Simplified method for the calculation of the parallax

P. Rocher (IMCCE)

In this note we will present a simplified calculation of the average equatorial parallax of the Sun. This simplification of calculation is done at the price of an important constraint for the observations. We will suppose that we have two simultaneous observations which provide us the distance between the two apparent centers of the planet Venus in front of the solar disc. We will indicate the approximations and simplifications which we carry out.

Let be two places of observations M_{1} and M_{2},
sufficiently distant; two observers note at the same moment *T*
the position of the apparent center of
the planet Venus in front of the solar disc. Then using these two
observations, it determines the distance which joins these two
apparent Venus centers. The measurement of this distance
expressed in solar radii, makes it possible to calculate the equatorial
mean parallax of the Sun
p_{0}. We will see that this measurement is far from being simple.

Figure 1. - Observation of the Venus transit from two sites at the same moment

Let *O* be the center of the Earth, *C* the center
of the Sun, *V* the Venus center and *V _{1}* and

If the two points *M _{1}* and

And **the following relation** :

**is false**. It is true only if the four points are coplanar.

Contrarily, the difference of the parallaxes is equal to the angular distance between the two apparent centers of Venus (figure 2).

Figure 2. – Apparent positions of Venus on the solar disc.

It is checked that this difference is equal to *D _{2 }*-

The value that the observers will measure is thus
*the distance Dp * between
the two apparent centers of Venus and it is the relation _{
}
which will allow us to calculate the parallaxes.

For that we will express the two parallaxes according to
the distances between the center of the Earth and the center of Venus and the Sun.
Let be *R** _{V}* the distance between the center of the Sun and the
center of Venus and

Figure 3. – Solar parallax related to the points M_{1}
et M_{2}.

As the terrestrial radius and the distance between the two points are small compared to the distances Earth-Sun and Earth-Venus, the parallaxes are given by the following approximate formulas :

^{ (1)}

Actually, the exact parallax are given by :

Then we have the following relation :

^{ (2)}

and

or

_{
}^{ (3)}

The measurement gives us the value Dp expressed in solar diameter and one must also measure the diameter of the Sun, because if the distance Earth-Sun is unknown, this diameter may not be calculated.

To know the solar parallax, it is thus necessary also to know the ratio of the distances Sun-Earth and Sun-Venus. However this ratio can be calculated thanks to the laws of Kepler.

The first law of Kepler says to us that the
planets describe elliptic orbits around the Sun and that the Sun
occupies one of the focus of these ellipses. At a given moment
the radius vector *R _{p}*
joining the center of the Sun to a planet

_{}
^{ (4)}

Where
*a _{p}* is the equatorial radius of the ellipse,

The third law of Kepler provided a relation between the equatorial radii of the orbits and the periods of revolution of the planets, thus for the same central body all the orbits of the planets which revolve around this central body check the following relation :

The laws of Kepler thus describe the orbits of the solar system except for a scale factor. The observation of the periods of revolution of planets gives us the ratio of the semi-major axis, thus the ratio of the semi-major axis of the orbits of Venus and of the Earth is equal to :

_{}^{ (6)}

and at any date *T*, the ratio of the radius vectors is equal
to

_{
}
^{ (7)}

Thus the laws of Kepler make it possible to
calculate the ratio of the radius vectors for any date *T*.

Our measurement allows us to calculate the value *p _{S,}* it is
thus appropriate now to pass from this value to the value of the
mean equatorial parallax of the Sun p

The mean equatorial parallax of the Sun p_{0 }is by
definition the angle under which one sees the equatorial radius of the
Earth from the center of the Sun when the Sun is at one astronomical
unit from the Earth.

We have thus the following relation :

_{
}^{ (8)}

*R* being the terrestrial equatorial radius
and *a*
the astronomical unit.

However the equation (1) gives us the value of the solar
parallax*p _{S }*according to the distance

It is enough to express this distance *D*
in terrestrial radius and the distance Earth-Sun in
astronomical unit to have a relation between *
p _{S}*
and p

_{
}
^{ (9)}

It remains only to
calculate the *D* on *R* ratio .
The *a/R** _{T }* ratio is
provided to us by the first law of Kepler (cf formulae 4).
However if one makes the vector product of the two vectors

_{
}^{ (10)}

However the product of the length of the first vector by the sine
of the angle between the two vectors _{
}
is equal to the distance *d*. In the same way the length of _{
}
is equal to the distance *R _{T}*

Figure 5. -
Solar parallax related to the points M_{1}
and M_{2}.

The resolution of
equation 10 gives us the value of *d*.

_{
(11)
}

Note : if the concept of vector product
is not known, one can use the scalar product of the same vectors, that
allows to calculate the cosine of the angle, then his sine using the
relation : _{}.

This calculation on the vectors requires to know the Cartesian
co-ordinates of the two points *M _{1}* and

This reference frame is defined by the plan of the terrestrial
equator at the date T of the observation (plane *Oxy*) and by the
direction of the northern celestial pole of the axis of rotation of Earth (*Oz*).
In this reference frame one can define a Cartesian reference frame (*x, y,
z*) and a polar reference frame (*a*,
*d*, R) the two angles having the
name of right ascension and declination. One passes from one system to the other
by the following relations :

_{}^{ (12)}

and the relations opposite

_{
(13)
}

The direction of axis
*Ox* at the date T is the direction of the vernal equinox at the
same date.

The ephemerides (i.e. the Kepler's laws) provide us the equatorial
geocentric coordinates of the center of the Sun (a, *d*);
thedistance is not know but it is not important

because the vector_{
}
may be replaced by its unit vector in equation 11.

The more complicated problem is the determination of the cartesian
coordinates of the points *M _{1}* et

Figure 6. –* Geocentric equatorial coordinates*.

The position of a point at the surface of the Earth is given
by its latitude and its longitude (geographic); the latitude is given referred
to the terrestrial equator, so that it is an angle such as the declination.
The longitude is given referred to a meridian origin (Greenwich meridian), so
that it is an angle similar to the right ascension, but with an origin different
from the one of the equatorial lestial coordinates. It is then necessary to
know at each date the angle between the direction of *Ox* axis and the
direction of the projection of the meridian origin in the equator plane (cf.
figure 6). This angle is related to the rotation of the Earth: it is named the
"sidereal time" of the Greenwich meridian and its increases by 360°
during 23h 56m 4s (sidereal revolution of the Earth).

So, it is sufficient to know the sidereal time of Greenwich
*T _{G}* at 0h UTC for the day of the transit to know the sidereal
time of Greenwich at any time

_{}
^{ (14)}

One will pass from the sideral Greenwich time to the sidereal
time of the site M having the longitude *l*,
by adding or substracting this longitude.

Attention, sidereal time increases when going towards east
from of the meridian

of Greenwich; it is thus appropriate to pay attention to the convention of sign
well used

to note longitudes.

If **the longitudes are counted negatively towards east**
then

the relation linking local sidereal time to the meridian line of the site of
longitude *l* and sidereal time
with the meridianof Greenwich is as follows:

_{
(15)
}

Attention les deux angles doivent être exprimés avec la même unité (degrés ou heures).

Alors les coordonnées cartésiennes d’un point *M _{1}*
de coordonnées géographiques (

_{}
^{ (16)}

The length_{
}
of the vector_{
}
(its modulus) and its coordinates (*X*, *Y*, *Z*) are given by :

_{}
^{ (17)}

_{
}
of the direction « center of the Earth - Sun » is given by :

_{}

The vector product_{
}
and its modulus are then :

_{}
^{ (19)}

and finally, using formula (11), one will get :

_{}
^{ (20)}

And the mean equatorial parallax is given (following (9) by :

^{ (21)}

We will take as an example the observation made at Antananarivo (Madagascar) and at Helsinki (Finland) at the date t=8h 30min on June 8, 2004.

The geographic coordinates of Antananarivo are :

Latitude :18° 52' south, longitude : 47° 30' east then j_{1}_{ }= –18,866667° and l_{1} = –47,5°.

The geographic coordinates of Helsinki are :

Latitude :60° 8' north, longitude : 25° 3' east, then j_{2}_{ }= 60,133333° and l_{2} = –25,05°.

The geocentric equatorial coordinates of the Sun at 8h 30m UTC are provided by the ephemerides :

Right ascension of the Sun *a _{s}*
= 76°49' 36.493"

Declination of the Sun *d _{s}*
= +22°53' 16.237"

The sidereal time at bGreenwich at a date *t* in UTC is
provided by the following formula :

*T _{G}* (

Then the sidereal time at Greenwich at 8h 30min is equal to :

*T _{G}* = 17h 6m
51,31s + 8h 31m 23,78s = 25h 38m 15,09s = 1h 38m 15,09s

It is necessary to convert it in degrees before calculating the local sidereal
time for the two cities.

*T _{G}* = 1h 38m
15,09s = 24,562875°.

From this, one will deduce the local sidereal time at 8h 30m at Antananarivo :

*T**l** _{1 }*= 24,562875 – (–47,5°) = 72,062875°

And the sidereal local time at 8h 30m at Helsinki is :

*T**l _{2}*

One will deduce from this the cartesian equatorial coordinates for these
two cities :

Antananarivo :

_{}

Helsinki :

_{}

The coordinates of the unit vector of the direction Earth-Sun are obtained through formula 18 :

The vector_{
}
has as coordinates :

_{}

Formula 20 allows us to calculate the value of *d* :

_{}

The ephemerides provide us the ratio between the radii vectors and the ratio between the distance Earth-Sun and the semi-major axis of the orbit of the Earth at the given date :

_{}

Now, we will just have to make an hypothesis on the measured
values, i.e. on *Dp * and on the solar diameter :

We will suppose that :

That gives the value of* Dp *:* *28,359"

Formula 3 provides us the value of the solar parallax :

_{}

And formula 21 provides us the value of the mean equatorial parallax :

_{}

The value that we finds is relatively close to reality, but it depends only on the measure of the distance from the apparent centers of Venus on the solar disc and on the size of the solar diameter. The apparent size of the solar diameter can be measured with a good precision; on the other hand the measurement of the distance between the apparent centers of Venus is not obvious, on a traditional photographic stereotype, the apparent diameter being around 20mm, the distance from the centers is then of 0,3mm and a precision of one thousandth corresponds to a measurement to 0,02mm.

In the preceding forms, one occulted a certain number of difficulties
to simplify the problem. Here the list of the complications which appear if
one wants to make a rigorous calculation:

1. Because of the mutual disturbances, the orbits of the planets do not follow
the laws of Kepler (valid only for two bodies) but more complex trajectories.

2. This is not the Earth which has a quasi-elliptic orbit around the Sun but
the barycentre of the system Earth-Moon.

3. Following the movement of the axis of rotation of the Earth (precession and
nutation), the origine Ox of the equatorial reference frame is not fixed relatively
to time.

4. The light being propagated with a finished speed, the positions of the Sun
and Venus at a given date *t* are not geometric positions but positions
of two bodies at the date *t – **t _{p}*,

We saw, in the sheet n°04b, that it exists two simplified formulae allowing to calculate directly the parallax from the comparison of the dates of the same contact seen from two different sites (Delisle method) or from the comparison of the duration of the transit observed from two different sites (Halley method).

We will study simultaneouly these two aspects from the preceeding numerical example.

The mean equatorial solar parallax
_{} is obtained by comparing
two identical contacts using the following simplified formula
(cf. formula 16 of the sheet n°04b) :

_{}

If one neglect the uncertainties and the errors, then, the formula becomes :

_{}

Same, the mean equatorial solar parallax is obtained by comparing the two identical duration, using the following formula (cf. formula 21 of the sheet n°04b) :

_{}

*i* and *j* are the index related to the same contacts
: *i* = 1, *j* = 4 for the external contacts and *i*
= 2, *j* = 3 for the internal contacts.

The coefficients *A*, *B*, *C* and the
term _{} are calculated for
each contact and are given by the following table :

Description of the contact |
A |
B |
C |
dD/dt "/min |

First external contact (index 1) |
2,2606 |
-0,0194 |
1,0110 |
-3,0846 |

First internal contact (index 2) |
2,1970 |
0,2237 |
1,1206 |
-2,9394 |

Last internal contact (index 3) |
-1,0929 |
-1,1376 |
1,9090 |
2,9391 |

Last external contact (index 4) |
-0,9799 |
-1,3390 |
1,8383 |
3,0842 |

We will take again the example of the same two cities with the observational hypothesis as follows :

City n°1 : Antananarivo (j_{1}_{ }= –18,866667° et l_{1} = –47,5°)

Date of the first observed internal contact (index
2) : t2 = 5h 35m 30s UTC.

Date of the last observed internal contact (index 3) : t3 = 11h 8m 4s UTC

Observed duration of the transit (internal): 5h 32m 34s.

City n°2 : Helsinki (j_{2}_{
}= 60,133333° et l_{2}
= –25,05°)

Date of the first observed internal contact (index 2) :
t2 = 5h 38m 38s UTC.

Date of the last observed internal contact (index 3) :t3 = 11h 2m 20s UTC

Observed duration of the transit (internal) : 5h 23m 42s.

In the formulae (22) and (23) the factors of the coefficients
*A*, *B*, *C* are identical and may be calculated separately:

**Calculation of the parallax using the first
contacts :**

The differences between the dates of the first internal contacts
is –3m 8s (–3,1333m), and the use of the values of the coefficients *A _{2}*,

_{}

That provides _{}.

**Calculation of the parallax using the duration of
the internal transits**

The difference in duration of the internal transits is
8m 52s (8,866m), and the use of the values of the coefficients *A _{2}*,

_{}

Attention, it is the value _{}
and mainly its sign which may be used.

That gives _{}.

Let us remind that these methods are not completely accurate and one should use more complete formulae for the reduction of the observations.