301	0.0017	5.7	0.8118	0.020	2.4
5	0.0305	0.0020	6.7	0.8408	0.020	2.3
6	0.0291	0.0038	13	0.8018	0.020	2.5
7	0.0326	0.0044	13	0.8220	0.012	1.5
8	0.0370	0.0041	11	0.8116	0.029	3.6
9	0.0232	0.0038	17	0.8210	0.030	3.6
10	0.0239	0.0030	12	0.8187	0.023	2.8
11	0.0291	0.0018	6.2	0.8637	0.030	3.5
12	0.0211	0.0045	21	0.8157	0.031	3.8

Static		Translational
$d_{E V} (10^{7} km)$	$D_{V} (10^{4} km)$	$d_{E V} (10^{7} km$	$D_{V} (10^{4} km)$
$7.610 \pm 0.060$	$2.086 \pm 0.065$	$4.231 \pm 0.048$	$1.160 \pm 0.037$
$7.586 \pm 0.097$	$2.11 \pm 0.30$	$4.187 \pm 0.077$	$1.16 \pm 0.17$

$d_{E V} (km)$	$D_{V} (km)$
$4.296 \pm 0.048$	$1.178 \pm 0.038$

d_{E V} = [\frac{2 (δ_{V} - δ_{E}) \frac{d_{E S}}{D_{S}}}{α + β + 2 (δ_{V} - δ_{E}) \frac{d_{E S}}{D_{S}}}] d_{E S} .

d_{E V} = \{\frac{[2 (δ_{V} - δ_{E}) + \frac{Δ l}{d_{E S}}] \frac{d_{E S}}{D_{S}}}{α + β + [2 (δ_{V} - δ_{E}) + \frac{Δ l}{d_{E S}}] \frac{d_{E S}}{D_{S}}}\} d_{E S} .

[1] 1 E-mail: acbertuola@gmail.com.

Brasil

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Geometrical aspects of Venus transit

(Aspectos geométricos do trânsito de Vênus)