This article is also available in Japanase, thanks to a translation by Mr. Kazumi Funakoshi
 

Albireo (Beta Cygni). Image by NASA

Introductory note:   Double stars are common in our galaxy. Born as twin fragmentations from lumps in a giant molecular cloud, these celestial objects appear as a single point of light to the naked eye, but most of them can be observed in its double nature with a telescope. Some of them can even be observed with a good pair of binoculars. However, some doubles are easier to observe than others. Two physical parameters are responsible for the theoretical difficulty in observing them: Difference in bright (magnitude) between the stars in the double system and (angular) distance between them. Needless to say, when separation between components is wide, the system tends to be easy to split, while when both stars are tight it requires some effort for distinguishing them from a single star. Difference in brightness is important, too: when one of the stars is a lot brighter than the other component in the system the eye accommodates for the higher bright levels, making it difficult to observe the fainter star. On the other hand, the most favorable scenario happens when both stars have similar brightness. In the following article, Fuzzy-Logic theories are introduced and applied towards modeling an intelligent system in order to know in advance how difficult is to split a given double star.   Additional note: You can download a free Excel-Based model to calculate Difficulty Indexes of double stars visiting the Diffucalc site.

  

       

Everybody would agree that a double star whose components are separated by 10 seconds of arc is an open double, while another one whose components are only 1 second of arc apart is a tight double star. So far so good. Now, let me approach you while asking a simple, innocent at first question: “Where in the range from 1” to 10” would you change the adjective “tight” to “open”?

After a while, you probably answer me “Well, I’m not pretty sure, but maybe in an intermediate point, let’s say, 5 seconds of arc”.

Your reply is rather logical, but it creates a new, uncomfortable problem: from your classification, a double star such as Cp 13 (separation = 5.1”) is an open double star, while Cor 148 (separation = 4.8”) is a tight double. This sudden change of behavior doesn’t fit well with the experience of double star observers.

Wilhelm Struve, one of the greatest double stars observers from all times, suffered also from similar problems while making a classification for doubles in his Catalogus Novus (1827) depending on separation: Type I for doubles with less than 4" separation, Type II for doubles between 4" and 8", Type III between 8" and 16" and Type IV for separations  between 16" and 32". He realized the problems arising for such a sharp classification and again divided Type I into Vicinae, Pervicinae and Vicinissimae. If you pay some attention to this sub-classification you will see that it put the problem to sleep for a while, but it doesn’t solve it.

Interestingly, we humans tend to communicate with other humans with linguistic terms that are not based directly on arithmetics, but in experience and knowledge. Imagine the following situation: Two amateur astronomers are talking about their last double star observing session:

- "John, yesterday I started the session observing Epsilon Lyrae, the double double. It’s a lovely system, you know, with those rather close components. Later I aimed the telescope to Otto Struve 298 in Bootes (separation = 1.0”, same magnitude = 7.5 for both components). I suffered a lot until getting the split."
- "I know Mark, STF 298 is a really tight double!"

John and Mark don’t even need to speak about numbers in order to express their observing experiences. Language allows them to exchange information easily, and terms like “rather close”, “very tight” and so on are very useful when speaking about double stars and related terms. What a pity mathematics doesn’t allow us to express such things, right?. Don’t be worry, let’s enter the world of Fuzzy Sets Theories.

The central idea about Fuzzy Sets Theories (“Fuzzy Logic” is the expression that usually gathers all things related to Fuzzy Sets Theories, from Fuzzy Control to Fuzzy Statistics) is based on the inexistence of the so named “law of excluded middles”: From the times of Aristotle, we have grown in digital mathematical models: people is either tall or short, cars are fast or slow, things are cheap or expensive, everything is black or white. Nevertheless, the real world is made from different shades of gray!. Let me introduce you a Fuzzy Classification for separation in double stars. Pay a bit of attention now because you will hear some new powerful concepts.

Fuzzyfication

Figure #1 represents a Linguistic Variable that expresses the concept “Separation in double stars”. A Linguistic Variable is simply a collection containing several Fuzzy Sets. Here the used Fuzzy Sets are “Very Tight”, “Rather Tight”, “ Tight”, “Open Tight” and “Bit Open”. In order to see what a Fuzzy Set is, let’s take as an example the expression “Very Tight”.

As you can see, there is a triangle located at left in the figure with the label “Very tight” over it. The base of this triangle goes from 1.0 to 3.0 seconds of arc, so instinctively, we can say we have defined “Very tight” as every double star whose separation between components is bounded between 1 and 3 seconds of arc. Now the fun begins: Observe that the upper vertex of the triangle is located just over a vertical line from 1.0 seconds of arc. In fuzzy sets theory, we say that the membership degree to the Fuzzy Set of “Very Tight” double stars is highest when separation equals 1.0 seconds of arc. Then, this membership degree starts to decline, let’s say walking down along the triangle’s hypotenuse, until reaching a membership degree of 0.0 when it reaches a separation of 3.0 seconds of arc. That’s the real core of Fuzzy-Logic: Everything is a question of grade!

See now the representation of the Fuzzy Set “Rather Tight”; it starts at 1.0”, reaches its maximum membership degree at 2.0” and ends at 4.0”. The rest of Fuzzy Sets from the Linguistic Variable “Separation” are described in a similar way. In order to fix our ideas and also to experiment a bit with them, let’s see how these concepts describe a “real” double star. Let’s use Struve 3050 for it.

Figure #2: Membership degrees for STF 3050 with respect to Separation

This nice double in Andromeda Constellation has a separation between components of 1.8 seconds of arc. As you can see from Figure #2, it has the following membership degrees:

0.8 to the Fuzzy Set of “Rather Tight” doubles
0.6 to the Fuzzy Set of “Very Tight” doubles
0.26 to the Fuzzy Set of “Tight” doubles

Using this approach, we can smoothly describe the concept Separation without sudden changes in classification. Also, under this model, Struve 3050 is a “Tight double”, a “Rather Tight” double and a “Very Tight” double star at the same time, although with different membership degrees. 

In order to evaluate difficulty in splitting a double star, observers also take into account the bright of the main and secondary components. This plays a crucial role for knowing in advance how difficult to split a star will be. Using the same strategies as above, we can easily build another Linguistic Variable for the concept “Difference of Magnitude”. Please, observe Figure #3 for a representation of this.

Rules, rules everywhere

If we ask John and Mark how they know in advance about the difficulty in splitting a double star, they will tell us some expert rules to apply. For example, they can tell us the following:

Since John and Mark are lovely people with lots of patience, they agree to write for us a table describing all the possible combinations from our Linguístic Variables for Separation and Difference of Magnitude. Separations are expressed in columns, while rows represent Difference of Magnitude. The table looks as follows:
 

As you can see, we already have Fuzzy Sets describing Separation and Difference of Magnitude, but nothing has been said yet about difficulty. Well, we could model a new Linguistic Variable describing a Difficulty Index (DI), going for example from 0.0 to 100.0, where a DI of 0 means a double splittable by virtually every telesscope, while a DI of 100.0 would imply the use of a good telescope under very good “seeing” skies. From the table given from John and Mark, we could directly label the Fuzzy Sets for Difficulty as “Very Easy”, “Rather Easy”, “Something Easy”, “Something Difficult”, Rather Difficult” and “Very Difficult”. Figure #4 shows a representation for the concept “Difficulty Index”.

Inference

This phase of the process is where a Fuzzy Logic based system “thinks”. Let’s take Algieba, Gamma Leonis for this. Algieba has their components separated by 4.4 seconds of arc, and they shine at 2.5 and 3.6 magnitude respectively, so the difference of magnitude is 1.1. Figures #5 and #6 show how to obtain the membership values for “Separation” and “Difference of Magnitude”. These values are as follows:

Membership degrees for Separation:
Tight: 0.87
Open-Tight: 0.35

Membership degrees for Difference of Magnitude:
Medium: 0.93
Rather Big: 0.07

So, as described by our model, Algieba is a “Tight” and “Open-Tight” double (separation) with a “Medium” and “Rather Big” difference of magnitude. Now, what rule from the rules expressed in Table 1a must we apply?. Since we have “fired” two Fuzzy Sets for every Linguistic Variable involved, we need to get all possible combinations:

Distance is Tight and Delta Magnitude is Medium
Distance is Open-Tight and Delta Magnitude is Medium
Distance is Tight and Delta Magnitude is Rather Big
Distance is Open Tight and Delta Magnitude is Rather Big

Visually, we can see on Table 1b that in order to evaluate the Difficulty Index for Algieba, four rules have been triggered. This is another crucial point in Fuzzy Logic techniques: While traditional, old Intelligent Systems fired only one rule at a time, Fuzzy Logic works in parallel processing mode.

Now, another question arises: what membership degree will be applied to the Fuzzy Sets “Something Easy”, “Something Difficult” and “Rather Difficult” describing the Difficulty Index?. Well, Fuzzy Logic offers the modeler several alternatives for describing a “Fuzzy-AND” operator. Nevertheless, one of the most used one is the “minimum value”. Let’s see this for the first rule for Algieba:

Now, let’s substitute the Fuzzy Sets in bold font by their membership degrees:

Here, the minimum value of 0.87 and 0.93 is 0.87, so, using something named “Modus Ponens” in Logic, the resulting membership value for the Fuzzy Set “Something Difficult” will be 0.87. In another words, for Algieba, the first rule becomes:

Doing the same for every rule affected for Algieba, we would obtain Table 1c:

That is, Algieba has a Difficulty Index expressed with the following membership degrees:

Something Easy: 0.35
Something Difficult: 0.87
Something Difficult: 0.07
Rather Difficult: 0.07

You can see this as a board of four members from a committee of experts in double stars evaluating the Difficulty of splitting Algieba. The first member of them says it is “something easy” to split, and has a “confidence degree” in his vote of 0.35. In the same way, every expert expresses his/her opinion and gives also a “confidence factor”. As you can see, Fuzzy Logic is a true democratic system, but newspapers need a verdict for Algieba!. Fortunately, the board of experts have some tricks to elaborate a final result, that is, a Difficulty Index (DI) between 0 and 100.
 

For coming back from the Fuzzy world to the “normal” world, we need to convert a result expressed as a collection of membership degrees to a numerical result. After all, almost every Fuzzy system needs a real value as an output-result. In order to accomplish this, there are several techniques available to fuzzy designers. The important concept is that we can associate the resulting membership degrees for difficulty as “weights”, locating these “weights” in the basis’ central point of every triangle representing the respective Fuzzy-Set. The following Figure (#7) will make things crystal clear:

You can see the weights represented by vertical arrows, with longitudes expressing how much “weight” (membership degree) is associated. Now, what if we look for a point on the horizontal axis that would equilibrate these weights?. Ok, that’s exactly the output for a Fuzzy-System using what is known as "Sugeno’s Singletons modelling" and it’s represented in the Figure by the a red arrow pointing upwards. This method of defuzzyfication has the advantages of being intuitive and easy to calculate, so it’s very fast in critical control systems. The formal expression for this defuzzyfication method is as follows:

where x is the final result (Difficulty Index), wi are the weights (output membership degrees) and xi are the points (horizontal axis) where the weights are applicated. Substituting the values from Algieba in this formula, we will have:

So,

That is the obtained Difficulty Index obtained for Gamma Leonis.