A full account of the computational techniques used to obtain the astrometry of a binary star from the contact times of an occultation can be found in the DOCUMENTS section. In this section we overview the basic concepts and the occultation geometry in the case of a double star.
Consider the light curve below, which shows the occultation of SAO 093396, a binary star with WDS code J03206+1911 (observed by Ricard Casas on 29 January 2023). The sequence corresponds to the ingress of component A, followed by component B. The time interval between the two jumps is Δt, which is indicated in the graph. This behaviour can be explained on the basis of the occultation geometry.
Geometry of the occultation
The figure below shows a schematic of the immersion of a binary star at the lunar limb.
In panel (a) the occultation geometry is shown. v is the lunar apparent speed , which is at an angle with respect to the North Celestial Pole (PNC). In this case the event takes place at the dark limb, which greatly facilitates the visibility of the event (emersion would occur at the illuminated limb which does not lead to accurate quantitative measurements).
Panel (a) General scheme of an occultation, in this case with immersion on the dark limb and emersion on the illuminated limb. v is the apparent velocity of the Moon, and the PNC direction is oriented towards the North Celestial Pole, which serves as the origin for the measurement of all angles. Panel (b) Detail of the immersion at the moment when the main component A is occulted (the case in which B is occulted first is identical, except that angle PA increases by 180°), with the definition of the angle β associated with the velocity of the Moon, the angle γ of the local limb, and the parameters r and PA associated with the double star (whose values are sought). d is the arc that the component B travels along its path to the point of occultation (antiparallel with respect to velocity v). d is related to the temporal interval measured in the light curve for the step, Δt, by the equation d=vΔt.
In panel (b), the geometry of the problem is indicated in more detail. It has been assumed that the main component, A, is the first to be occulted. d is the path of star B until it is occulted, occurring in the direction antiparallel to velocity v. The limb can be assumed to be flat on the scale associated with the distance r between the components (unknown), but in reality, the limb could be different from that followed by the circular reference Moon due to the irregular lunar topography. In any case, the angle γ between the PNC and the limb is known (since it is obtained from the contact angle of the occultation and lunar topography). β is the angle between the PNC and the apparent velocity of the Moon, while PA is the position angle of the double (unknown). β and γ are essentially known for each observer (this is not entirely true, but the values can be refined).
By measuring d for each observer (d=vΔt, where Δt is the measured time interval for the step in the light curve), it is possible to obtain r and PA with at least two observations. The corresponding observers must be well separated geographically, so that their values of γ (or the contact points of the occultation on the limb) are significantly different.
Another possibility is to measure the occultation of the same star in two different lunations by the same or another observer. This case is possible, as a high percentage of stars undergo an occultation in successive lunations. In addition, the difficulty associated with the visibility of the Moon on the horizon, its phase, etc., must be taken into account.
From panel (b) of the figure, we can see that r is related to d, but through the angles PA, γ, and β. Depending on the values of these angles, and for the same fixed value of r, the value of d can vary from 0 to ∞. It will be zero if both components enter the limb at the same time, that is, PA=γ or PA=γ+180°. Conversely, if γ=β, it may be that only one component is occulted, and d=∞ (ideal grazing occultation). These are ideal situations, but it could be that, for a certain observer, d is so small that it is practically undetectable, and the light curve does not show any step, even though in reality the components are well separated.
The figure below presents different situations in an immersion (emersions are symmetrical). In the left and right panels, a schematic of the same occultation as seen by different observers is shown, with a different orientation of the limb. While the relative orientation of the two stars with respect to the sky is the same for both observers, the orientation of the segment that connects them with respect to the local lunar limb varies markedly, which implies that the arcs d1 and d2 measured by both observers are different. From the two measurements d1 and d2 (or better said, from the related values of Δt1 and Δt2), the parameters of the double star r and PA can be obtained.
Two observers, 1 and 2, observe the same occultation (in this case, assuming that the North Celestial Pole is pointing upwards, an immersion). The orientation of the limb with respect to the sky in the region of immersion is very different, but the relative position of the two components is logically the same, which results in different arcs d1 y d2 for the steps observed by the two observers, even with a possible reversal in the order in which the two components are occulted.
Limb roughness
Let us now consider the effect of limb roughness. Although the shape of the Moon is extraordinarily spherical (and therefore its projected visible limb is practically circular), surface irregularities and topographic features can influence the analysis of the results. As a reference, an apparent arc of 1" at the average distance to the Moon translates to about 1900 metres, and one of 0.01" means 19 metres. These values are close to the spatial resolutions of the lunar missions LRO LOLA and Selene (Kaguya), based on radar-derived altimetry. These altimetry data allow to correct the limb with respect to a perfect circle at the points where the two occultations occur.
According to the previous scale, a hill with a height difference of about 20 metres projects onto the sky in an arc of 0.01", which is compatible with the separations r between the closest (but measurable) pairs. Therefore, it is necessary to take into account the roughness of the limb for stars with separations r ranging from a fraction of an arcsecond to tens of arcseconds (left and central panels of the figure below). For larger separations, another scale comes into play, the lunar curvature (right panel). For example, 1' is projected onto the limb as 112 km, which is about 6% of the lunar radius: in the reduction of observations, it is necessary to consider limb roughness at small scales (fractions of 1" and a few arcseconds) and intermediate scales (up to tens of arcseconds), and lunar curvature at large scales (1' and above).
On the other hand, the limb profile that the Moon presents to an observer strongly depends on the orientation of the Moon. However, it should be noted that current knowledge about the position of the Moon and its rotation state is very high, and in any case more than sufficient to carry out data reduction that makes sense and includes the lunar limb component.
Left panel: Scales on the order of 0.01" in arc projected onto the plane of the sky imply distances on the lunar limb (tangent and perpendicular directions) of about 20 metres. At this scale, the lunar surface exhibits a roughness that has an impact on the reduction of measurements, as the typical scale of these profile variations are compatible with the separation between the components of some double stars measurable with the lunar occultation technique. Central panel: At larger scales (on the order of 1", which implies projected distances of a few kilometres), roughness also plays a role, with the coarser details of the topography contributing most. Lunar curvature only contributes a few metres to vertical variation and can be disregarded. Right panel: At even larger scales, curvature comes into play, with the effect being more significant.
Relative magnitudes of the star components
Finally, let us calculate the brightness drop in the light curve as a function of the magnitudes of the components of the double star. Suppose the magnitudes of the components of the double star are mA and mB, with mA < mB and Δm=mB-mA. Considering the relationship between magnitude and flux, we have:
mA = -2.5 log IA and mB = -2.5 log IB,
where IA and IB are the respective fluxes (relative to a standard flux; see the light curve at the beginning of the section). The total flux will be I=IA+IB. Rearranging the variables, the combined magnitude of the double star is:
m = -2.5 log (IA+IB)=-2.5 log (10-mA/2.5+ 10-mB/2.5)
If the secondary star B is occulted, the flux will drop from I=IA+IB to IA, and the flux drop ratio, visible in the light curve, will be:
IA/I=10-mA/2.5/(10-mA/2.5+10-mB/2.5) =1/(1+10-Δm/2.5).
If the primary star is occulted, the flux drop will be equal to:
IB/I=1/(1+10+Δm/2.5).
These relations allow to determine the magnitude difference between the components from the height of the steps in the light curve.
The value of the ratio for both cases is represented in the figure below. Note that, if we set the limit of flux drop at 5% or 95%, the magnitude difference Δm between the components cannot exceed 3.2 magnitudes. That's why the proposed double stars for this project tend not to exceed this magnitude difference. Finally, it should be noted that if the stars have the same magnitude, Δm=0, the flux drop will be exactly 50% (to see this, we just need to substitute Δm=0 in any of the two expressions above, for IA/I o para IB/I).
Flux ratio in the light curve according to the magnitude difference of the components of the double star. From a difference of 5 magnitudes, the flux difference is negligible. We consider a magnitude difference limit of 3.2, for which the flux ratio (partial occultation flux over the nominal value) is 0.05 (if the primary star is occulted and the secondary is visible) or 0.95 (if the primary star is visible and the secondary is occulted).