The Day the World Discovered the Sun (34 page)

Ideally, one could take a time-lapse photograph of Venus as it traced a chord across the sun's face in, say, Vardø and compare that to the same time-lapse photograph of the Venus transit as seen by Chappe in Baja, Mexico, or by Cook and Green in Tahiti. The angular separation of the two Venus transits would be
θ
, and some smart cartography could yield
b
.

However, this approximation is too simplistic to do justice to the precision of the transit measurements and the needed precision of the calculations. For starters, it doesn't take into account the fact that
b
is measured over the curved surface of the earth. And, of course, there was no such thing as photography in 1769.

The approximation needs to be better.

Like many complex problems, there is no single correct way of arriving at an answer. Perhaps the most straightforward conceptually is the “educated guess” approach.

Namely, take the best result of the solar parallax from the 1761 Venus transits, and then blindly calculate the transit time one would expect to observe at any given latitude and longitude in the 1769 transit. Then factor
out
the estimated solar parallax and factor
in
the 1769 observations of transit times and known latitudes and longitudes.
⁵

Mathematically, if solar parallax being calculated is π
_Sun
and the estimated solar parallax is π
_est
, then the ultimate expression would take the form:

Where
dt
_O
is the observed difference in Venus transit times, discussed below, and
dt
_C
is the calculated difference in transit times.

Before getting into theoretical calculations, it's important to spell out precisely what data the three teams at the core of this book—Hell and Sajnovics at Vardø, Chappe at San José del Cabo, Cook and Green at Tahiti—brought home.

Table A.1. Results of 1769 Venus transit expeditions as measured by teams led by Maximilian Hell, Jean-Baptiste Chappe d'Auteroche, and James Cook

So one of the two possible
dt
_o
s here would be
t
₃
-t
₂
for Chappe subtracted from
t
₃
-t
₂
for Hell: 5
^h
53
^m
14
^s
- 5
^h
37
^m
23
^s
= 15
^m
51
^s
. The other would be
t
₃
-t
₂
for Cook subtracted from
t
₃
-t
₂
for Hell: 5
^h
53
^m
14
^s
- 5
^h
30
^m
04
^s
= 23
^m
10
^s
.

What remains, then, is the calculation of
dt
_C
, whose full derivation exceeds the scope of this appendix. Nevertheless, some essential equations and numerical results in these calculations can still yield a numerical answer.

It begins with so-called direction cosines of the three observers' locations on earth. Consider them
x, y
, and
z
components of unit vectors pointing to each observer's station (Vardø, San José del Cabo, Tahiti)—where this particular Cartesian coordinate system has its origin at the center of the earth and its
x-y
plane as the earth's equator and its
x-z
plane the Greenwich meridian.

Here, for example, are the direction cosines for Hell's observatory at Vardø:

Since the direction cosines describe unit-length arrows locating the various Venus transit observing stations, then similarly splitting up the
dt
_C
calculation into its three Cartesian components might help to simplify this difficult problem. For instance,
dt
_C
between Vardø and Tahiti would then take the form:

Where the coefficients
A, B
, and
C
(whose units are in seconds) represent components of the differential transit time weighted to expected values of its x,
y
, and
z
contributions. The coefficients are not particular to any location on earth but rather to the geometry of the sun's position in the sky and Venus's typical path across the sun's face during the June 3, 1769, transit.

The (present-day) French astronomer François Mignard derived the first approximations of
A, B
, and
C
as follows—in this case, particular to each contact. (So for the following formulas, one would calculate separate coefficients for the 1769 transit's external ingress, internal ingress, internal egress, and external egress.)

The variables
X
and
Y
(and their respective time derivatives) concern the position (and components of its speed) of Venus on the sun's disk, as seen through a telescope. The origin of the
X-Y
coordinate system is the sun's center, with
X
directed toward increasing right ascension
⁶
and
Y
toward celestial north. The values
r
_Venus
and
r
_Sun
represent the distances in AU from the earth to the respective bodies; thus 0.28 and 1. And
H
_Sun
and
δ
_Sun
are the right ascension and declination
⁷
of the sun as measured at Greenwich.

Despite the intimidating formulas above, there are just two key points to equations 8–10. First is the significance of any observer's additional Venus transit measurements—beyond timing, latitude, and longitude of the observatory. For example, Chappe's almost obsessive chronicling of every moment of Venus's transit—its angular speed across the sun's disk, its angular distance from the edge of the sun—would translate easily into
Ẋ
,
Ẏ
,
X
and
Y
in the above formulas and thus help any solar parallax calculation enormously.

The second point is to demonstrate that
A, B
, and
C
are each directly proportional to the initial estimate of solar parallax from equation 5. So
π
_est
merely cancels out in this first approximation, leaving
π
_Sun
independent of one's initial guess for the solar parallax.

The above represents only a first approximation of the coefficients
A, B
, and
C
, however. Mignard performed more detailed calculations of the three values and found—again, specific to the June 3, 1769, transit—
A
= 476.5 seconds,
B
= 376.5 seconds, and
C
= 516.1 seconds.
⁸

At last we may be able to derive our own value of the solar parallax, π
_Sun
, from equations 5–7 and the data in
Table A.1
.

Sticking with the Vardø and Tahiti measurements, then, equations 6 and 7 yield:

Equation 5 along with the
dt
_O
enumerated on p. 237 produces:

Mignard used the modern-day value (8.794 arc seconds) for deriving his
A, B
, and
C
coefficients, which, as shown above, to first approximation is irrelevant to the final solar parallax calculation anyway. So, using just Hell's and Cook's 1769 Venus transit measurements and a reasonable facsimile of the information available to an eighteenth-century astronomer, we calculate a solar parallax value of: