Sur l'extrême petitesse du diamètre apparent des étoiles fixes

1998: Refereed Papers in Stellar Interferometry

Sur l'extrême petitesse du diamètre apparent des étoiles fixes
E. Stéphan
Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 78, 1008-1012 (1874).

Translated by Peter R. Lawson.

Astrophysics. - On the extreme minuteness of the apparent diameter of fixed stars. Note from Mr. Stéphan.

(Commissioners: Le Verrier, Fizeau, Janssen.)

In a previous Communication (Comptes rendus, Vol. 76, p. 1008), I had the honour to remind the Academy of an idea previously put forth by Mr. Fizeau in the form of a simple insight and which, up until then, seemed to have remained forgotten, even though it contained the foundation of matters of significant consequence. This idea can be formulated as follows: In many cases, by giving rise to certain interference phenomena, we can increase the sensitivity of ordinary optical instruments.

Guided by the illustrious physicist, I sought to deduce from this original concept some precise notions on the apparent diameters of the fixed stars, and in the Note referred to above, I made known to the Academy the results of some preliminary experiments whose general principle it is now useful to recall.

We know that if a telescope is pointed at a bright point source and masked by a screen with two small holes in it, in the focal plane are formed a set of alternatively dark and bright fringes, and a very elementary theory gives us the angle over separation l that the first two dark fringes would subtend as viewed from the center of the objective. This angle, measured in seconds of arc, is given by the ratio 103.1/l (with an assumed wavelength of 0.0005 mm, and where l is expressed in millimeters).

If the luminous source possesses appreciable dimensions, its different points give rise to sets of fringes that are each superimposed one on the other. Therefore if the diameter is greater than or equal to 103.1/l, the coverage is complete, and the fringes disappear. If, on the other hand, the fringes persist, we must conclude that the diameter of the source is smaller than the above ratio.

In practice, when we work with a faint luminous source, for example when we view a star, we are obliged to use apertures of a sufficiently large size; we can show that, even in this case, providing the apertures are of each of equal shape and composed of two halves that are symmetric with respect to an axis, the two axes being perpendicular to the line that joins their mid-points C and C $^\prime$ , fringes are formed whose spacing is the same as if the openings were reduced to points at C and C $^\prime$ .

The problem can be posed in the following manner:

Given two equal segments ADBE and A $^\prime$ D $^\prime$ B $^\prime$ E $^\prime$ of the same plane wave, where the edges are each composed of symmetric halves, with respect to the two axes AB and A $^\prime$ B $^\prime$ , that are each parallel to each other and perpendicular to the line CC $^\prime$ that joins their centers, determine the intensity of light received an infinite distance away, along the line OL, which lies in a plane perpendicular to the wavefront and passing through CC $^\prime$ . Let us define the abscissa (x) to be the line (CC $^\prime$ ) and the ordinate (y) the perpendicular to the above, taken from the middle O of (CC $^\prime$ ).

Let

$\theta$ be the angle that the line OL makes with respect to the normal to the wavefront;
l = CC $^\prime$ , the distance between the centers of the apertures;
A = ED, the width of the apertures along the x axis;
h = AB the width of the apertures along the y axis;

We let sin2 $\pi t/T$ (T being the period of the vibration) describe the rate of the vibration which, having been emitted by the element O of the plane wave, would arrive at time t at a plane far removed from the origin, and perpendicular to the direction OL. The rates sent simultaneously to the same plane by two elements of the apertures taken symmetrically with respect to the origin of the y axis are respectively

( $\lambda$ being the wavelength);

The resulting rate can be expressed as

therefore the luminous intensity, along the direction OL, is proportional to the quantity

the integral extending over all elements in the aperture ADBE.

Now let

be the equation of the line ADB; we have, by a initial integration with respect to x,

y₁ and y₂ being the ordinates of the points A and B.

The integration which is the last part of the above expression cannot be carried out in all cases; but the value of this integral can be written as

$\mu$ being a unitless fraction.

This expression for I² tells us that the global maximum of illumination takes place in the direction normal to the wavefront. Away from this direction, on one side and the other of OL, follow relative maxima separated by minima completely devoid of light.

These minima are divided into two series defined by the following formula:

$\begin{eqnarray*}\mbox{Series A} & & \sin\theta = {2k+1\over 2} {\lambda\over l},\\ \mbox{Series B} & & \sin\theta = k {\lambda\over \mu a}, \end{eqnarray*}$

k being any whole number.

If we consider the first arc of these two series, we see that

For Series A, the separation of two consecutive dark fringes is $\lambda/l$ ,
For Series B, the separation of two consecutive dark fringes is $\lambda/\mu a$ .