modeling the law of times

7/28/2019 Modeling the Law of Times

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Modeling the Law of Times

Julio Plaza del Olmo

July 2013

Abstract

A model to describe the Law of Times has been developed. The model is based

on a geographical-astronomical factor, describing the probability of a phenomenon

of being visible, and a second factor related to the social habits of the population,

accounting for their availability to witness a phenomenon. These two factors can

accurately reproduce the main peak at 21-22 h, and also account for sightings oc-

curring at dawn. However, it is not capable of reproducing the secondary peak at

2-3 h, but an ad hoc term is introduced to describe it.

Three catalogs with different scopes and geographical coverage are analyzed using

this model. Differences between them can be understood in terms of the input

parameters of the model. Issues related to Daylight Saving Time, time zones and

duplication of entries in catalogs are discussed, and solutions are proposed to make

all data comparable. It is also found how extra factors, such as technology, can

influence the shape of the curve, and how different contributions to a catalog can be

separated and studied independently. In this regard, an interesting finding is how

technology is affecting the time distribution of hoaxes and also the secondary peak.

Finally, past interpretations of the Law of Times are reviewed and discussed under

the scope this model. One of the main conclusions of this work is that the Law of

Times is not a property of UFOs, but only a consequence of sighting conditions:visibility and social habits of population

Correspondence: [email protected]

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Modeling the Law of Times 2

1 Introduction

From the very beginning of ufology, the compilation of UFO cases has been an important

activity to try to find patterns that can explain the phenomenon. As a result, there are

different catalogs on which statistical analyses have been done, and can still be done.

One of the patterns was found by Jacques Vallee and published in 1966 [1] [2], and has

come to be known as the Law of Times. In subsequent years, this pattern was replicated

by Vicente-Juan Ballester Olmos [3] and Ted Phillips [4] using different catalogs, which

confirmed the original discovery and lead to the conclusion that a real phenomenon was

taking place. Furthermore, this Law has also been replicated recently with pre-1880

UFO cases[5], which suggests that the same kind of phenomenon was already happening

in the past.

Figure 1 shows Vallees original graph, along with several other Laws of Times for differ-

ent catalogs. They have different scopes and geographical coverages: worldwide landing

reports (VALLEE, extracted from [6]); worldwide photographic records (FOTOCAT)[7];

landing reports in Spain and Portugal (ALLCAT)[8]; all kinds of reports in Spain, Por-

tugal and Andorra (CUCO) [9]; and all kinds of reports worldwide (HATCH, extracted

from [10]).

How similar these graphs are, can be quantified using a correlation coefficient, shown

in Table 1[11]. Despite their different scopes and origins, they all show a basic common

pattern: a peak at about 21-22 h. But there are also differences, and it is remarkable

to note differences between CUCO and ALLCAT, both covering the same geographical

area, besides sharing a certain number of cases, since most of ALLCAT is included in

CUCO.

FOTOCAT ALLCAT VALLEE HATCH CUCO

FOTOCAT 1.0 0.46 0.72 0.78 0.72

ALLCAT - 1.0 0.84 0.85 0.90

VALLEE - - 1.0 0.96 0.96

HATCH - - - 1.0 0.99

CUCO - - - - 1.0

Table 1: Correlation coefficients for time distributions of Figure 1. From ref. [11]


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Figure 1: Top: Law of Times, as originally plotted by Vallee [2]. Bottom: Law of Times for several

catalogs[11]


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Figure 2: UFO and IFO Laws of Time for FOTOCAT(left)[11] and CUCO(right)

Another interesting feature is the comparison between UFO and IFO Laws of Times, as

they show the same time distribution (Figure 2)[11], that leads to the logical conclusion

that both must be affected by very similar factors, if not the same factors.

In the past, some efforts were made to quantify the differences among catalogs. It

does not seem possible to define a Satisfactory Law of Times to compare them with[12];

however, the development of a mathematical model capable of describing and reproduc-

ing time distributions based on known factors should allow finding a way to comparecatalogs, and understand similarities and differences that can shed some light on the

factors influencing UFO phenomena.

In the next sections, we will derive a model using simple and reasonable assumptions.

Then, some qualitative results will be shown to see how the model can reproduce some

basic features of the Law of Times. In section 3, we will look at some catalogs, and

find out how the model can reproduce them. This will allow us to find deviations from

the theoretical model that can be understood in terms of other factors. It also includes

finding issues in catalogs that have to be solved by pre-processing the data. Finally, wewill briefly discuss past interpretations of the Law of Times, and compare them with the

interpretation provided by our model.

2 Mathematical derivation of the Law of Times

The starting point to derive an equation describing the Law of Times (LoT), is supposing

that every day, at every hour, there is a constant number of luminous events happening,


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Ne. These events appear randomly, and may have any origin: lights in the sky, meteors,

reflection of sunlight on balloons or clouds, planes, lights from a car, satellites, the

moon, stars, planets. . . even a genuine flying saucer or any unknown phenomenon can be

an event. The nature of the event is not important at this stage, we just have to suppose

that luminous phenomena appear either in the sky, or at ground level.

But the brightness of the event, i.e. its magnitude, must be enough so that it is not

eclipsed by atmospheric luminosity. Obviusly, an event is more likely to be seen at night

than during the day. Therefore, from the number of events Ne, only a fraction of them

will be bright enough to be seen by the naked eye, so we can define the number of visible

events, Nv(h, d) as:

Nv(h, d) = Ne Pv(h, d) (1)

Pv(h, d) being the visibility, the probability of an event being visible at hour h, on day

d.

However, for an event to be witnessed, visibility is not the only condition. There has to

be somebody present to see it. Thus, the number of witnessed events, Nw(h, d), depends

on a witnessing probablity, Pw(h, d), defined as the fraction of visible events that are

actually witnessed:

Nw(h, d) = Nv(h, d) Pw(h, d) = Ne Pw(h, d) Pv(h, d) (2)

To construct the time distribution, we need to calculate the total number of witnessed

events at hour h, Nw(h), adding all witnessed events at that hour,

Nw(h) =d

Nw(h, d) =d

Ne Pw(h, d) Pv(h, d) (3)

and divide by the total number of events:

P(h) =Nw(h)

Nt=

Nw(h)h Nw(h)

(4)

P(h) =

d Pw(h, d) Pv(h, d)

h [

d Pw(h, d) Pv(h, d)](5)

Equation 5 is the final expression that replicates the Law of Times. Ne is a constant,

therefore it cancels out after the division. The problem of modeling the LoT is now


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transformed into finding mathematical descriptions for the visibility (Pv), and witnessing

probability (Pw)

2.1 Visibility

Visibility is the probability of an event to have a magnitude great enough to be visible

to the naked eye. An event can either emit or reflect light, and whether it is visible or

not depends on the atmospheric luminosity, that is determined by the night/day cycle,

geographical coordinates and season of the year.

Visual Limiting Magnitude (VLM)[14] is defined as the faintest magnitude that thenaked eye can see. At nighttime, this value is around 5.5. During the day, it is about

4. One must remember that the lower the magnitude, the brighter the object.The transitions between daytime and nighttime create sunrises and sunsets, moments

when the VLM will change between these two extreme values. Twilight [15] is the period

of time when the position of the sun changes from the horizon (altitude of 0o) to an

altitude of18o (under the horizon). Night starts when its altitude is below that value.A simple model for the transition between daytime and nighttime is to consider that

VLM changes from 4 to 5.5 linearly with the altitude of the sun between 18o and 18o(Figure 3), to take into account that light also vanishes while the sun approaches the

horizon during sunset (and vice versa during sunrise). Reference [13] provides us with

an on-line calculator for VLM, and we can see that our approximation is reasonable.

Finally, VLM does not only depend on the hour of the day, but does depend on the

geographical location and the day of the year.

After calculating the VLM, the next step is to assign a value for the visibility probabil-

ity. Let us suppose that whenever an event appears, it has a random magnitude following

a normal distribution with a mean value and standard deviation . We define Visibility

as the probability of an event to have a magnitude equal or lower (brighter) than the

visual limiting magnitude at the time, day and location where the event appears (Figure

4). Mathematically, it is determined by the Cumulative Probability Function.

Pv(h, d) =

V LM(h,d)

122

e(v)2

22 dv 12

1 + erf

V LM(h, d)

22

(6)


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Figure 3: Top: Altitude of the sun for solstice days, at 40oN, 0oE (UTC). Bottom:Modeled VLM for

winter solstice (black line), and calculated from ref. [13] (dashed red line) (UTC)


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Figure 4: Top: Distribution of Events Magnitude, and Cumulative Probability Function. Bottom:

Color-Mapped Visibility as a function of hour and month. The darker the color, the higher the visibility.


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Perhaps it would be more accurate to consider contrast instead of brightness, specially

if we think in daylight hours. Contrast can be defined in different ways, but in any case,

it is related to the difference in luminosity between the object and the background. That

relation is basically the same we are using to define visibility at any time (night or day)

in this work. Thus, the definition of Visibility based on Magnitude can also be applied

to daylight time.

Briefly, Pv(h, d) is a rather complex function that has to be calculated in several steps:

Calculate the elevation of the sun with respect to the horizon. The input parameters

are: latitude, longitude, day and UTC hour.

Calculate VLM as a function of sun elevation.

Calculate Pv as a function of VLM and magnitude distribution.

Taking into account the annual periodicity of night/day cycles helps to simplify and

accelerate calculations, as the summation can be done over the days of the period of

interest. For instance, if the interest is in reproducing a catalog covering several years,

the summation on d can be done only over the 365 days of a single year (or, for more

accuracy than is probably warranted, over four years to account for leap years).

P(h) =

31Decd=1Jan Pw(h, d) Pv(h, d)

h

31Decd=1Jan Pw(h, d) Pv(h, d)

(7)But the summation can also be done over a single month to have a Monthly Law of

Times:

P(h) = 31Juld=1Jul Pw(h, d) Pv(h, d)

h

31Juld=1Jul Pw(h, d) Pv(h, d)

(8)Other periodicities such as bimonthly or seasonal periods can also be considered, to

construct a Winter LoT, a Summer LoT, and so on. . . The possibilities the model offers

are flexible to analyze different situations.


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2.2 Witnessing Probability

This is the most difficult factor to model, since it is meant to represent a social habit.

The most intuitive idea is to represent the fraction of the population that is awake

as a function of time. The more people who are awake, the higher the probability that

somebody can witness an event. We can suppose that sleep habits are the same everyday.

This might be a correct guess for short periods of time, but there could be differences

among the seasons of the year, because of different lengths of the day and night. People

could go to bed later because of longer days in summer. Moreover, this habit could have

regional differences, influenced by culture, and the adaptation to local daylight and night

times. This is a possibility that will not be taken into account as a first approximation,

but we should keep it in mind. In any case, we can think that the parameters will be

mean values over the considered period of time. Thus, equation 5 can be then rewritten

with Pw independent of the day, and removed from the summation on d:

P(h) =Pw(h)

d Pv(h, d)

h [Pw(h)

d Pv(h, d)](9)

To model this function, we will suppose a normal distribution for the Wake-Up time

of the population, and another normal distribution for the Go-to-Bed time.

Pwu(h) = exp

(ha h)

2

22a

(10)

Pgb(h) = exp

(hb h)

2

22b

(11)

ha : Mean Wake-Up time.

a : Standard deviation of Wake-Up time.

hb : Mean Go-to-Bed time.

b : Standard deviation of Go-to-Bed time.

p0: Minimum percentage of the population that is awake at night.

The percentage of the population that is awake can be calculated as the Cumulative

Distribution of the Wake-Up distribution minus that of the Go-to-Bed distribution:


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Figure 5: Solid black line: Witnessing Probability. Red/blue dashed lines: Wake-Up and Go-to-Beddistributions.

Pw(h) = p0 + (1 p0)h

0Pwu(h

)dh h0

Pgb(h)dh

(12)

Pw(h)

p0 + (1

p0)

1

21 + erf

h ha

22

a

1

2

1 + erf

h hb

22b

(13)

A minimum of population may remain awake at night, p0. We can think that this

value represents people working at night, or awake for any other reason. Figure 5 shows

Pw, as well as the meaning of the parameters.


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Figure 6: Law of Times as the multiplication ofPv and

Pw. Main features are: 1-peak rise, 2-peak

maximum, 3-peak decrease and 4-morning peak/valley

2.3 Qualitative analysis

Once Pw(h) and Pv(h, d) have been modeled, the LoT is related to the multiplication of

both functions (Figure 6): a peak is formed right after sunset.

During daylight hours, most of the population is awake and there is a high probability

of witnessing an event. However, the VLM has a value of

4, meaning that only very

bright events will be visible. Therefore, a low percentage of cases can be reported during

those hours. The opposite reasoning is valid during most of the nighttime: a VLM of 5 .5

makes even faint events visible, but the fraction of people awake is low, again yielding a

low percentage of reported cases.

The main feature of the Law of Times is the peak. It is the consequence of an increase

in visibility due to sunset, as well as still having a high percentage of the population awake

(Figure 6-1). The combination of both factors causes the peak to reach its maximum


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Figure 7: Peak maximum and sunset time in Belgium from Ref. [16]


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Figure 8: Reproduction of Figure 7 using the LoT Model. Calculated at 50o

N, 0o

E (UTC). Note thatvertical axis is inverted respect to the original graph.

value (Figure 6-2). Then, as people go to bed, there is a decrease in reported events

(Figure 6-3).

In 1980, Gregor and Tickx published the paper OVNI: un phenomene parasolaire?[16].

A catalog of over 4000 cases worldwide was analyzed, although a regional analysis low-

ered the number of cases, yielding noisy plots. In any case, it was enough to deduce an

important feature of the LoT: a correlation between peak maximum and sunset was ob-served. This led them to the conclusion that UFO sightings were related to the elevation

of the sun relative to the horizon.

Figure 7 shows one of the graphs from Gregor and Tickx, illustrating the variation of

the peak maximum, compared to the time of sunset throughout the year in Belgium.

Figure 8 shows a reproduction of the same plot, using the model to produce a Monthly

LoT. It has been calculated using eq. 8, varying only the month over which the summa-


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Figure 9: Left: Monthly LoT from NUFORC-California Right: Monthly LoT from Model

tion in d is done, while all other parameters remain unchanged. The qualitative behavior

is essentially the same.

We have analyzed a list of UFO sightings for California, taken from NUFORC[17],

with 9225 cases. No particular revision was made in order to reject cases, because of

duplication, hoaxes or any other reason. The list was used as it is. There are enough

entries to construct monthly histograms with about 700 cases per month. There has been

no correction for Daylight Saving Time either, and its effect can be seen in the monthsfrom March/April to September/October in Figure 9-Left. Colors map the percentage of

cases, with black and blue for lower values, and yellow and red for higher values. Figure

9-Right shows again a simulation of a Monthly LoT.

Gregor and Tickx showed that the variation of peak maximum was correlated to sunset,

and NUFORC data allow us to verify another feature: the increase in percentage of

cases of the main peak during summer months. This increase in percentage is not to

be confused with an increase in the absolute number of cases. It can be understood by

thinking that in summer time, nights are shorter. As observations are more frequent atnight time, in summer they tend to be grouped in a shorter time frame before people

go to bed. On the contrary, in winter, with longer nights, observations can be scattered

over a longer time frame. Hence the increase in peak maximum value (in percentage)

during summer months.

The model indicates that a fourth feature could exist in time distributions, that should

appear at the moment when Pw and Pv cross again at sunrise (Figure 6-4). The model

predicts either a small peak or a valley:


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If people wakes up when it is still dark, the percentage of cases should increase,

creating a morning peak. This situation would be typical of winter.

If sunrise starts before a significant number of people wakes up, a small morningvalley would be observed. This situation could happen in summer.

This last feature could be difficult to reproduce in the model, since it is near the

daylight baseline, although in Figure 9 there seems to be a small morning valley during

summer, which can also be seen in the model simulation.

2.4 The secondary peak

2.4.1 Addition of catalogs

Before going any further, it is interesting to know a property of catalogs, which is inde-

pendent of any model or distribution. Given n catalogs, each with Ni (i=1 . . . n) entries

and individual LoTs Pi(h), it can be shown that after joining all the catalogs together in

a single one with a total of NT =

Ni entries, the total Law of Times can be expressed

as a weighted addition of the individual time distributions:

P(h) =ni=1

Pi(h)

NiNT

(14)

where the weight factors are just the proportional contributions to the total catalog,

Ni/NT. Mathemathical proof of this property is given in appendix A.

2.4.2 Adding a secondary peak

One feature not reproduced by the model is a secondary peak that appears at about 2-3am. Sometimes, this peak is evident. Other times, it is hidden under the tail of the

main peak, revealing itself only as a change in slope. Although its origin is unknown at

the moment, it is possible to add its contribution to our model, as a Gaussian function

whose area is the fraction of cases that create this peak in the catalog (Fs).

Ps(h) =Fs

s

/2exp

2 (h hs)

2

2s

(15)


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Using the property of catalog addition, we may split any catalog in two: one for the

main peak, and a second one for the secondary peak. When adding both, Fs is by

definition the weight factor for the secondary peak, whereas (1Fs) is the weight factorfor the main peak. The total time distribution can then be expressed as:

P(h) = Pmain(h) (1 Fs) + Ps(h) (16)

Pmain is eq. 5, and Ps is just a Gaussian function (eq. 15), with a mean value hs and

standard deviation s. A different way to deduce equation 16 is given in appendix B.

3 Application to catalogs

After development of the model, we are in a position to understand why catalogs have

basically the same shape, and also to understand why there are differences among them.

We can reproduce the LoTs for catalogs by optimizing the parameters fed into the model,

and then analyzing if their values are reasonable. This optimization is done by minimizing

the following function:

2 =h

[PC(h) PM(h;a)]2 (17)

where PC is the LoT of the catalog, PM the model, h is the hour of the day, and a are

the parameters to optimize. A value of 2 = 0 means a perfect match between data and

model.

Parameters for visibility are related to geographical position and event magnitude

distribution. The former are fixed by the geographical location of sightings. The latter

have to be optimized, but it is difficult to evaluate how reasonable they will be. Even ifcatalogs record any data related to brightness, it will be a subjective value given by the

witnesses. Perhaps some study could be done on IFO cases with known magnitudes, such

as stars or planets. For witnessing probability, sociological surveys or statistics should

provide independent data to validate the model, and/or to fine-tune its mathematical

expression.

But first of all, we also have to determine whether data contained in catalogs are

really comparable or not. From the point of view of the LoT model, it is important to


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determine how much light there was at the moment of the sighting. If we think carefully

about the raw data, we may easily identify some potential issues:

Influence of Daylight Saving Time

Influence of geographical position

Influence of Time Zones

Multiple entries for a single case

All these factors may influence the time distribution. But as explained in section 2.4.1,

their influence depends on the proportion of the related entries in the whole catalog.

3.1 Daylight Saving Time

Catalogs register the official local time at the place of the sighting. However, in most

countries official local time changes for certain periods of time. In the Northern Hemi-

sphere, time is adjusted forward one hour from March/April to September/October in

order to adjust daylight to human activity. In the Southern Hemisphere, the forward

adjustment is from October to March.

These changes lead to artifacts, and to deviations respect to what is expected from

the LoT model. In principle, using the catalog addition property, we can separate the

catalog in two, one using standard time, covering the months of standard time; the

other one with Daylight Saving Time (DST), covering the months of summer time; and

add them with their corresponding weight factors. However, not every country observes

this adjustment, and not every country has always observed it. Therefore, it is much

simpler to correct the time of sighting in catalogs to treat them all in standard local time.

From here on, unless specified otherwise, the time distributions in figures will show the

standard time, after correcting the DST time where needed.

Figure 10 shows a simulation of the DST effect, and Figure 11 shows the effects on two

catalogs. The most significative difference is a delay in the main peak maximum, which

is shifted about one hour.


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Figure 10: Dashed lines: Standard Time (October to April) and Daylight Saving Time (April to

October). Red line: addition of dashed lines with 0.4 and 0.6 as weight factors.

Figure 11: Left: CUCO. Local official time (black), and local standard time (red). Right: ALLCAT.

Local official time (black), and local standard time (red)


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Figure 12: Left: Selected east and west regions of Spain. Right: Time distributions for east and west

regions of Spain (CUCO)[9]

3.2 Effect of longitude

Geographical parameters for the model are latitude () and longitude () of the location

of the sighting. These parameters, along with the local time and UTC offset, are used

to determine the altitude of the sun. As this altitude is dependent on the geographical

position and local time, it is expected to find some dependence of the LoT on them,

especially with : given two different locations with the same local time, sunset occurs

later in the one to the west, and hence, the main peak will be delayed with respect to

the other location to the east.

We have used the CUCO catalog[9], containing over 5000 cases in Spain, Portugal and

Andorra, where the sighting time has been recorded. We have constructed the LoT for

two different regions: eastern Spain (728 cases) and western Spain (1134 cases). Figure

12 shows these two regions, and their respective LoTs; a time delay can be seen for the

western region, as predicted.

In any catalog, sightings do not happen in a single location, but in several different

places, creating a sighting distribution. For our western and eastern distributions, the

latitude range is about the same in both regions, from 37o to 43o N. But in longitude, the

eastern region covers from 1.6o E to 2.3o W, whereas the western region goes from 3.7o

to 7.4o W. Figure 13 shows the distribution in . If we think again in terms of adding

catalogs, we may conclude that having a distribution on and can be seen as adding

multiple catalogs for each location, with weight factors that are the relative frequency of


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Figure 13: Longitude distributions of West and East regions

each pair (, ). Thus, in the end, the distribution of and can be described by their

mean values, which can be used as input parameters for the LoT model.

Figure 14 shows an optimization of the model to the time distributions, and Table

2 shows the optimized values for the parameters. Mean difference is of 5.7o, which

means a sunset difference of about 22 minutes. The model shows a small morning peak

(Green dashed line). Secondary peak is mixed with both morning and main peaks, and

as a result, secondary and morning peaks are not evident in the time distribution. Also

note that the optimized parameters for Pw are quite similar for both regions. Figure 15

shows the Witnessing Probability, as well as the distribution of Events Magnitude.

The optimization of Pw parameters shows that the mean Wake-Up hour is about

8 : 30, and Go-to-Bed is near midnight. Are these values reasonable? As we said earlier,

sociological studies should give us some feedback. As a first approximation, one might

think that this Wake-Up hour is more or less in the middle of the typical rush hour, and


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Figure 14: CUCO LoTs (black and red lines) and optimization of model (blue lines). Dashed green

lines show the contribution of main and secondary peaks. Top: West of Spain. Bottom: East of Spain


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Figure 15: Top: Witnessing Probability. Bottom: Events Magnitude distribution.


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Fixed Parameters West East

Mean Latitude 37.37o N 39.64o N

Mean Longitude 6.1o W 0.4o W

UTC Offset +1 +1

Witnessing Prob. (Pw)

Wake-up time ha 8 : 33 8 : 33

Wake-up deviation a 3.24 2.62

Go-to-bed time hb 23 : 42 23 : 51

Go-to-bed deviation b 1.21 1.46

Awake population at night p0 0.0% 0.0%Visibility (Pv)

Mean Events Magnitude 1.046 0.693

Events Magnitude Deviation 3.06 3.65

Secondary Peak (Ps)

Percent of cases Fs 10.6% 11.4%

Mean hour hs 2 : 07 2 : 30

Deviation s 2.37 1.97

Table 2: Parameters for the optimization of Figures 14

one would expect people to wake up earlier than that. But we also have to realize that

ha is a mean hour, meaning that half of the population is already up. Therefore, even if

the value seems a bit late, it still makes some sense.

Mathemathically, the function is useful to describe the model. But, its iterpretation

might be open to debate. Does it actually represent awake population? Or could it be

any other social factor closely related?

3.3 Secondary peak side-effect: CUCO and ALLCAT

In the last section we compared two separated regions of CUCO catalog to understand

how geographical longitude has an effect in the time distribution. Let us now compare

CUCO and ALLCAT catalogs. They both cover the same geographical region, Spain and

Portugal. We are going to compare data from continental Spain. Portugal and Canary

Islands belong to a different time zone, and we will deal with that issue later.


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CUCO is an all-category catalog, and for this section we are taking into account 4376

entries. ALLCAT[8] is a landing-category catalog with 614 entries of interest. Figure 16

shows the time distribution for both catalogs. ALLCAT plot is more noisy, reflecting

the fact that it contains less entries than CUCO. But it is enough to observe similarities

and differences.

They both show the usual secondary peak at about 2-3 a.m., although ALLCATs is

higher. Daylight cases are also very similar. The main difference is in the position of

the main peak, with a time delay between them. It is important to know that most of

ALLCAT is included in CUCO, accounting for the 6% of CUCO entries.

Let us analyze the geographical distribution of cases. Figures 17 maps them, showing

the percentage of cases in areas of 0.5ox0.5o in and . Notice that both catalogs

have a hot spot near the city of Seville. However, CUCO also has other hot spots near

Madrid, Barcelona and Basque Country. As a result, their mean geographical position

Figure 16: Law of Times for CUCO and ALLCAT


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Figure 17: Geographical distribution of sightings for CUCO and ALLCAT. Color scale is the same in

both graphs. Red crosses mark the mean latitude and longitude of each distribution

is different, marked with a red cross in both maps. ALLCAT mean position is more to

the west (4o46 W) than CUCOs (3o36 W).

Figure 18 shows again an optimization of the model, and the value of parameters are

listed in Table 3. Parameters are not much different from those of the previous section,

something that already was expected for CUCO. The largest difference is in the secondary

peak. Visually, ALLCATs is higher, and it is confirmed after the optimization process,if we look at the percentage of cases included in the secondary peak (Fs): 11% vs 23%

Looking at both geographical distributions, the difference in mean longitude is only

about one degree. That means a delay in sunset of 4 minutes. That is not enough to

justify the apparent delay between CUCO and ALLCAT. The reason for ALLCATs main

peak rise delay is a side effect of the higher secondary peak. A property of the LoTs is

that they have a constant area under the curve because of the way they are constructed.

The summation over all data must always be 100%. This means that wherever there

is an increase(such as a higher contribution of the secondary peak), there has to be a

decrease somewhere else as to keep a constant total value of 100%.

Let us have a look at the main and morning peaks optimizations of CUCO and ALL-

CAT. Substracting the effect of the secondary peak, and normalizing to an area of 100%

(Figure 19), shows that both curves are much closer (solid lines) than it seems. When

having into account the secondary peaks, the main peaks are reduced in different propor-

tions, due to the different secondary peak contributions, and creating the visual illusion

of a delay in ALLCAT respect to CUCO (dashed lines).


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Fixed Parameters CUCO ALLCAT


Mean Longitude 3.6o W 4.77o W

UTC Offset +1 +1


Wake-Up time ha 8 : 34 8 : 36

Wake-Up deviation a 3.10 2.20

Go-to-Bed time hb 23 : 22 23 : 43

Go-to-Bed deviation b 1.34 1.34


Mean Event Magnitude 1.09 0.63

Event Magnitude deviation 3.53 3.96

Secondary Peak (Ps)

Percent of cases Fs 11.6% 23.1%

Mean hour hs 2 : 14 2 : 24

Deviation s 2.69 2.77

Table 3: Parameters for the optimizations of Figures 18

Figure 18: CUCO (Left) and ALLCAT (Right) LoTs and model (blue lines).


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Figure 19: Left: Modeled main peaks of CUCO and ALLCAT, normalized to 100% area. Dashed lines:

Secondary peaks. Right: Real catalog data after substraction of secondary peaks, and normalized to

100% area.

3.4 Time zones and longitude translation

Time zones exist to adjust the local time to the daylight. Earth rotates at an angular

speed of 15o/h. That means that in two places 15 degrees apart in longitude, the relative

position of the sun will be the same with a difference of one hour. For that reason, local

time is adjusted adding or substracting hours depending on the geographical location,creating time zones. Coordinated Universal Time (UTC) is the time reference, taken at

0o longitude.

Ideally, each time zone covers 15o. UTC+0 time zone spans from 7.5o W to 7.5o E;

UTC+1 spans from 7.5 E to 22.5o E, and so on. However, most countries use a time zone

because of other factors (political or economical, for instance). For this reason, most of

western Europe is included in the CET time zone (Central European Time, UTC+1),

when, because of their longitude, should use the UTC time zone. Such is the case of

France and Spain, while Portugal, and Great Britain, being in about the same longitudes,observe UTC time. Furthermore, some countries use more than one time zone. Canary

Islands (Spain) observe UTC time, while Madeira and Azores (Portugal) are in UTC-1

and UTC-2 time zones. United States is divided in 5 different time zones and Russia

uses up to 7 time zones (Figure 20).

If a catalog includes data from different time zones, a question arises on how the data

should be treated, and whether the raw data is comparable or not. Such is the case

of CUCO, with data from continental Spain and Portugal, and their islands (Canary


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Figure 20: Time zones in the world[18]

Islands, Madeira and Azores). Let us use again the western Spain LoT of section 3.2

(Figure 12), and compare it with the LoT of Portugal (452 cases). Even though Portugal

is located to the west of Spain (mean = 8.58o W), time distribution is advanced in

time, as if it was to the east of Spain (Figure 21).1

In fact, we can consider that Portugal is to the east of Spain. Let us think about 12

p.m. in continental Spain. As Spain is one hour ahead of its ideal time zone (or ahead

of its solar time), the sun is aligned with the 15o E meridian. In Portugal, the local hour

is 11 a.m. One hour later, the sun will be aligned with the 0o meridian. It will be 12 p.m.

in Portugal, but 1 p.m. in Spain. When constructing the LoT, we are counting in the

same bin both the spanish 12 p.m. and portuguese 12 p.m. However the relative position

of the sun with each country is different. At 12 a.m. in Spain, the relative position of the

sun is 15o to its east. If we could move Spain 15o W from its location, we would be ableto use the UTC+0 time zone (Figure 22), and keep the local time as well as the relative

position of the sun respect to Spain and Portugal at the same time. Thus, respect to the

behaviour of LoT model, Portugal looks like being to the east of Spain with respect to

the light/darkness conditions, and the main peak will therefore rise earlier.

1From 1966 to 1976, Portugal used a UTC+1 time zone with no DST. Before and after that period,

Portugal used a UTC time zone with DST. Times have been corrected to have all times in UTC time

zone, and corrected the DST when needed.


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Figure 21: Time distribution of Portugal, and western Spain as defined in Figure 12

In the previous sections, we have been optimizing the data using the real mean values

for longitude and time zone UTC+1. With this translation property, we can optimize

the same time distributions and get the very same parameter values using longitudes 15o

to the west, and a time zone UTC+0.

We can define Longitude Z (Z) as the longitude at which local time can be considered

with no UTC offset. It can be calculated as:

Z = 15 UT C (18)

If we apply this transform to CUCO, we obtain the Figure 23. We can see that

Portugal appears to the east of Spain. But we can also see that Canary Islands do not

suffer any change, since they are already using a UTC+0 time zone, and appear in their

real position in the same longitude ranges than translated Spain.

With this transform, we can use the local times in catalogs without any pre-proccessing.


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Figure 22: Top and Middle: Position of the sun respect to Spain and Portugal. For the same local

hour in both countries, the sun is aligned with two different meridians. Bottom: Translation of 15o to

the west makes Spain use the same time zone that Portugal, and keep the same local time and relative

position respect to the sun.


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Figure 23:Z distribution of CUCO. Portugal appears to the east of Spain. Canary Islands appear at

the same longitudes than continental Spain.

However, for a geographical distribution analysis, we should transform the real to Z.

Some countries have a UTC offset that makes local time become syncronized with their

natural solar time. That means that a longitude translation to UTC time zone will yield

values between 7.5o E and 7.5o W. For those one hour ahead of their solar times, Z will

be between 22.5o and 7.5o W. Finally, the behaviour of any catalog (local, regional or

worldwide) will be as though the world has been compressed into a region from 22 .5

o

Wto 7.5o E and can be described in terms of this Longitude Z.

3.5 The influence of technology: FOTOCAT

In Figure 1, we compared several Laws of Times from different catalogs. FOTOCAT

had the most different shape, with the lowest correlation coefficient respect to the other

catalogs. However, we must realize that FOTOCAT is influenced by a very specific


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factor.

FOTOCAT[7] is a photograph and footage catalog, which means it has a strong tech-

nological factor. We are going to work with a set of 2247 IFO cases. These have been

originally classified in 7 categories:

1. Hoax: Fakes and manufactured flying saucers.

2. Camera and film related: Development flaws, lens flares, artifacts related to

the camera,. . .

3. Aerospatial: Aircrafts, condensation trails, balloons, helicopters, satellites, reen-

tries, airborne debris,. . .

4. Meteorological and geophysical: clouds, mirages, ball lightning,. . .

5. Astronomical: Bolides, stars, planets,. . .

6. Biological: bugs, birds, persons,. . .

7. Miscellaneous: automobiles, debris, ground lights,. . .

We can see from this classification that some of the explanations are solely depen-

dent on technology: development flaws, lens flares, flying-by birds or bugs, blurred

objects,. . . They do not depend on visibility and are only seen after taking the image

(i.e. they were not seen initially by the photographer, and hence, not photographed

on purpose). On the other hand we have cases which are basically in the scope of our

model: planes, satellites, distant lights, clouds,. . . Those are events likely to have been

seen by the photographer, and so, photographed on purpose. The distribution of expla-

nations is plotted in Figure 24, and we can see what the main contributions to IFO cases

are: Aerospatial ( 33%) is the highest contribution, followed by hoaxes ( 22%) and

technology related explanations ( 13%).But, we can define a broader classification of IFOs as follows:

1. Hoax: This is the same category as the previous Hoax category (512 cases, 22.9%).

2. Accidental image: Composed of the previous categories 2 and 6 (camera and film

related; and biological). This category joins cases of UFOs most likely not seen at

the moment of taking the images: film flaws, flares, flying-by birds or bugs. . . These

cases are not covered by the model developed in this work (500 cases, 22.3%)


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Figure 24: FOTOCAT IFO explanations. Left: Original defined categories. Right: Re-classification

of categories

3. On-Purpose image: Composed of the previous categories 3 (Aerospatial), 4 (Me-

teorological and geophysical) and 5 (Astronomical). This category joins UFOs that

were likely to be seen, and hence photographed on purpose. They also are cases

covered by the model developed in this work (1227 cases, 54 .8%)

Most cases from category 7 (Miscellaneous) have been re-classified into Accidental

and On-Purpose new categories. With this division, we can describe the total time

distribution as the weighted addition of these categories (Sec. 2.4.1):

PIF O(h) = FHoax PHoax(h) + FAcc PAcc(h) + FOnP POnP(h) (19)

with FHoax = 0.229, FAcc = 0.223, and FOnP = 0.548. On-Purpose images explain

about half of the cases, and Hoaxes and Accidental Images take care of the other half.

It is, with no doubt, a strong contribution to the total, and since they do not follow our

model assumptions, they produce significant deviations in the shape of the total time

distribution.

Let us look at the individual time distributions of each of these categories in Figure 25-

Left. As expected, Hoaxes and Accidental images do not follow the usual LoT. However,

the On-Purpose images curve resembles much more the usual shape. Compare it with

both the Total FOTOCAT IFO and CUCO curves in Figure 25-Right. The slope in the

rise of the main peak is more similar to CUCO, but also notice that the secondary peak

in On-Purpose images ditribution seems to be absent.


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Figure 25: Left: Law of Times for each IFO category. Right: Total IFO cases, On-Purpose cases, and

CUCO.

Hoaxes and Accidental images contribution to the total is higher during the day. If we

think on how somebody would create a hoax or fake, it is reasonable to think that one

would make a photograph or video clearly showing a fake object, and clear images are

easily taken during daylight.

For Accidental images, the probability of having flares, development flaws, taking

blurred objects, etc. . . is directly related to the number of images taken. More pho-tographs means more oportunities to have a flaw or artifact. It is just obvious that more

photographs are taken during daylight than during the night because of available light

but also because more people is awake. In that sense, it is important to see that the

three categories have a decrease in the percentage of cases at about the same time at

night, clearly showing when people go to bed.

Our main focus is the On-Purpose images distribution, and how it compares to CUCO

the whole catalog: Spain, Portugal and their islands. The number of cases are 1227

vs 5135. The most remarkable difference is the shift of FOTOCAT main peak to earlier

hours. In order to explain this shift, we check the geographical distribution of the entries

in Figure 26. Left image shows the distribution using the natural longitude, which helps

us to find hot spots. The hottest spot is in Norway, due to Hessdalen Valley entries (a

total of 102 entries, 8.3%). Other hot spots can be seen in Chile and Central Europe.

After translation to Z, the new distribution compresses all countries to a region

between 10o E and 25o W. The mean position is marked with a red cross, at = 40.33o

N, Z = 7.061o W. CUCO Z distribution was shown previously in Figure 23, and its


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mean position is = 40.35o N, Z = 17.98o W. The mean Z of FOTOCAT is 10.9

o to

the east of CUCO, which means a time difference of 43 minutes (earlier for FOTOCAT).

However, the shift in Figure 25-Right is longer than those 43 minutes. Also, the high

peak at 19 h looks like an anomaly.

Therefore, there are still some issues with the On-Purpose images data that must be

solved before trying to compare with CUCO. Looking at the data, we see that FOTOCAT

has an important number of duplicated cases. If for a single event, people take multiple

images, FOTOCAT is counting these as independent entries. Not correcting these du-

plications leads to artifacts, an also may mislead statistical analyses[10],[19]. Multiple

sigthings for a single case are introducing a bias in the data, breaking the randomness

that was one of the starting points to develop the model.

In the case of FOTOCAT, for a total of 308 multiple entries, some of the most important

contributions are:

March 5th, 1979. Poseidon Missile, seen from Canary Islands (Spain). 27 entriesbetween 18:50 to 20:07.

February 28th, 1963. Rocket launch, seen from Arizona (USA). 27 entries at

18:40.

August 24th, 1990. Flares from military exercise, seen from Germany. 11 entriesat 22:30.

June 12th, 1974. Missile launch, seen from eastern Spain and France. 11 entriesbetween 21:00 and 21:25.

September 20th, 1977. Rocket Launch of Kosmos 955, seen from Russia andFinland. 11 entries at 4:00.

June 14th, 1980. Rocket booster, 4th stage of Kosmos 1188 ignition, seen fromArgentina and Venezuela. 10 entries between 18:05 (Venezuela) and 19:05 (Ar-

gentina).

September 17th, 1985. Balloon, seen from Argentina. 10 entries between 10:00and 10:45.

March 13th, 1997. Flare dropped over test range, seen from Arizona (USA). 9entries at 22:00.


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Figure 26: Left: FOTOCAT geographical distribution of On-Purpose Images. Right: Z distribution

of FOTOCAT On-Purpose Images.


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Figure 27: Contribution of duplicated cases to the total distribution (Blue dashed line), and distributionafter removal of duplications (Red dashed line).

These duplicated (or multiplied) entries have been removed, leaving only one for each

sighting. However, sometimes it is not clear how this removal should be done. The

main criterion is that entries with the same date, same explanation, similar hour and

similar region are considered as duplicated cases. But when entries cover several hours,

which one should be kept in the data? Furthermore, when the same sighting is made

from different countries, regions, and even different time zones, which ones should beremoved?

With a lack of a clear criterion respect to some entries, we have removed duplicated

cases, and left only one of them. The result is shown in Figure 27. Rise time of the

main peak is slightly corrected to a later time, but it seems like there could be still

some more entries to be corrected in some way. A deeper study of the catalog, a more

strict reclassification, and a better criterion for duplicate cases may yield a different time

distribution that might improve the shape of the curve; but for the scope of this work,


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Figure 28: Left: Optimized curve for CUCO (red line). Right: Optimized curve for FOTOCAT

On-Purpose images (red line).

detecting these kind of issues, discussing how they could be corrected, and checking that

corrections seem to go in the right direction is just enough.

If we compare the optimized curves for FOTOCAT and CUCO (Figure 28), and look

at the values of the parameters in Table 4, we find that the different mean Z seems

one of reasons to explain the time difference in the rise of the main peak. We can also

observe that there is no need to include a secondary peak in FOTOCAT, something thatcould be important.

Data gives us a hint of the existence of a morning peak, which is reproduced by the

model, but it is not really clear whether it really exists, or it is just some random noise

in the graph. Another important difference is in the mean Event Magnitude (1.26 vs0.90), since for FOTOCAT it means that registered events are brighter than CUCOs

(Figure 29). We must not forget that in FOTOCAT images are taken by cameras and

other imaging devices, and hence a technological factor is always present. Sensibility of

devices are not the same as that of the naked eyes. And also, the Limiting Magnitude

can be different for different cameras or devices.

Nowadays, it is easy and quick to take photos or videos at any time. But, was it so

easy to have a camera ready at any time in the 60s, 70s, or 80s? Technology evolves in

time. From old manual cameras to compact cameras, they have been easier to use with

time. Film sensibility, development processes, and optics have also improved. And then,

digital technology made film cameras become obsolete. Could this evolution have any

influence the time distribution in any way? Furthermore, as an indirect but interesting


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Figure 29: Event Magnitude distribution for CUCO and FOTOCAT.

example of technology influence, we can think about how digital image edition makes

extremely easy to create hoaxes at any time. Will, in a few decades, the time distribution

of future photographic catalogs be different from FOTOCAT?

3.6 An attempt to identify the secondary peak

One of the interesting results of FOTOCAT is the lack of a secondary peak. From thevery beginning, the secondary peak is outside the scope of the model. It is an ad hoc

addition in order to be able to reproduce the catalogs we have used until now. In last

section, we used only the IFO entries of FOTOCAT to understand the structure of its

Law of Times, and finally, find the appropiate subset of data to compare with CUCO.

In the process, we identified the extra factors (hoaxes and accidental images) needed to

completely describe FOTOCAT IFO cases, and their time distribution.

We are going to proceed in the same way with CUCO, to try to find the origin of the


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Fixed Parameters CUCO FOTOCAT


Mean Longitude Z Z 17.98o W 7.06o W

UTC Offset +0 +0


Wake-Up time ha 8 : 30 8 : 58

Wake-Up deviation a 3.76 6.72

Go-to-Bed time hb 23 : 59 23 : 01

Go-to-Bed deviation b 0.96 1.09


Mean Event Magnitude 0.90 1.26Event Magnitude deviation 3.70 3.40

Secondary Peak (Ps)

Percent of cases Fs 12.1% 0%

Mean hour hs 2 : 09 -

Deviation s 2.55 -

Table 4: Parameters for the curve optimizations of Figure 28

secondary peak. IFO cases are 1547, and we have defined 5 different categories for the

explanations available in the catalog:

1. Misperception: Planes, stars, planets, satellites, balloons, lights,. . . (1016 cases,

65.7%)

2. Hoaxes: Hoaxes, fakes, jokes, rumours, low reliability cases (227 cases, 14.7%)

3. Psicological: Hallucinations, psicological causes (47 cases, 3.04%)

4. Technological: Lens flares, film development flaws, radar echoes (36 cases, 2.33%)

5. Other: Marked as explained in the catalog, but the explanation is not registered

(221 cases, 14.3%)

We can see in Figure 30 the time distribution of each of the categories. Misperception

is the main contribution, and clearly shows the typical LoT shape. So does the Other


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Figure 30: Left: Distribution of explanations. Right: Contribution of each category to the total.

category. These cases are just marked as explained, but no explanation is given in the

catalog. However, the curve also has the typical shape. These two categories together

represent 80% of cases, and hence, the overal shape of the total IFO distribution is then

highly determined by these categories.

The Technological category has a very low count of cases (36). It is composed of images

(33) and radar echoes (3). It resembles FOTOCAT Accidental images category, since

photographs are mostly taken during daylight. In fact, some the these entries are alsoin FOTOCAT, and classified as Accidental images.

Psychological category has only 46 cases. They seem to happen during night time.

Even if these causes are related to the mind, one may wonder if there is a trigger, an

external stimulus that causes an hallucination, and hence, in the first stage it would also

be a misperception case. In any case, the weight of this factor is only a 3%, so it does

not have much influence in the total time distribution.

Unlike the technological category, the distribution of Hoax category is totally different

to FOTOCAT Hoax category. We can see here an indirect but clear influence of technol-ogy. Even if the goal of a hoax is basically the same (either fooling somebody, getting

attention, or anything else), technology inluences how the goal is achieved: if an image

has to show clearly of a flying object (so daylight is the best time to take the image),

trying to fool or confuse somebody is easier done at night. Therefore, Hoax distribution

is concentrated at night hours. The count is not high (227 cases, 14.7%), so even if a

rather high peak at 2 a.m. is shown, the graph has to be taken with care as it could

be only noise, and the real shape of hoaxes might be a broad gaussian peak covering all


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Figure 31: Time distribution for HOAX, MISP categories, and Total IFO.

night hours (Figure 31).

It is difficult to draw any clear and definitive conclusion about the secondary peak

origin. It is not present in FOTOCAT, so it seems that a social factor, like a different

awake population distribution for night hours, is not involved. On the other hand,

technology is somehow influencing it.

Data suggest that hoaxes and fakes may have some contribution, but on the other hand,

they also contribute to the main peak. It seems like the hoax distribution is mimicking

the misperception distribution (Figure 31). It makes sense if we think that it is easier to

confuse people on purpose at night, in the same way people get confused by real common

lights. On the other hand, fake reports, when no real event took place, could have some

other time distribution. We have also considered cases with low confidence, and rumours.

Could some cases identified (or classified) as hoaxes be in fact misperception cases? Could

some misperception cases be hoaxes after all? A revision of IFO cases to confirm the


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Figure 32: Left:Poher and Vallee intepretation of the Law of Times.[20] Right: Equivalent interpre-

tation of the Law of Times according to our model.

misperception or hoax nature of the cases, as well as larger IFO catalogs would be useful

to find the real shape of hoaxes time distribution and find an answer on their effect on

the secondary peak.

4 Interpretations of the Law of Times

4.1 Poher and Vallee

The Law of Times was first found in the 60s, using Close Encounters statistics. It is

only natural that interpretations for the time distribution were put forward to explain

the nature of the graph, and its relation with UFO phenomena.

Poher and Vallee [20] assumption was that UFO landings happened at night following

a gaussian distribution. The decrease of landing reports was because of people staying

at home at night, not being able to have a close encounter with UFOs. That meant that

there was a significative amount of unreported landings. Figure 32 shows this ideal UFO

activity, and the expected deficit of reports. It was calculated that the rate of potential

encounters to the actual number of reports could be of 14 to 1.

This interpretation means that UFO activity is independent of day/night cycle, and

independent of the position of the sun. It just assumed a specific fixed dependence with

the time. But this interpretation cannot reproduce the behaviour detected by Gregor

and Tickx[16] (Figure 7), nor explain graphs such as Figure 33, where we plotted the


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Figure 33: Law of Times for December cases in Californa. Data taken from NUFORC[17]

Law of Times for December cases in California. As December has the longest nights in

the year2, there is a flat zone in the time distribution, where visibility is at its highest

value, while people is still up for some time before going to bed.

On the other hand, Poher and Vallee, took into account a social factor similar to our

Witnessing Probability. They studied sociological statistics to find when people was out

of home, and found the graph reproduced in Figure 34.

The graph represents the percentage of working population not at home. This was

a reasonable qualitative approximation to the problem. We can compare it with the

optimizations found for CUCO and ALLCAT (section 3.3) of the Witnessing Probability.

We know that, whatever the most accurate interpretation is for Pw, mathematically it

reproduces the time distribution for the main peak.

We can interpret our model in similar terms to Poher and Vallee (Figure 32-Right),

2In Northern hemisphere


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Figure 34: Black line: Percent of working population not at home (Ref. [6]). Red and blue lines:optimized Pw for CUCO and ALLCAT

thinking in what the potential number of events would be if people did not go to bed.

Our assumption is that events have a constant activity during the day. During daylight,

there are an important number of them that go unseen, since they are not bright enough.

At night, most of them are visible. Unlike Poher and Vallee, we have not made any

assumption on the nature of the events, meaning that some of them could be just ordinary

events, and reports the consecuence of misperceptions. This also means that peoplecan sometimes identify what they are witnessing, and hence do not report it. Finally,

reported events are all those people fail to identify for any reason (fail to identify a

natural phenomenon, or actually witnessing a strange phenomenon). In any case, the

time distribution appears as a consecuence of observational factors, and is not a feature

related to events.


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Figure 35: Histogram for |

| parameter.

4.2 Process Theory

Another interpretation was given by Miguel Guasp, after proposing a UFO Process The-

ory[21]. This theory was developed with the intent to provide a tool to work on catalogs

and extract useful data about UFOs. The initial assumption was that real objects com-

ing from outer space arrived to a specific point on earth. But, during the entry into the

atmosphere, their trajectory changed depending on mission-specific variables, , and

, to reach their final destination. Applying this theory to the 1968 wave in Spain (a

small catalog of 29 cases), it was found a | | distribution resembling the Law of Times(Figure 35), and therefore, this parameter was describing some aspect of those objects.

Process Theory asummed that objects travelled to earth using the shortest trajectory.

Since there are more points outside earths orbit than inside, objects would be more likely

to arrive at earth to the dark side, i.e. at night. The sun would be blocking the trajectory

of objects (Figure 36), and to avoid it, some objects may make a change in trajectory


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Figure 36: Origin of the Law of Times, according to Process Theory.

that would take them to the dark side. UFO activity would be again a gaussian shaped

curve, centered about midnight. The apparition of the secondary peak would be caused

by the relationship with the parameter distribution.

This interpretation relates UFO activity to the position of the sun. Thus, if we modeled

it mathematically, it is expected to find some kind of variations with latitude, longitude,

and also with the day of the year, most likely in the same sense than our model. But, it is

questionable what the shape of that ideal UFO activity would actually be. Given a point

on earth, the sky overhead covers a solid angle of 360o in azimut, and 90o in altitude.

But if we think in objects coming from whithin our solar system, we can restrict possible

origin points to the ecliptic plane (i.e., from a point of view on earths surface, an arc in

the sky of 180o). Only when the sun is up, there would be a decrease in possible origin

points. The apparent size of sun is 0.5o, only 0.2% of the arc in the sky. Such a reduction

seems too small to produce a gaussian, or a significative difference between night and


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day, even if we assume that objects deviate to avoid the sun and arrrive to the dark side.

We can think on increasing the avoid zone, due to gravitational issues. This would

make a larger difference between night and day.

Whether the gaussian shape is correct or not, the proposed relationship between

and the LoT is at first sight a striking result. As we said, the starting point for Process

Theory was the asumption of an origin or departure point for the objects, such as Mars

or any other planet. Objects, following the shortest path to earth (and discarding travel

time) would ideally arrive to a point P on earth, that was calculated from the celestial

coordinates of the departure point, and the time of the sighting. However, UFOs arrived

at points A. Thus, the , and parameters were introduced to account for deviations

during the entry in the atmosphere (Figure 37).

Given a point P with coordinates (p, p) for latitude and longitude, and point A with

coordinates (A ,A), was calculated as:

tan =

p Ap A

(20)

The intention of defining these parameters was to find some quantity directly related

to UFO behaviour, and thus, being able to characterize them. This procedure was usedon a dataset of 29 sightings in Spain (23 of them in August, September and October),

with the result shown above. However, if we really think about the math involved, we

will realize that this parameter is not characteristic of any object. accounts for the

relative position between P and A points. P is describing the position of the departure

point, and was treated as a new origin of coordinates to which A coordinates were related.

If we fix the departure point (let us think about Mars as departure point), P as will move

westwards as earth rotates3, since it describes the location at which the Mars is at its

zenith at the hour of sighting.

P will move over earth surface at an angular speed of 15o/h to the west. If we calculate

for every hour, knowing that p in eq. 20 is calculated as

p(h) = 15 h (21)

3Process Theory considered P as a new origin of coordinates, and hence, A was moving Eastwards as

the day advanced.


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Figure 37: Definition of atmospheric entry parameters, according to Process Theory. Black lines

represent equator and 0o meridian, as the coordinate system to calculate and

We get Figure 38, that can be compared to the from the 1968 wave. All (h, ) values

are inside a region limited by the two extreme positions of Mars for 1968.

Since and time are directly related through the rotation of earth, any distribution in

one of the variables will be reflected in the other. We can make a qualitative comparisonwith CUCO: the main peak is aligned with a higher density of the (h, ) points, while a

lower density corresponds with the low percentage of cases during daylight. At this point,

Process Theory questioned causality: Was the time distribution causing the distribution

of ? Or was the Law of Times the effect of distribution? The latter implies that is

a variable describing the objects behaviour.

But, from the point of view of our model, the Law of Times can be explained by


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Figure 38: Parameter

as an effect of earth rotation. Solid lines represent the limit values for

in1968. Most of the points are aligned with the line for August (dashed), since 10 out of 29 cases were in

that month.

visibility and witnessing probability factors. distribution is then a consequence of the

time distribution, describing a geographical relation with some arbitrary-chosen celestial

coordinates, but not related to any objects coming from outer space at all.

5 Conclusions

To the authors best knowledge, this is the first time a quantitative model has been

developed to explain the Law of Times. The model can succesfully reproduce the main

peak at 21-22 h, and predicts the existence of a morning peak.

The model was derived using only two factors: an geographic-astronomical factor, and

a social factor. The first one accounts for the conditions of sightings (i.e., the light con-

ditions are determined by the position of the sun respect to the location of the sighting).


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The second one accounts for the availability of anybody to see it. These two factors are

enough to describe time distributions. We made simple and reasonable assumptions, also

thinking of using variables that could be verified independently. Geographical variables

can be fixed looking at the distribution of reports. Sociological variables can be validated

using sociological studies. Only Event Magnitude variables seem difficult to validate at

the moment.

We used the model to understand the differences among a few catalogs. We identified

some issues to take into account, and also proposed some solutions for them: geographical

distributions and their influence, time zones and Z translation, duplication of reports,

isolation of factors. . .

One key point of the model is randomness. Events appear randomly in a constant

uniform distribution, and are witnessed randomly. One could think in a reporting factor,

an attention factor, an identification factor, or any other factor that may change the

count of seen events. Those factors are important of one is interested in the absolute

number of events. But we have focused our atention in relative numbers, the distribution

of cases in 24 hours, and the results indirectly show us that all of those factors are

independent of the time of day, and the day of year, making them constants. As such

constants, after the division in eq. 5 they dissapear from the expression, and there is noneed to care about them. Even if any of these factors had to be finally taken into account,

they will represent second-order corrections, since the main features are succesfully

explained.

But, if someone, after looking at the time distribution is biased to look at the skies at

a specific hour in search for UFO events, he will most likely start reporting more events

than predicted by the model at that hour. This person would have, in fact, broken the

randomness, and an extra factor would be needed to describe the new time distribution.

The same happens if events cannot be explained in terms of the factors present in thismodel: deviations from the model will appear that will have to be identified. We had a

clear example with FOTOCAT IFO cases analysis, strongly influenced by a technological

factor.

It is important to think on the starting point for the derivation of the model. Poher and

Vallee, and Process Theory, assumed an ideal UFO activity to explain the origin of the

LoT. We also did it, but assuming a totally different ideal distribution of events. Another

important difference is that the other theories began assuming a specific nature of the


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UFOs. We did not. Events may be either natural or strange phenomena. The model only

adresses why people see things at the times they see them, but does not address

their nature; whether or not the sighting becomes an UFO report is not dependent on

any strangeness factor, or any other that may indicate its natural or unknown origin.

That origin can only be known after a study of the individual reports.

However, some thoughts can be derived about the nature of events within our model.

An event may be anything: lights, stars, planets, reflections on balloons or clouds,. . . the

only way those can be reported as UFO sightings is by misperception. The model is telling

us that these misperceptions are happening continuously, and their time of sighting is

solely related to the aforementioned factors. That does not rule out the sighting of any

strange or unusual phenomenon, but since UFO and IFO LoTs show basically the same

shape, it is just straightforward to think that the vast majority of UFO reports may also

be misperception cases which are still unexplained.

As we said above, other factors can also influence the time distribution, like photo-

graphic cameras or video recorders: lens flares, flying-by birds or bugs, dust, development

flaws... do not depend on astronomical factors, but on a technological one. Hoaxes is

another factor which is not dependent on astronomical or wake/sleep cycle, although it

can be considered as some sort of social factor and can also be indirectly influenced bytechnology. Strong contributions of these other factors to a catalog do appear in time

distributions as deviations from the model.

To deal with those deviations, we developed an interesting property of catalog addition.

Deviations from the model can be identified, isolated and studied independently, as we

did with FOTOCAT IFO cases. It is also the case of the secondary peak, whose origin

is still unknown. There is no a priori rational justification for its inclusion, but we tried

to isolate it in CUCO studying the IFO cases.

In the same way, if a real UFO activity is taking place, as those proposed in the past,it should appear in the time distribution. A difficulty for this is if that UFO activity is

so low that is totally hidden by unexplained, but natural-origin reports. But, as UFO

cases are solved and become IFO cases, and if a UFO activity really exists, it has to pop

up at some time.

As a last comment, we can have a look at a catalog of 205 pre-1880 cases, compiled by

Vallee and Aubeck[5], in Figure 39. It is interesting to see that the same basic distribution

appears, despite of the noise. We can clearly identify the main peak, and what seems


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Figure 39: Law of Times for 205 pre-1880 cases, extracted from [5]. Secondary peak is missing.

to be the morning peak, but no secondary peak. It shows that visibility and witnessing

factors can also be applied to past cases.

But for the secondary peak, it becomes more puzzling. It did not appear either in

FOTOCAT, but we can see it in all the other catalogs shown in this work, which are

(mostly) post-Arnold sighting. It is also present in CUCO IFO cases distribution, sug-

gesting that it has an explainable origin, and some kind of connection with hoaxes was

suggested by the analysis of data.

In we stick to the idea of some connection to hoaxes, it makes sense that technology

influences how they are made, and the different hoax distribution in FOTOCAT affects

the secondary peak apparition. For pre-1880 cases from Vallee and Aubeck, we can

think that communications were radically different than todays. After Kenneth Arnolds

sighting, an interest in UFO phenomena emerged, also fed by mass media. News spread

quickly over the world. That may have inspired people to create hoaxes and jokes. But


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before that time, news were far more slow to spread. It was much more difficult to create

a global interest in such a phenomenon, and hoaxes would be much less frequent.

A study of the evolution in time of the secondary peak could be a start point; and

then try to relate it with the rate of hoaxes in time.

6 Acknowledgements

I would like to thank Vicente-Juan Ballester Olmos for getting me interested in the Law of

Times. Also for the interesting discussions, suggestions and recommended bibliography.

I also want to thank Vicente-Juan and Juan Pablo Gonzalez for FOTOCAT, ALLCAT

and CUCO catalogs, that have been extensively used in this work. Thanks are also for

Miguel Guasp, and Jacques Vallee for valuable comments.

Finally, thanks to Richard Heiden and Tom Strong for all the corrections and comments

to this text. Mistakes still present in this work are only my fault.


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A Catalog addition

Given a catalog with a total of N entries, the Law of Time is constructed as :

P(h) =N(h)

N

Suppose we have n different catalogs, each with Ni (i=1. . . n) entries, and individual

time distributions Pi(h). If we join all of them in a single catalog, we will obtain a new

catalog with NT =

ni=1 Ni entries. The new time distribution will be constructed as:

P(h) =N1(h) + . . . + Nn(h)

NT=

N1(h)

NT+ . . . +

Nn(h)

NT

It is easy to operate to arrive at:

P(h) =N1(h)

N1

N1NT

+ . . . +

Nn(h)

Nn

NnNT

The joint time distribution is just a weighted summation of the individual time distri-

butions:

P(h) = P1(h)

N1NT

+ . . . + Pn(h)

NnNT

and the weight is nothing else but the proportional contribution to the total catalog.


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B The secondary peak

The LoT Model is based only in an astronomical factor, and a social factor. Those factors

do not explain the secondary peak that appears in most of the time plots around 2-3 am.

Sometimes, the peak is clearly visible. Some others, it reveals itself as a change in the

slope of the main peak. Even if its origin is unknown, it is possible to add a gaussian

peak to the model to reproduce it.

Given a catalog, at a specific hour there are Nw(h) witnessed events that make up the

main peak of the LoT. Let us add the number of secondary events, Ns(h), of unknown

origin, that would make up the secondary peak. The total number of reported cases athour h is expressed as :

N(h) = Nw(h) + Ns(h)

being N(h) the number of total cases at hour h. The total number of catalog cases

is given by the summation of N(h) over all hours, and will have two contributions: one

from the main peak and another from the seconday peak.

N =h

Nw(h) +h

Ns(h)

We construct the LoT of the catalog, P(h) , as:

P(h) =N(h)

N=

Nw(h)h Nw(h) +

h Ns(h)

+Ns(h)

h Nw(h) +

h Ns(h)

The second term is the one giving origin to the secondary peak time distribution,

Ps(h), that could be expressed as a gaussian function centered in hs, with a standard

deviation s, and area Fs, being the fraction of cases that make up the secondary peakrespect the total of cases.

Ps(h) =Ns(h)

N Fs

s

/2 exp

2(h hs)

2

2s

Operating now on the first term, we need to re-order the expression:

Nw(h)

h Nw(h) +

h Ns(h)

=Nw(h)

h Nw(h)

1 +

h Ns(h)

h Nw(h)

1


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We have recovered the original expression of the LoT Model without secondary peak,

which is modified by the value given inside the square brackets. We need to express this

last part in terms of already known or defined values:

1 +

h Ns(h)h Nw(h)

1=

h Nw(h)

N= 1

h Ns(h)

N

as

Nw(h) = N

h Ns(h). This last summation is nothing else but

h Ps(h),

whose value, thanks to the previous definitions, is just the fraction of cases that make

up the secondary peak respect to the total, i.e. Fs.

Finally, the time distribution with a secondary peak can be written using the original

expression of the LoT Model that generates the main peak:

P(h) = Pmain(h) (1 Fs) + Ps(h)

P(h) =

Pw(h)

d Pv(h, d)

h Pw(h)

d Pv(h, d)

(1 Fs) + Fs

s

/2 exp

2(h hs)

2

2s

(22)

Three parameters have been added to describe the secondary peak:

Fs, fraction of cases making up the secondary peak respect the total number ofcases.

hs, mean hour of the secondar peak.

s, standard deviation.


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References

[1] Jaques Vallee and Janine Vallee. Les Phenomenes Insolites de lEspace.Table Ronde,

1966. In english, Jacques Vallee and Janine Vallee. Challenge to Science. The UFO

Enigma. Neviller Spearman (London), 1966, pg. 151-154

[2] Jaques Vallee. The pattern behind the UFO landings. Flying Saucer Review, Special

issue #1 The Humanoids. Oct.-Nov. 1966, pg 26.

[3] V.J. Ballester Olmos and Jacques Vallee. Type I Phenomena in Spain and Portugal.

Flying Saucer Review, Special Issue N.4 UFOs in Two Worlds, August 1971, Pg.61.

[4] M. Swords, The LAW of the TIMES: is there a Close Encounters Pattern?, 2010.

http://thebiggeststudy.blogspot.com.es/2010/02/

law-of-times-is-there-close-encounters.html

[5] Jaques Vallee and Chris Aubeck. Wonders in the Sky. Jeremy P. Tarcher/Penguin,

New York, 2010.

[6] J. Allen Hynek and Jaques Vallee. Edge of reality, Henry Regnery (Chicago), pg. 7.

[7] FOTOCAT catalog . Worldwide UFO Photograph and footage catalog compiled by

V.J. Ballester Olmos. http://fotocat.blogspot.com

[8] ALLCAT catalog . UFO Landings in Spain and Portugal. Compiled in 2013 by V.J.

Ballester Olmos (unpublished)

[9] CUCO Catalog . Catalogo Unificado de Casustica OVNI (Unified Catalog of UFO

Cases) of all-kind UFO cases in Spain, Portugal and Andorra. Compiled by J.P.

Gonzalez and supported by Fundacion Anomala, with the data from 23 differentcatalogs.

[10] Peter A. Sturrock, Time Series Analysis of a Catalog of UFO Events: Evidence of a

LocalSiderealTime Modulation, Journal of Scientific Exploration Vol.18:3 (2004).

http://www.scientificexploration.org/journal/jse_18_3_sturrock.pdf

[11] Vicente-Juan Ballester Olmos (2013). UFO Reports by Time of the Day.

http://tinyurl.com/UFO-Reports-by-Time-of-the-Day

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http://thebiggeststudy.blogspot.com.es/2010/02/law-of-times-is-there-close-encounters.htmlhttp://thebiggeststudy.blogspot.com.es/2010/02/law-of-times-is-there-close-encounters.htmlhttp://fotocat.blogspot.com/http://www.scientificexploration.org/journal/jse_18_3_sturrock.pdfhttp://tinyurl.com/UFO-Reports-by-Time-of-the-Dayhttp://tinyurl.com/UFO-Reports-by-Time-of-the-Dayhttp://www.scientificexploration.org/journal/jse_18_3_sturrock.pdfhttp://fotocat.blogspot.com/http://thebiggeststudy.blogspot.com.es/2010/02/law-of-times-is-there-close-encounters.htmlhttp://thebiggeststudy.blogspot.com.es/2010/02/law-of-times-is-there-close-encounters.html


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[12] V.J. Ballester Olmos and Miguel Guasp. Quantification of the Law of the Times

DATA-NET, June 1972, pg 2-8. In Spanish: Cuantizacion de la Ley Horaria Sten-

dek, IV, 14, Septiembre 1973, pg 7-11.

[13] Limiting Magnitude Calculations.

http://cleardarksky.com/others/BenSugerman/star.htm

[14] Limiting magnitude, Wikipedia.

http://en.wikipedia.org/wiki/Limiting_magnitude

[15] Twilight, Wikipedia. http://en.wikipedia.org/wiki/Twilight

[16] Eric Gregor and Henri Tickx.OVNI: un phenomene parasolaire? Inforespace, special

issue 4, 1980, pages 3-44

[17] National UFO Reporting Center. http://www.nuforc.org/

[18] http://www.timeanddate.org/

[19] Jaques Vallee. Are UFO Events related to Sidereal Time?. Arguments againts a

proposed correlation.

http://www.ufoskeptic.org/Vallee_LST.pdf

[20] Claude Poher and Jaques Vallee. Basic Patterns in UFO observations. AIAA 13th

Aerospace Sciences Meeting. (1975)

[21] Miguel Guasp. Teora de procesos de los OVNI, 1973
http://cleardarksky.com/others/BenSugerman/star.htmhttp://en.wikipedia.org/wiki/Limiting_magnitudehttp://en.wikipedia.org/wiki/Twilighthttp://www.nuforc.org/http://www.timeanddate.org/http://www.ufoskeptic.org/Vallee_LST.pdfhttp://www.ufoskeptic.org/Vallee_LST.pdfhttp://www.timeanddate.org/http://www.nuforc.org/http://en.wikipedia.org/wiki/Twilighthttp://en.wikipedia.org/wiki/Limiting_magnitudehttp://cleardarksky.com/others/BenSugerman/star.htm

modeling the law of times

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