Ian J Deary
email: i.deary@ed.ac.uk

Wendy Johnson

Causal perceptions drive analytic process -1 Intelligence and Education: Causal Perceptions Drive Analytic Processes and Therefore Conclusions

Keywords: intelligence, education, epidemiology, health education, intelligence, IQ, health, epidemiology

Causal perceptions drive analytic process -2

Summary

Background

Educational attainment is associated with many life outcomes, including income, occupation, and many health and lifestyle variables. Many researchers use it a control variable in epidemiological and other social scientific studies, often without specifying exactly what environmental effects or set of personal characteristics is being controlled. Other researchers assume that geneticallyinfluenced intelligence drives educational attainment, and think that intelligence is the appropriate control variable. Researchers' different and often unstated causal assumptions can lead to very different analytical approaches and thus to very different results and interpretations.

Methods, results, and conclusions

We document several examples of this important variation in the treatment of education and intelligence and their association. We recommend: greater clarity in stating underlying assumptions and developing analytical approaches; and greater objectivity in interpreting results.

We discuss implications for study designs. Causal perceptions drive analytic process -3

Brighter people tend to get more schooling, and the longer-schooled tend to be brighter. These simple facts elicit surprisingly different interpretations among the many epidemiologists and social scientists who measure education and intelligence for research use. Their different interpretations contribute to differences in methodological and analytical treatments that can have profound impacts on study design, methodological choice, results, and interpretation of results. Implicit interpretation of the association between these two variables is common throughout epidemiological and other social science research. With regard to health and other outcomes, this observationally ambiguous association involves the statistical issues of mediation, moderation, confounding, and direct and indirect effects. These issues are always troublesome because their treatment depends not only on timing of available measurements but understanding of causal pathways. The issues involved in the association between these particular two variables, however, are especially important to the newly emerging field of cognitive epidemiology. One or the other-especially education, due to its greater availability in datasets-is very commonly used as a control variable; intelligence and education are closely inter-related, and they may be measured with varying degrees of precision. Moreover, there is probably some form of longitudinal cascade between them, quite possibly with reciprocal causal and selection effects (1); yet, the optimal longitudinal data sequence to understand the processes involved in these reciprocal and selection effects is often unavailable. At the same time, because they are not perfectly correlated, neither education nor intelligence is a perfect proxy for the other. It is thus often important to understand objectively which (if either) exerts a causative effect on an outcome.

Does higher intelligence beget better educational outcomes?

In longitudinal studies that measure psychometric intelligence first and educational attainments later (thus assessing that causal chain), there is a moderate to strong correlation between the two, as assessed by years spent in full-time education, the highest qualification obtained by a person, or the scores obtained on educational assessments [START_REF] Jencks | Who gets ahead?: The determinants of economic success in America[END_REF]. For example, in a study of about 70,000 children in the United Kingdom, the general factor from the Cognitive Abilities Test (CAT) battery taken at age 11 correlated about 0.8 with the general factor of grades on the General Certificate of Secondary Education (GCSE) examinations taken at age 16 [START_REF] Deary | Intelligence and educational achievement[END_REF]. The general factor of the CAT test had very similar loadings from the three domains of verbal, non-verbal (abstract), and quantitative

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Causal perceptions drive analytic process -5 reasoning. Older studies have reported correlations ranging from 0.60 to 0.96 (e.g., 7, 8, 9). The conclusion from such studies might be that intelligence has stronger causal effects on educational results than vice versa. Does more education beget higher intelligence? Most studies of the influence of education on intelligence have not been longitudinal, but they have carefully examined the relation between length of schooling and intelligence, thus attempting to assess the reverse causal chain. Findings generally support the observation that more time in school does lead to greater intelligence. For example, Baltes and Reinert (10) compared the intelligence scores of three cross-sections of German 8-to 10-year-olds who were separated in age by 4-month intervals.

The intelligence tests used were assessments of induction, verbal comprehension, numerical facility and processing speed from the German Begabungstestsystem, which was based on Thurstone's theory and classification of Primary Mental Abilities. Since the German school system at the time required the entering children to be 6 years old by April 1, it was possible to compare the scores of children whose birthdays fell either just before or just after that dividing point, so that the children were effectively the same chronological ages but had a 1-year difference in schooling. Baltes and Reinert found that 8-year-olds who had received an extra year of schooling performed more like the least schooled 10-year-olds than the least schooled 8-yearolds. They noted, also, that the test most affected in this way-the Grundrechnen test of numerical facility-"is heavily loaded with material that is covered in the grade levels used".

Tests of more fluid skills were less affected, for example, the Buchstabenzaehlen test of letter counting which assessed processing speed, and which contained material much less based on taught materials. Schmidt [START_REF] Schmidt | Socio-economic status, schooling, intelligence, and scholastic progress in a community in which education is not yet compulsory[END_REF] reported analogous results from a South African community of East Indian immigrants who had varying exposure to school that was not dependent on ability. There, the correlations between schooling and two measures of nonverbal intelligence and one measure of verbal intelligence ranged from 0.49 to 0.68. The conclusion from such studies might be that education influences the development of intelligence. However, this requires the caveat that the so-called 'intelligence tests' should be scrutinised to examine the extent to which they contain materials that appear in the taught curriculum. So, it is possible that intelligence causes differences in educational outcomes, or that education causes intelligence differences, or a bit of both. Indeed, it is probably more complex than this. Readers can find further detailed consideration of possible non-linear effects of schooling on mental test scores, and the parts played by measurement error in intelligence and education measurement in a rather technical paper by Hansen et al. [START_REF] Hansen | The effect of schooling and ability on achievement test scores[END_REF]. In this paper, too, is the interesting idea of using a latent trait of 'ability' that might underlie both schooling and scores on achievement tests that are often used as indicators of intelligence.

Within epidemiology, educational attainment or performance in young adulthood is often implicitly or explicitly assumed to be an outgrowth of social position in childhood, reflecting causal familial environmental effects. For example, in a study modeling the effects of education and childhood and adult socioeconomic position (SEP) on midlife cognitive function, Singh-Manoux et al. ( 13) concluded (p. 577) that, "A major part of the effect of education on cognition… is also indirect. In these data, the influence of education on cognition is mostly through its influence on adult SEP…. The total impact of socioeconomic circumstances [both childhood and adult] on cognitive abilities… is substantial…. This merits the appropriate modeling of the impact of socioeconomic circumstances."

Ironically, despite their note regarding appropriate modeling, in their analysis they assumed rather than tested the directions of the 'causal' arrows linking education and cognitive function.

In their study, cognition was a latent trait comprising the following manifest variables: verbal

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Causal perceptions drive analytic process -7 memory, Alice Heim 4 Test Part 1 (verbal and numerical reasoning), Mill Hill vocabulary, phonemic fluency, and semantic fluency.Education was assessed by allocating subjects to five levels of attainment, from no formal education to higher university degree.

Different views about education and intelligence and their association in epidemiology

Intelligence and education are commonly used as possible causes and mediators of other outcomes. Epidemiologists, sociologists, psychologists, economists, social geographers and demographers include intelligence and education as possible influences upon a variety of human factors, including health and illnesses, late-life cognitive function, social mobility, and subsequent status attainment. Among such researchers there are striking differences in how the association between intelligence and education is viewed and treated analytically.

Examples assuming education is causal. Richards and Sacker ( 14) studied life-course contributions to scores on the National Adult Reading Test (NART) at age 53, a measure which they took to be an index of cognitive reserve, or peak cognitive ability in adulthood. The NART asks the subject to pronounce 50 words that do not follow the normal rules of graphemephoneme association or stress. Therefore, the subject has to have met the words previously to know how to pronounce them. Among the contributors to differences in NART scores at age 53 were cognitive ability at age 8 (the sum of four tests) and educational attainment by age 26 (a variable with five levels). A path analysis showed standardized path weights of 0.45 between cognition at 8 years and education by 26 years, and 0.24 between education by age 26 years and NART at age 53. Here is some verbatim discussion of these results by the authors,

The second path to midlife cognition, that via educational attainment, is easier to conceptualize, since there is clear evidence that schooling per se can lead to cognitive gains, even in late adolescence... Indeed, data from the 1946 birth cohort show that academic performance of the primary school (i.e., its record in sending pupils to selective

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Causal perceptions drive analytic process -8 secondary schools) was predictive of increased cognitive performance... Furthermore, it has been shown in the British 1958 birth cohort that the academic performance of the school is one of the major contributors to social class differences in childhood cognitive function... (p. 621).

The study results suggest at least two possibilities that are ignored in this interpretation.

First, there was a strong association between cognition at age 8 and education by age 26. As childhood cognition or IQ is subject to genetic influences, there may be genetically-influenced individual differences that contribute directly to educational attainment. Second, though this study measured both direct and indirect (via education) paths between cognition at age 8 and NART at age 53, it is likely that the tests given at age 8 did not fully capture all the cognitive variance that was present in the NART at age 53. This is because there are individual differences in the rates at which intelligence develops in children [START_REF] Moffitt | The natural history of change in intellectual performance: who changes, how much, is it meaningful?[END_REF][START_REF] Wilson | Synchronies in mental development: Epigenetic perspective[END_REF], and there is some evidence that the intelligences of brighter children may develop more slowly relative to their mature levels than those of the less bright [START_REF] Shaw | Intellectual ability and cortical development in children and adolescents[END_REF] relative to their mature levels. It is possible that education by age 26 acted as a surrogate marker for cognitive variance that was not picked up by the age 8 tests. That is, some of the differences in what is actually cognitive development from age 8 to age 53 might be indexed by education as a proxy measure. Moreover, the NART is a measure of intelligence that reflects accumulated knowledge to a particularly high degree. As knowledge is one of the clear benefits of education, performance on the NART may be especially subject to this effect. Thus, in two ways, the association between education and NART in the Richards and Sacker [START_REF] Richards | Lifetime antecedents of cognitive reserve[END_REF] study might be caused at least in part by inherent cognitive ability per se. Richards's and Sacker's discussion of education ignored these possibilities and considered only the possible environmental effects of the educational setting. Epidemiologists also routinely use education as an indicator of socioeconomic position in studies of health, or adjust for education as a possible confounding (causally prior) factor in studies of the determinants of health outcomes. For example, Singh-Manoux and Marmot [START_REF] Singh-Manoux | High blood pressure was associated with cognitive function in middle-age in the Whitehall II study[END_REF] followed this practice, concluding that,

The effect of blood pressure on cognition was stronger among women, and was stronger for some measures of cognitive ability than others… Confounding factors of age, educational level, occupational position, smoking, alcohol consumption, use of antihypertensive medication, diagnoses of diabetes, and cardiovascular disease were controlled in the analyses.... (p. 1312) Cognition and education were assessed as described above [START_REF] Singh-Manoux | Socioeconomic position across the lifecourse: how does it relate to cognitive fucntion in mid-life?[END_REF]. It is rare to see full discussion or even statement of the assumptions underlying these adjustments for educational level and occupational position. Sometimes, the assumption is that education represents childhood social position. For example, in an important study that compared education and adult occupational social class, Davey Smith et al. [START_REF] Smith | Education and occupational social class: which is the more important indicator of mortality risk?[END_REF] commented that, "The stronger association of education with death from cardiovascular causes than with other causes of death may reflect the function of education as an index of socioeconomic circumstances in early life, which appear to have a particular influence on the risk of cardiovascular disease" (p. 153). They noted, however, that this way of assessing early life socioeconomic circumstances was far from universally used. "In the UK, studies of socioeconomic differentials in mortality have generally relied upon occupational social class as the index of socioeconomic position, while in the US, measures based upon education have been widely used" (p. 153). Other times, it is at least recognised that education might lie between mental ability and health outcomes on a causal path. For example, Osler et al. [START_REF] Osler | Cognitive function in childhood and early adulthood and injuries in later life: the Metropolit 1953 male birth cohort[END_REF] stated, "The attenuation towards the null of this association [between prior IQ and later injuries] with adjustment for educational attainment suggests that an overall increase in educational level may result in a reduction in adult injury risk" (p. 7). Here, education is cast as

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Causal perceptions drive analytic process -10 the causal factor, though it might be acting as a surrogate for other aspects of the earliermeasured cognitive ability. Some economists, too, have examined education as a variable related to health, without considering the role of intelligence in the creation of educational variance. For example, the large study of the US censuses of 1960, 1970 and 1980 found that education was related to mortality.

People with less education had greater mortality rates [START_REF] Lleras-Muney | The relationship between education and adult mortality in the United States[END_REF]. The conclusion was that, "education has a causal impact on mortality" (p. 189), and that "we need to consider education policies more seriously as a means to increase health" (p. 215). Some possible mediating variables were mentioned, including stress, depression, and hostility, but the place of intelligence as a possible influence on educational outcomes was not mentioned. On the other hand, other economists have been nuanced in looking at the contributions of intelligence and education to health. In an analysis of the National Longitudinal Survey of Youth 1979, there was an interaction between them: "the causal effect of schooling on health is greatest for individuals with low cognitive ability" [START_REF] Auld | Schooling, cognitive ability and health[END_REF].

Examples assuming education is an outcome of intelligence.

A contrast to the treatment of education as causal is that by Herrnstein and Murray [START_REF] Herrnstein | The bell curve[END_REF] in The Bell Curve. They argued that education should not be statistically controlled at all in examining the association between adolescent cognitive ability and later life outcomes, because intelligence is a determinant of education. Their argument was that there is movement of people into higher levels of education based upon prior intelligence differences which are in part caused by genetic variation. This is actually consistent with current teachings of statistical practice in epidemiology [START_REF] Mcnamee | Confounding and confounders[END_REF], but both interpretations and statistical approaches rely on causal models of the processes involved that should be tested rather than assumed. 25) noted the potential implications of adjusting for education in evaluating the impact of childhood IQ on premature mortality in middle age. Rather than conceptualise education as an indicator of childhood social position-a confounding factor-they explained how it also might be a mediating factor or a proxy indicator of intelligence.

Higher IQ test scores may lead to educational success, and entry into well remunerated, high-status employment with a concomitantly high salary. An alternative, but often ignored, explanation is that educational attainment may represent a proxy for IQ, rather than the converse. That is, people with higher IQs stay longer within education, gaining more and higher qualifications. In this study, IQ at age 11 was moderately strongly correlated with subsequent educational attainment (r = 0.61; p = 0.001)… including education in our statistical models may be regarded as overadjustment (p. 243-244). Such a treatment, though perhaps also arbitrary, at least makes the alternative causal accounts explicit. The IQ tests in this study were two of the Moray House Test series, which mainly contain items requiring verbal reasoning, but not material that is explicitly taught in school. Education was assessed using qualifications, which were classified into six categories, from none to postgraduate qualifications.

We have now shown that intelligence and education are correlated, and given illustrations of how education is sometimes assumed to be causal in epidemiology without considering that it might be in part an outcome of intelligence, and might even share genetic as well as environmental influences with it. Next, we examine the extent to which this is found.

Intelligence and education: do they share genetic and environmental influences?

One way to resolve some of the confusion over the causes of the association between intelligence and education is to examine the transactions among the genetic and environmental influences contributing to them. As we have already noted, the presence of genetic influences on intelligence is well established. These influences increase from less than 50% of variance in Causal perceptions drive analytic process -12 childhood to around 70% in adulthood [START_REF] Bouchard | Familial studies of intelligence: A review[END_REF][START_REF] Deary | Genetics of intelligence[END_REF][START_REF] Plomin | Intelligence: genetics, genes, and genomics[END_REF]. The variance accounted for by shared environmental influences on intelligence declines from early childhood to a near-to-zero contribution in adulthood. Non-shared environment contributes a sizeable minority of the influence through most of life, though this term also contains error of measurement.

Multivariate variance decompositions can take this exploration further. They can estimate the environmental and genetic contributions to the correlation between two measured variables such as intelligence and education, and the extent to which the two variables share common genetic and environmental influences. For example, the national test of educational achievement used in the Netherlands at age 12 (the CITO test; 29) correlated between 0.41 and 0.63 with intelligence test scores gathered at ages 5, 7, 10 (using the Revised Amsterdamse Kinder Intelligentie Test) and 12 (using the Wechsler Intelligence Test for Children). The additive genetic contributions to variance in the CITO were about 60%, and genetic influences were the principal reason for the correlations between the intelligence test measures and the CITO. Similar results were obtained by Johnson, McGue, and Iacono [START_REF] Johnson | Disruptive behavior and school grades: genetic and environmental relations in 11-year-olds[END_REF][START_REF] Johnson | Genetic and environmental influences on academic achievement trajectories during adolescence[END_REF] in an adolescent sample, where a latent variable representation of school grades formed the measure of achievement and intelligence was measured using abbreviated Wechsler Scales (the children's scales for the under-16s, and the adult scales for those aged 16 and older). Almost 70% of the educational variables's variance could be attributed to genetic influence, and over 56% was common to genetic influences on intelligence. Even after other predictors of school grades including engagement, family risk, and disruptive behaviors were included in addition to intelligence, 34% of the genetic influences on school grades were shared with intelligence. In a Swedish twinbased study, intelligence was assessed at military conscription-using tests of reasoning, synonym detection, viusospatial perception, and mathematics/physics-and education was based Causal perceptions drive analytic process -13 on seven categories from less than nine years of education to doctoral studies. The genetic correlation between intelligence and education was greater than 0.5, and varied little (from .53 to .56) across the range of intelligence, and the shared environmental correlation between the two variables was 1.0 [START_REF] Johnson | Educational attainment: family background buys it in Minnesota but not in Sweden[END_REF]. This evidence of shared sources of influence is useful for epidemiologists to know and recognise in discussing results.

In fact, the causes of the association between intelligence and education might be more complex [START_REF]Genetic and environmental influences on behavior: capturing all the interplay[END_REF]. Analyses of educational attainment at age 24 in the USA, based on data from the the Minnesota Twin Family Study, showed that the genetic and environmental contributions to educational outcomes can differ at different levels of intelligence [START_REF] Johnson | Genetic and environmental transactions underlying educational attainment: something for everyone but everything for no one[END_REF]. The genetic variation in educational attainment increased four-fold from low intelligence (people two standard deviations below the mean intelligence level) to high intelligence (two standard deviations above the mean).

By contrast, the shared environmental variation increased more than ten-fold across the same range of intelligence. In simpler terms, this means that, in this particular geographical and temporal setting, one's rearing environment (including family resources, broadly conceived) was a much more important source of variance in educational outcomes at lower than at higher levels of intelligence, where genetic sources were much more important. A similar set of analyses was conducted in Sweden, with importantly similar and different results. At higher levels of intelligence, as was found in the Minnesota twin sample [START_REF] Johnson | Genetic and environmental transactions underlying educational attainment: something for everyone but everything for no one[END_REF], genetic variance in educational outcomes were greater than at low levels of intelligence. For shared environment variance, however, the two countries had opposing results: in Sweden, there was more shared environmental variance at higher than lower levels of intelligence [START_REF] Johnson | Educational attainment: family background buys it in Minnesota but not in Sweden[END_REF]. One should not forget, however, the genetic and shared environmental correlations between intelligence and educational attainments, which were strong in both locations. Causal perceptions drive analytic process -14 Genetic influences common to educational attainment and cognitive ability are also found among older people. In a sample aged between 50 and 70 years, a common genetic factor accounted for 40% or more of the variance in two measures of cognitive ability (Mini-Mental State Examination and Iowa Screening Battery for Mental Decline) and 21% of the variance in educational attainment [START_REF] Carmelli | Genetic mediation of the relationship of education to cognitive function in older people[END_REF]. The authors of this study concluded with the following comment, At present, clinicians are taught to discern cognitive loss when a diagnosis of dementia is considered, and final diagnostic criteria specify that a decline in ability must have occurred before a definite diagnosis of dementia is made. Because in most situations no data on premorbid level of function are available, the general practice is to use education and occupational attainment as substitute measures of premorbid levels of function. In this regard, education adjustment seems useful and necessary, and the present finding of a common genetic factor supports this practice (p. 52).

This provides a marked contrast to the quotation from Richards and Sacker [START_REF] Richards | Lifetime antecedents of cognitive reserve[END_REF], above. Whereas Richards and Sacker viewed education as an environmental contributor to peak cognitive ability, these researchers [START_REF] Carmelli | Genetic mediation of the relationship of education to cognitive function in older people[END_REF] viewed education as a proxy for peak prior cognitive ability precisely because it captured at least some of the genetic influences on intelligence. Importantly, although the statistical approach these researchers [START_REF] Carmelli | Genetic mediation of the relationship of education to cognitive function in older people[END_REF] recommended is the opposite of that recommended by Hernnstein and Murray [START_REF] Herrnstein | The bell curve[END_REF], their conceptions of the role of education in cognitive function are the same. This emphasizes that the appropriateness of statistical approaches are dependent not only on the accuracy of the causal conceptualizations underlying their use, but also on the specific timing of measurement of the variables involved.

Conclusions and recommendations

These examples illustrate the diversity of assumptions that underlie approaches to study design involving education and intelligence among epidemiologists and other health and social scientists. At the same time, they highlight the impact that such assumptions can have on study Causal perceptions drive analytic process -15 design, results, and interpretation of results. Because these assumptions are often unstated and unacknowledged, these examples also demonstrate that part of the difficulty in disentangling the possible causal associations linking these two variables can be traced to less-than-objective examination of all of the causal possibilities during study design and interpretation. Some of these difficulties can be remedied by greater attention to, awareness and statement of, underlying assumptions, and the consideration of reasonable alternatives by all researchers making use of education and intelligence and other closely related variables. This is important if we are to understand how cognitive function is involved in the development, maintenance, improvement, and deterioration of physical health.

We are far from being the first to state that one must be suspicious about inferences after statistical tests to assess confounding, or mediation. We concentrated narrowly on this matter with respect to how education and intelligence are treated in epidemiology because these closely related variables are critical to understanding the role of cognitive function in epidemiology. And we tried to argue that knowledge about causal background enhances analytical decisions and interpretations. This point is made well, in the context of birth defects epidemiology, and more generally, by Hernán et al. [START_REF] Hernán | Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology[END_REF].

In order to make the points above, we have not always gone into detail on how Causal perceptions drive analytic process -16 distinctions will be crucial for forming meaningful causal hypotheses about education and intelligence and how they combine to influence people's lives.

It is clear that not everyone derives the same benefit from any given educational opportunity and that the same educational opportunities are not available to everyone.

Distinguishing between the processes involved in education and intelligence is difficult because it requires measurement that can simultaneously establish causal attributions through precise timing and identify both genetic and environmental influences and their relations to the timing of measurement. The data necessary to do this with respect to education and intelligence are not often available. There are clear implications of the above points for study design.

First, the temporal cascade between intelligence and education will be clearer when repeated measures of each are available. This would allow longitudinal models to examine the direction and strengths of the mutual causal influences. Second, genetically informative designs-such as twin studies-can help to uncover the environmental and genetic aetiologies of the correlations between intelligence and education, and the other life outcomes with which both are associated. It will be especially interesting when specific genetic variants are found that are associated with intelligence differences, as these can also be examined to discover whether they are associated with educational differences. Third, it should be kept in mind that, even though intelligence and education are correlated, one can still act as a moderator of the other with respect to life outcomes, such as health [START_REF] Auld | Schooling, cognitive ability and health[END_REF]. Therefore, study designs powerful enough to include interactions between the two are desirable. Fourth, where it is possible to do so, multiple assessments of intelligence and educational outcomes at a single time point will alleviate the problems of measurement error through the construction of latent variables [START_REF] Deary | Intelligence and educational achievement[END_REF][START_REF] Hansen | The effect of schooling and ability on achievement test scores[END_REF]. Fifth, a mind that is kept open to the various plausible interpretations of analyses which involve education and Causal perceptions drive analytic process -17 intelligence helps, even when the above design strengths are not available. Sixth, we should not be blinkered by considering only intelligence and education. It should be kept in mind that there might be other variables that contribute to the association between intelligence and education.

Possible candidates could be personality traits and their influences on coping styles and motivations. Therefore researchers should consider measuring such constructs.

Key messages

We illustrate that the use of education and intelligence measurements comes with different views about their environmental and genetic origins, and the reasons for their being correlated We show how this influences approaches to analyses and the interpretations of results

We provide some information about the reasons for intelligence and education being correlated

We provide some suggestions for study designs and argue for researchers to consider all likely interpretations of results involving education and intelligence

  Causal perceptions drive analytic process -4Intelligence and education: clearly correlated, but what is the direction of causation?Intelligence and education have been studied together since the earliest empirical research on these topics. Spearman (2) found teachers' estimates of intelligence to be correlated with school exam results.Binet (3) developed what we now know as IQ tests to identify those children who would not benefit from normal education. When intelligence and educational outcomes-often assessed as years of full time education or as highest achieved qualification, but also by school grades or educational achievement test scores-are measured at about the same time, a typical correlation is about 0.5 (4). Like any other correlation, a cross-sectional correlation between intelligence and education demands an open mind with regard to causal interpretation. Perhaps more intelligent people gain access to more and higher-level education. Perhaps exposure to more education causes higher intelligence test scores. The problem is one that is basic to epidemiology: what is person and what is situation, what is genetic and what is environmental, and what is cause and what is effect? Influences may flow in both directions, and longitudinal studies can help to quantify their relative magnitudes.

  Causal perceptions drive analytic process -6

  Causal perceptions drive analytic process -9

  Causal perceptions drive analytic process -11Batty et al. (

  educational assessments can differ. What are referred to as 'educational outcomes' can refer to quite distinct empirical phenomena: e.g., years of schooling completed, highest credential obtained, subjective assessments of academic performance (e.g. class rank), and standardized tests of academic achievement in some content domain. These have different correlations with intelligence test scores, because all result from somewhat different personal traits and circumstances, and they are measured with varying degrees of accuracy. Making such

  Causal perceptions drive analytic process -23

Acknowledgements

Ian J. Deary and Wendy Johnson are members of the University of Edinburgh Centre for Cognitive Ageing and Cognitive Epidemiology, part of the cross council Lifelong Health and Wellbeing Initiative. Funding from the Biotechnology and Biological Sciences Research Council (BBSRC), Engineering and Physical Sciences Research Council (EPSRC), Economic and Social Research Council (ESRC) and Medical Research Council (MRC) is gratefully acknowledged (Grant #G00700704/84698). The authors have no conflict of interest.


József Vanyó
email: vanyoj@general.elte.hu

Bruno Escribano

Julyan H E Cartwright

Diego L González

Oreste Piro

Tamás Tél

A minimal dynamical model for tidal synchronization and orbit circularization

Keywords: tidal friction, locking, resonance, spin-orbit coupling

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model [7]; here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spinorbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.

to its bare bones with a minimal model that captures just the qualitative dynamical aspects of the problem.

To our knowledge there are only two approaches modelling tidal interaction as a fewbody problem. Hut [START_REF] Hut | Tidal evolution in close binary systems[END_REF] investigates a close binary system where the primary is modelled by three gravitationally interacting point masses (the secondary is point like). Equilibrium tides are described, and the effect of weak friction is taken into account by means of the introduction of a constant time lag. In the approach of Hurford and Greenberg [START_REF] Hurford | Tidal evolution by elongated primaries: Implications for the Ida/Dactyl system[END_REF] the primary is modelled as a sphere coupled to two point masses via damped springs which do not keep their straight shape but can become curved. The point-like secondary can move along a spherical orbit only and the primary is assumed to spin uniformly.

Although our model has a secondary consisting of two point masses only bound with a damped spring, no specific restrictions are applied. We can thus study dynamical tides, the nonuniform rotation of the secondary, resonances between rotation and orbital motion, and the effect of circularization. This simple model fundamentally differs from others where the secondary is described as a spatially extended object. The two-point-mass feature is nevertheless similar in spirit to restricting those models to the l = 2 normal mode, as done e.g. in Mardling [START_REF] Mardling | The role of chaos in the circularization of tidal capture binaries. I. The chaos boundary[END_REF], [START_REF] Mardling | The role of chaos in the circularization of tidal capture binaries. II. Long-time evolution[END_REF] and Witte et al [START_REF] Witte | Orbital evolution by dynamical tides in the solar type stars[END_REF]. It is perhaps due to this feature that we find chaotic dynamics as Mardling [START_REF] Mardling | The role of chaos in the circularization of tidal capture binaries. I. The chaos boundary[END_REF] did although, owing to the presence of dissipation, in our case chaos is unavoidably of a transient character. A somewhat related problem is that of tethered satellite systems [START_REF] Levin | Dynamic Analysis of Space Tether Missions[END_REF][START_REF] Sidorenko | A Spring-mass model of tethered satellite systems: properties of planar periodic motions[END_REF] but dissipation is typically ignored in this approach.

Our minimal model also enables us to investigate the nature of resonances in detail. We introduce a new definition of resonant states based on a resonance function which distinguishes resonances p : p and 1 : 1, and characterizes the different types of temporal behaviour. A necessary condition for the existence of a resonance is formulated: the energy of the secondary should be constant on average. The response of the dynamics of the secondary to the temporal change of orbital elements can be monitored.

The model is well-defined and easy to treat numerically in its original form [START_REF] Escribano | Dynamics of tidal synchronization and orbit circularization of celestial bodies[END_REF]. Nevertheless, we carry out a large distance Taylor expansion (l ≪ r) over the exact equations of motion. The approximate equations enable us to identify and isolate the different effects of tidal interactions. Special attention is paid to chaotic cases where the applicability of the expansion method is nontrivial. Our finding is, that the exact and approximate equations might lead to different resonances (quasi-attractors) owing to sensitivity to the initial conditions, but if the two solutions reach the same asymptotic state, even the second-order approximation provides a rather accurate description of the dynamics over long times. We can also obtain insight into apapsis precession and the temporal change of orbital elements via the approximate equations.

The paper is organized as follows. The model is introduced in Section 2. The succeeding Section is devoted to a discussion of resonant and chaotic states, and contains the definition of the novel resonance function. The energy condition for resonant states is also given there. The approximate equations of motion are derived, and their application to the energy condition is worked out in Section 4. The change of orbital elements and the comparison of the exact and approximate equations are the subjects of the following two sections. Finally, a discussion follows that also summarizes the results which are expected to be valid beyond the particular model. Technical details are relegated to the Appendix.

The model

We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring; see Fig. 1(a). The spring and damper or dashpot are in parallel in a Kelvin-Voigt configuration [START_REF] Munk | The Rotation of the Earth: A Geophysical Discussion[END_REF].

This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. We use polar coordinates r, β for the centre of mass of the secondary, with l as the instantaneous length of the spring and φ as the rotational angle characterizing the orientation of the secondary. Both angles β and φ are measured from the x-axis in an inertial reference frame. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected.

To describe the conservative part of the dynamics we construct a dimensionless Lagrangian, with the initial semi-major axis as the unit of length and the corresponding Keplerian period as the unit of time, in terms of the generalized coordinates q = (r, β , l, φ ):

L = ṙ2 2 + r 2 β 2 2 + l2 8 + l 2 φ 2 8 + 1 2r 1 + 1 2r 2 - ω 2 8 (l -l 0 ) 2 , (1)
where

r i = r 2 + l 2 /4 + (-1) i+1 rl cos (φ -β ) 1/2 , i = 1, 2
, is the distance of the ith component of the secondary to the primary and ω is the vibrational frequency of the spring. After including dissipation proportional to l the equations of motion are

r = r β 2 - 1 2 rR + - 1 4 lR -cos (φ -β ), (2)
β = -2 ṙ β r - 1 4 l r R -sin (φ -β ), (3)
l = l φ 2 -ω 2 (l -l 0 ) -2γ l - 1 2 lR + -rR -cos (φ -β ), (4) φ
= -2 l φ l + r l R -sin (φ -β ), (5)
where γ is the dissipation constant and the shorthand notation

R ± = 1 r 3 1 ± 1 r 3 2 (6)
has been used. The set of generalized coordinates can be split into two parts. The first subsystem is related to the orbital motion (r, β ) and the second subsystem describes the internal dynamics (rotation, vibration) of the secondary (l, φ ).

In order to understand the temporal evolution of the full system we can study the energy and angular momentum transfer between the degrees of freedom and subsystems. The angular momenta of the first and second subsystems are defined as

N 1 = r 2 β and N 2 = 1 4 l 2 φ , (7)
respectively. The angular momentum is transferred between the two subsystems but the sum is conserved as we only have central forces

(N 1 + N 2 = N = const.). Because l ≪ r, in practice N 1 ≈ N = const.
The total energy (E) is split into a centre-of-mass energy E c (energy of the first subsystem), a vibrational energy E vibr and a rotational energy E rot :

E c = 1 2 ṙ2 + (r β ) 2 - 1 r , (8)
E vibr = 1 8 l2 + ω 2 (l -l 0 ) 2 , (9)
E rot = 1 8 l 2 φ 2 - 1 2 1 r 1 + 1 r 2 + 1 r . ( 10
)
With the above equations we can monitor how the energy is transferred from the centre-ofmass orbit to vibration and rotation and how the total energy decreases through dissipation. We also define the energy of the secondary as

E sec = E vibr + E rot . (11)
The instantaneous angular momentum and the energy transfer are given by the time derivatives of these quantities ( Ė, Ṅ).

In order to follow the orbital elements in continuous time, we recall some formulae from the Keplerian problem. The instantaneous dimensionless semi-major axis and the eccentricity can be expressed as

a(t) = - 1 2E c (t) , ε(t) = 1 + 2E c (t)N 2 1 (t) = 1 - N 2 1 (t) a(t) . (12)
respectively. According to [START_REF] Landau | Course of Theoretical Physics -Mechanics[END_REF] the change of E c plays an important role in the change of the semi-major axis and eccentricity. The angle w between the vector pointing to the centre of mass of the secondary and the semi-major axis of the orbit can be obtained from the Keplerian form r = p/(1 -ε cos w) as

w(t) = arccos 1 -N 2 1 (t)/r(t) ε(t) (13)
because p = N 2 1 . So the angle of apapsis -see Fig. 1(b) -is

β apapsis (t) = β (t) -w(t). ( 14
)
In this article t i denote the instants when the secondary is at the apapsis (apapsis passages). If a quantity ( f ) is followed over such a long time that the change of the orbital elements is significant, we plot the quantity only at the instants t i . In such cases we use the notation f (t i ). To study the long term behaviour of f , we use the time average of the time derivative of f over a suitable time period. The choice of the suitable time period depends on the state of the secondary. Generally, it is one period (q = 1) but if the system is in a p : q resonance, the average is taken over q periods.

< ḟ > q = 1 t i+q -t i t i+q t i ḟ (t)dt. ( 15
)

Resonant and chaotic states

In dissipative cases (γ > 0) there is only one attractor, the 1:1 resonance, and a circular orbit [START_REF] Escribano | Dynamics of tidal synchronization and orbit circularization of celestial bodies[END_REF]. The approach towards the attractor typically occurs through a series of resonances.

Because of the change of orbital elements we introduce the following resonance functions

ρ(i) ≡ R(t i ) = 1 + 1 2π t i+1 t i α dt, i = 1, 2, 3 . . . . (16)
The integral term yields the overall rotational angle of the secondary between times t i and t i+1 relative to the r axis in units of 2π; see Fig. 1(a). The unit term express the fact that the radius has carried out approximately one rotation. Because of the small apapsis precession, R or ρ differ slightly from the entire rotational angle of the secondary (in units of 2π). The resonance function is thus a characteristic of the rotation after eliminating the apapsis precession. If the system is in a p : q resonance, function ρ is periodic in i, its period is q and

q ∑ i=1 ρ(i) = p. ( 17
)
The reason for the definition of the functions R is that they are more convenient to plot than ρ(i). In the presence of constant apapsis precession, R would be periodic. Because of the time dependence of the orbital elements in our problem, R is not strictly periodic but is almost so. In a p:q resonance, during q orbital periods the rotation of the secondary relative to the r axis is pq and the average of the relative rotation is (pq)/q. According to Eq. ( 16) the relative rotation during one period is R -1. Generally, the relative rotation is not exactly equal to the average of the relative rotation but almost so: R -1 ≈ (pq)/q. The average of R is therefore p/q and it typically takes on q different values around the average. To understand better the features of R, consider two examples. In a 3:2 resonance, during two orbital periods the relative rotation of the secondary is one. During one of the periods the relative rotation is a little smaller than 1/2 and during the other it is a little larger than 1/2. Thus in the first case R < 0.5 + 1 = 1.5 and in the second case R > 1.5. When plotting R versus t i , if the points are quite dense, we can see two lines near to the value 1.5. The other example is a 2:2 resonant state (see right side of Fig. 2(a) and the mid part of Fig. 2(b)), which is different from 1:1 resonance. In the former during two orbital periods the relative rotation of the secondary is two. During one of the periods the relative rotation is a little less than 0 (negative) and during the other the relative rotation is a little greater than 0 (positive). Thus in the first period R < 1 and in the second R > 1. In a 1:1 resonance R = 1. Although the average of R is the same in both cases ( R = 1), in a 1:1 resonance the position and the angular velocity relative to the r axis are always the same at apapsis. The 1:1 resonant state and the 2:2 should thus be distinguished from each other because they have quite different features; for example, the amplitude of the libration is larger and the changes of orbital elements are faster in a 2:2 resonance. In this article the term p : q resonant state will be used for resonances different from 1 : 1 only. Fig. 2(a) shows an example of the time-dependent resonance function where we also plot the temporal change of eccentricity. It is easy to separate the different regimes in the figure . During the first regime (t=0-300000) the rotation of the secondary is chaotic. The orbital motion drives the secondary. Depending on the orbital elements the driving can cause chaotic rotation. The orbital elements are changing and the type of rotation caused by the driving is also changing. By t ≈ 300000 the driving cannot cause chaotic rotation and the chaotic regime ends.

The second regime is a resonance. If the system is in a p : q resonance the temporal behaviour of the secondary is periodic, thus the average of its energy and the dissipation are constants: < Ėsec > q = 0 and

< Ė > q = 1 t i+q -t i t i+q t i - 1 2 γ l2 dt = C p:q = const < 0. (18)
A constant energy of the secondary can only be maintained if the system pumps the dissipated energy from the orbit to the secondary which means

< Ėc > q =< Ė > q -< Ėsec > q = C p:q . ( 19
)
This is an energy condition for the existence of a p : q resonant state. The subscript p : q expresses the fact that the value of C is determined mainly by the type of resonant state.

Of course C also depends on the parameters -e.g., on l 0 -and on the orbital elements. The resonant state ends -because of the change of orbital elements -when the system cannot pump enough energy to the secondary. If the energy condition [START_REF] Ott | Chaos in Dynamical Systems[END_REF] is not valid, the resonant state loses its stability. The third regime (near t ≈ 750000) is quite short. At the beginning of this regime a couple of stable resonant states are present. According to our simulations they behave as 2.0 Fig. 3 Lifetime distribution over a piece of the φ , φ plane with l = 0. The initial value of l is chosen as l = l 0 /(ω 2 -φ 2 ). Parameters are ω = 10, γ = 10, l 0 = 10 -4 , ε = 0.1 (The secondary is at the apapsis). Lifetimes are given in units of orbital periods. The filamentary structure is a manifestation of transient chaos.

quasi-attractors with fractal basin boundaries. If the state of the secondary is close to a boundary between the basins of quasi-attractors, the orbit can exhibit long-time chaos.

The fourth regime is also a resonance. This time the mechanism of the disappearance of the resonance is not the same as in the second regime. The simple quasi-attractor changes into a strange (chaotic) quasi-attractor as a consequence of the change of the orbital elements.

Since all quasi-attractors are simple resonances, any type of chaos present in such dissipative systems can only be of transient type. As is known from the theory of transient chaos [START_REF] Ott | Chaos in Dynamical Systems[END_REF], [START_REF] Tél | Chaotic Dynamics, An introduction based on classical mechanics[END_REF] the chaotic set is in such cases a non-attracting set, called a chaotic saddle. Orbits initiated close to the stable manifold of the saddle converge towards the chaotic set, stay a long time around it, and will be led eventually to a quasi-attractor along the unstable manifold of the saddle. The presence of a chaotic saddle can be demonstrated by plotting a lifetime distribution over the set of initial conditions. This is done in Fig. 3 where a clear fractal filamentation of the longlived regions can be seen. Although this set of points is similar in appearance to a chaotic attractor, which is an unstable manifold, we emphasize that what we see here is the stable manifold of a chaotic saddle. We have to distinguish two types of transient chaotic regimes. The first (second) type ends without (with) a change of the orbital elements. In Fig. 2(a) the second and fourth chaotic regimes (the short ones) are of the first type and the first and the third chaotic regimes (the longer ones) are of the second type.

The typical behaviour of the eccentricity is a decrease toward zero. In every regime but the 1 : 1 resonance the function of eccentricity is piece-wise concave (see Appendix 8.1). The semi-major axis also decreases and approaches a given value, a ∞ , determined by the initial conditions (see Appendix 8.2). The further evolution of the system in this case is that it falls into 1 : 1 resonance around t ≈ 1.1 × 10 7 , and the eccentricity tends to zero exponentially as the orbit circularizes. Fig. 2(b) shows another example illustrating that the 1 : 1 resonance can lose stability if the eccentricity is not so small. A similar loss of stability can be found in a recent paper [START_REF] Sidorenko | A Spring-mass model of tethered satellite systems: properties of planar periodic motions[END_REF]. Although the authors study a conservative model, a loss of stability of the 1:1 resonance has been found due to the decrease of the eccentricity. In that model the dynamics of the secondary is three-dimensional and the point masses of the secondary are unequal, but because the orbital elements do not change, the loss of stability is not the result of a temporal evolution as in our case. The figure also shows the centre-ofmass energy E c ; our numerical experience is that the change of E c is exponential only in a 1:1 resonance [START_REF] Escribano | Dynamics of tidal synchronization and orbit circularization of celestial bodies[END_REF]. In other resonant states the temporal change in E c is approximately linear (for an explanation see Eq. ( 19)). Another important feature illustrated in Fig. 2(b) is that the decrease of the centre-of-mass energy is, in general, more intense in higher resonances than in 1 : 1 resonance. In conservative cases (γ = 0), we do not have any attractor, but the system can still synchronize in several metastable resonances. The spring alone is enough for the secondary to interact with its orbit and temporarily store energy in the form of rotation and vibration. Eventually the energy will go back to the orbit and the system will return close to its initial state. In Fig. 4(a) we show how the energy is transferred from orbit to rotation and back. Stronger springs (higher ω) behave more like rigid bodies and hence produce fewer resonances. Weaker springs show a wide variety of synchronizations that can last from a few orbital periods to thousands of orbital periods. A very elastic secondary can easily synchronize in rarer resonances. For instance, we can have resonances with rotation velocity slower than the orbital velocity, or retrograde rotation: with rotation in the opposite sense to the orbit. Fig. 5(b) compares the varying resonances that are produced with different springs using R(t) defined in Eq. ( 16). We can note how a more elastic secondary is more often synchronized and its resonances remain stable for longer times.

Approximate equations of motion

To calculate the approximate equations of motion we Taylor expand the Lagrangian up to 4th order in l/(2r):

L = 1 2 ṙ2 + r 2 β 2 + l2 4 + l 2 4 φ 2 - 1 8 ω 2 (l -l 0 ) 2 + 1 r 1 + l 2r 2 P 2 (cos α) + l 2r 4 P 4 (cos α) .
The Taylor-expanded equations of motion can be written as q = q(0) + q(1) + q(2) +. . . where q = (r, β , l, φ ) and

q(0) =          r β 2 - 1 r 2 -2 ṙ β r l φ 2 -ω 2 (l -l 0 ) -2γ l -2 l φ l          , (20) q
(1) =       0 0 - l r 3 1 -3 cos 2 α - 3 2 1 r 3 sin 2α       , q(2) =        3 8 l 2 r 4 1 -3 cos 2 α 3 8 l 2 r 5 sin 2α 0 0        , . . . (21)
Another form of the Taylor-expanded equations of motion is 0th order 1st order 2nd order

r =r β 2 - 1 r 2 + 3 8
l 2 r 4 1 -3 cos 2 α β =-2 ṙ β r + 3 8
l 2 r 5 sin 2α l =l φ 2 -ω 2 (l -l 0 ) -2γ l - l r 3 1 -3 cos 2 α φ =-2 l φ l - 3 2 1 r 3 sin 2α
In general, the nth order approximation contains the nth power of l/r in the equations of r, r β , l and l φ (accelerations).

In the 0th order approximation ( q = q(0) ) the system of equations splits into two parts. The first pair of equations (r, β ) forms the equations of motion of the pure Keplerian problem and the second pair ( l, φ ) describes the problem of two rotating point masses connected by a frictional spring.

It is worth emphasizing that in the 1st-order approximation ( q = q(0) + q(1) ) the secondary has no feedback on the centre-of-mass motion at all. That is why the first order approximation has important qualitative differences from the exact model: (i) The orbital elements are not changing; there is no circularization. (ii) It is a driven nonlinear system in which the drive is not changing in time. Depending on the orbital elements the attractor(s) can be simple or strange ones. Of course in the case of a strange attractor the dynamics of the secondary is permanently chaotic and in the case of simple attractors the dynamics can be either regular or transiently chaotic depending on the initial conditions. (iii) The orbit pumps energy into the secondary but the orbital energy remains constant. (iv) The total angular momentum is not constant. (v) The dimension of the phase space of this problem is only 5. In spite of these differences, it is important to consider this approximation because a couple of its features are reflected in the exact model.

We can also define approximations 1a and 2a. Approximation 1a is the 1st-order approximation without the term q (1) 3 = -l/r 3 1 -3 cos 2 α , i.e., an equation corresponding to rigid-body rotation of the secondary, provided l = const. In this approximation the action of the primary on the spring is neglected. Approximation 2a is the 2nd-order approximation without the term q

(2) 2 = 3/8l 2 /r 5 sin 2α in the equation of β . The term kept in the equation of r describes a central force perturbation proportional to 1/r 4 provided l and α are constant. The force is then 3/8l 2 /r 4 1 -3 cos 2 α and an analytic expression (Eq. ( 32)) exists for the apapsis precession.

In general, the trajectories calculated by the approximate and exact equations deviate from each other because of the chaoticity of the dynamics; in such cases we expect only the statistical behaviour to be the same. On the other hand, if the motion is regular -as in a resonance -then the approximate equations are expected to be rather accurate.

The Taylor-expanded equations give us a facility to understand the response of the secondary to the changes of orbital elements. By using the second-order approximate equations

Ėc = ṙ r + 1 r 2 + r ṙ β 2 + r 2 β β = ṙq (2)
1 + r 2 β q

(2) 2 [START_REF] Sidorenko | A Spring-mass model of tethered satellite systems: properties of planar periodic motions[END_REF] and the time average of the change of the centre-of-mass energy can be written as

< Ėc > q = 1 t i+q -t i t i+q t i (ṙq (2) 1 + r 2 β q (2) 2 )dt = 1 t i+q -t i 3 8 t i+q t i ṙ r 4 l 2 1 -3 cos 2 α + β r 3 l 2 sin 2α dt ( 23
)
We use for illustrative purposes a 5 : 2 resonance and the initial eccentricity is 0.1. Fig. 6 Fig. 7 The shift of function α(t) with eccentricity. The line styles mark the eccentricity values as in Fig. 6. Fig. 7 shows the function α(t) at ε = 0.9, 0.8, 0.7, 0.65, and 0.63. As the figure displays, the response of the secondary is a shift of α(t). The shift becomes faster as the end of the resonance is approaching.

Temporal evolution of the orbital elements

In order to obtain a continuous-time estimate of the orbital elements we use equations for

ε = 1 ε ( Ėc N 2 1 + 2E c N 1 Ṅ1 ), (24) ȧ
= 1 2 1 E 2 c Ėc = 2a 2 Ėc . ( 25
)
These equations appear in several models which treat the secondary as a spatially extended object, see e.g. [START_REF] Witte | Orbital evolution by dynamical tides in the solar type stars[END_REF]. The point is that Ėc and Ṅ1 cannot be expressed by ε and a only, and the set of equations is thus not closed. Special approximate formulas for Ėc and Ṅ1 are typically needed obtain closure. We are not forced to do so since the numerical values of these quantities easily follow from (2)-( 5) and ( 7), [START_REF] Gladman | Synchronous locking of tidally evolving satellites[END_REF]. In fact we consider ( 24), (25) only as diagnostic equations investigated along with the numerical solution of our model.

For later purposes we also write down an equation for β apapsis , used in the same diag- nostic sense. By taking the time derivative of ( 14), [START_REF] Levin | Dynamic Analysis of Space Tether Missions[END_REF] and using ( 7) and ( 12) we find:

βapapsis = β + ε 3r 2 ṙ β 2 + 2r 3 β β + ε 1 -r 3 β 2 ṙε a(1 -ε 2 ) . ( 26
)
The leading-order behaviour of these quantities follows from the second-order approximate equations. Substituting

Ṅ1 = 2r ṙ β + r 2 β ≈ r 2 q (2) 2 , (27)
and Eq. ( 22) into ( 24) and ( 25), we obtain

ε = 1 ε ṙq (2) 1 + r β 2 q (2) 2 N 2 1 + 2E c N 1 r 2 q (2) 2 , ( 28
) ȧ = 2a 2 (ṙq (2) 1 + r 2 β q (2) 2 ). ( 29
)
In a similar way, one finds from ( 26)

βapapsis = 2r 3 β εq (2) 2 -r 3 β 2 -1 ε ṙε a(1 -ε 2 ) . ( 30
)

Apapsis precession

The apapsis precession ∆ β is obtained by multiplying βapapsis by the instantaneous period

T (t) = 2πa(t) 3/2 : ∆ β (t) = 2π a 3 (t) βapapsis (t). ( 31
)
This equation provides the instantaneous tendency for producing a given apapsis precession. This can, in certain phases of the motion, be of opposite sign than after a full period, i.e., than the average of ∆ β (t) over a period. A well-known result [START_REF] Landau | Course of Theoretical Physics -Mechanics[END_REF] for ∆ β in the presence of a dimensionless perturbative force of the type -κ/r 4 , κ ≪ 1 yields

∆ β pert (t) = 2π κ(t) N 4 1 (t) (32)
where κ(t) = 3 8 l 2 (t) 1 -3 cos 2 α(t) , under the assumption that κ(t) changes slowly. In a time independent naive approximation when κ(t) is considered to be constant with α ≈ 0 and l = l 0 ,

N 1 = N ∆ β naive = 3π 2 l 2 0 N 4 . ( 33
)
In order to test the accuracy of the naive and the perturbative approximations of the apapsis precession we compare the numerical results in the case of the simplest, 1:1 resonance. Fig. 8(a) shows ∆ β naive from Eq. ( 33), the measured ∆ β (measurement has been made after a period but the value is represented by a continuous line), ∆ β pert (t) from Eq. (32), its average, and the time average of ∆ β (t) from Eq. (31) all evaluated in approximation 2a. Our conclusion is that the naive and the perturbative approximations can only estimate the magnitude of the apapsis precession; they are not suitable to give the exact value. The Fig. 8 (a) Comparison of the different methods for the calculation of the apapsis precession in the case of a 1:1 resonance in approximation 2a: numerically measured ∆ β (heavy continuous), average of ∆ β (t) from Eq. (31) (thin continuous, running very close to the heavy line), ∆ β pert (t) from Eq. (32) (thin dashed, oscillating), average of ∆ β pert (t) (heavy dashed), ∆ β naive from Eq. (33) (heavy dotted). Parameters and initial conditions:

ω = 10, γ = 1, l 0 = 10 -3 , ε 0 = 0.1, φ (0) = 0, φ (0) = 5. (b)
ω = 10, γ = 1, l 0 = 10 -3 , ε 0 = 0.1, φ (0) = 0, φ (0) = 5.
value of the time-averaged ∆ β (t) is practically the same as that of the measured ∆ β . The difference between the averages of ∆ β (t) and ∆ β pert (t) may be because κ is also oscillating with an amplitude which is about one fifth of its average value. Therefore, the perturbative result based on the slow variation of κ is an educated guess only. Fig. 8(b) shows that the apapsis precession, Eq. ( 31), evaluated with the 2nd order approximation and the exact equations coincide (heavy continuous line). This means that the equations of the 2nd order approximation are sufficient for calculating the apapsis precession to high accuracy. The figure also shows the function ∆ β (t) -exact and 2nd order approximation (heavy dashed), 2a order approximation (heavy dotted) -and the numerically measured apapsis precession in approximation 2a (thin continuous line). The result is about one half of the measured apapsis precession of the 2nd order approximation and of the exact simulation. This implies that the contribution of the perturbation 3/8 l 2 /r 4 sin 2α in the equation of β to the apapsis precession is practically as important as the well known central force contribution, Eq. (32), of approximation 2a.

Here we have concentrated on the case of the 1:1 resonance on time scales when the system is still far away from the circularized attractor. Since the interesting time dependencies all die out in the long-time limit, the approximate equations are expected to work even better for later times. The behaviour of the approximation scheme around other resonances is also similar. Altogether, the 2nd-order approximate equations seem to provide a rather accurate description of the entire dynamics, provided that the resonant state under investigation is the same as that predicted by the exact equation. The observation that the approximate equations lead in some cases to different resonances or 'transient attractors' on intermediate time scales appears to be a consequence of the fact that the dynamics between resonances is chaotic and the errors attributable to using a perturbation expansion become drastically amplified.

We can study the influence that the different parameters -γ, ω, l 0 , ε -have on the apapsis precession ∆ β apapsis . Looking at Eq. (30), we can see that ∆ β apapsis does not depend on γ directly. Numerical results indicate that an indirect dependence cannot be found either for a wide range of γ (γ < 2). βapapsis (30) depends on q

(2)

1 (through ε) and q

(2)

2 , and the ldependence of both terms is quadratic, so βapapsis ∼ l 2 . Using the approximation for l (Eq. (43) of the Appendix 8.3) we expect that the l 0 and ω dependencies can be written as

∆ β apapsis = c 1 l 0 1 -c 2 /ω 2 2 , (34)
where c 1 and c 2 are fitting parameters. Numerical fits confirm this conjecture. The l 0dependence is perfectly quadratic and Fig. 9(a) shows that expression (34) as a function of ω is perfectly consistent with the measured data. Fig. 9(b) shows that the eccentricity dependence is basically also quadratic, but as ε → 0, the apapsis precession seems to converge to a nonzero value.

Comparison of approximations in chaotic cases

To produce Fig. 10(a), the exact equations of motion have been integrated with 35 different initial conditions and the time averages of the angular velocity have been calculated in every period. The initial eccentricities of all 35 integrations were the same (ε = 0.1). Fig. 10(a) shows the average of the angular velocities versus time. There is a plateau at the end of the curve, since all 35 cases are in resonance. The height of the plateau is 1 implying that all the cases are in a 1:1 resonance. We integrated the exact and the 2nd-order equations of motion with the same initial conditions. Angular velocity φ versus time can be seen in Fig. 10(b). Here and even in the case of the highest-order approximation the trajectories deviate from each other in the transient chaotic regime (t > 300). Based on a study of a couple of similar cases we conclude that the 2nd-order (and higher-order) approximations have the same characteristic features as the exact equations of motion.

We have also studied the statistical behaviour of the different approximations. The same type of plot as Fig. 10(a) has been made to all orders of the approximations in Fig. 11(a). We conclude that all approximations other than 1a exhibit the same statistical behaviour as the exact case. From the graphs the characteristic time, τ 1 , needed to reach a 1:1 resonance can be estimated. This state is not yet the circularized attractor since ε ≈ ε 0 after such short period of time. It takes typically a much longer time -denoted by τ in [START_REF] Escribano | Dynamics of tidal synchronization and orbit circularization of celestial bodies[END_REF] -for the attractor, where ε = 0, to be reached; in this example τ 1 ≈ 2000 and τ ≈ 10 6 . The plateau of the green, 4th-order curve is slightly higher than 1. The reason is that one of the 4thorder cases is in a 5:2 resonance and not in 1:1. Among the 1st-order simulations there were two in a 5:2 resonance, which is why the yellow curve is higher than the green one. It is hardly observable but the dark blue, 3rd-order curve is also higher than 1; this is owing to the presence of a 5:4 resonance. Fig. 11(b) and (c) show a comparison of the exact and the approximate cases with different initial eccentricities. Based on these, we conclude that the higher the eccentricity the greater the time τ 1 . In Fig. 11(c) the asymptotic value is not which suggests that a 1:1 resonance has not yet been reached by the end of the integration; the curve is not smooth because most of the trajectories are not in resonance. Fig. 12 shows the time dependence of the eccentricity ε(t) in the different approximate equations. In all cases a 1:1 resonance has been reached by t ≈ 3 000. In the first order approximation there is no feedback of the secondary on the Keplerian problem, therefore ε(t) = ε 0 = const. The asymptotic slopes of the other curves show the rate of decrease of eccentricity. These rates are the same in all cases with the exception of approximation 2a which indicates that the contribution of the term 3/8 l 2 /r 5 sin 2α missing from the equation for β is important.

Fig. 12(b) shows the eccentricity averaged over 35 different initial conditions in the different approximate equations. It is a general observation that in resonances different from 1:1 the eccentricity decays much faster. This is the reason why in cases in which there are also other resonances than 1:1 in the ensemble (the 3rd and 4th order) the averaged eccentricities decrease faster. If we average only the 1:1 resonance cases than we obtain the light and the dark blue curves. As expected, the curves of the 2nd and 3rd order are close to each other, and close to the curves of the 4th order and to the exact numerical solution which practically coincide. Fig. 12(b) indicates that in exceptional cases even the 4th-order approximation can lead to a resonance -a 'transient attractor' -different from the exact one. The good agreement between the approximate and exact equations only holds under the condition that the same resonance has been reached.

Discussion

Spin-orbit resonances between a large primary body and a smaller secondary body are generally studied using models designed for quantitative analysis of a specific instance or a particular part of the problem [START_REF] Macdonald | Tidal friction[END_REF][START_REF] Goldreich | Q in the solar system[END_REF][START_REF] Gladman | Synchronous locking of tidally evolving satellites[END_REF][START_REF] Rasio | Tidal decay of close planetary orbits[END_REF][START_REF] Murray | Solar System Dynamics[END_REF][START_REF] Celletti | Hamiltonian stability of spin-orbit resonances in celestial mechanics[END_REF][START_REF] Correia | Mercury's capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics[END_REF][START_REF] Dobbs-Dixon | Spin-orbit evolution of short-period planets[END_REF][START_REF] Celletti | Dynamics of the conservative and dissipative spinorbit problem[END_REF][START_REF] Efroimsky | Tidal torques: a critical review of some techniques[END_REF][START_REF] Sidorenko | A Spring-mass model of tethered satellite systems: properties of planar periodic motions[END_REF]. The complexity of these models tends to obscure the fundamental physical principles involved in this interaction. We have instead taken a qualitative dynamical-systems approach, using a minimalistic model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. Despite its simplicity, the model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors. Apart from its applicability to tidal synchronization and orbit circu-larization of celestial bodies, our model is also relevant to the problem of the dynamics of tethered satellites, which has recently provoked much interest and a number of space missions [START_REF] Levin | Dynamic Analysis of Space Tether Missions[END_REF][START_REF] Sidorenko | A Spring-mass model of tethered satellite systems: properties of planar periodic motions[END_REF].

One interesting open question is whether a pseudo-energy -or energy-like -approach would provide new insight into the problem. In this case we have completely defined the gravitational and mechanical energies -the centre of mass plus rotation and vibration of the secondary -and thus the dissipated energy can be obtained simply by subtraction from the initial energy; the same is true for the evolution between two arbitrary states. As the dissipation is strictly passive the total energy is necessary a decaying function, reaching a constant value only in a situation in which the dissipation term vanishes along the trajectory. In the approach of Sarasola et al. [START_REF] Sarasola | Energy-like functions for some dissipative chaotic systems[END_REF], energy dissipation is associated with a local change in phase space volume and on this basis they define a partial differential equation that an energy-like function must satisfy. The same depends on a part of the decomposition of the vector velocity field; the divergence free or rotational part in contrast with the part that carries the whole divergence. Once H is obtained we can calculate the energy derivative along a trajectory of the system which is due to the part carrying the divergence of the vector field. Probably this is the unique new information that can be obtained in our case: an analytical expression instead of the indirect calculation by subtraction of the mechanical plus gravitational energy. Nonetheless the pseudo-energy approach is of interest and we intend to pursue it in future work.

The main conclusions that may be drawn from the present investigations are: The 1:1 resonance is the only stable attractor. Dissipation drives the system towards this synchronization and circularizes the orbit. All other resonances are metastable, or transient attractors, and dissipate energy at a faster pace, and the approach towards the circular orbit is also faster.

If we look to our solar system, we find that all large moons are locked in 1:1 spinorbit resonance with their planets. Of the planets, however, only Mercury is locked into spin-orbit resonance with the Sun, and this is a 3:2 spin-orbit resonance [START_REF] Correia | Mercury's capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics[END_REF]. The presence or absence of locking can be explained in our model through the strength of the coupling [START_REF] Escribano | Dynamics of tidal synchronization and orbit circularization of celestial bodies[END_REF], which is largest for the moons, and smaller for the planets, of which Mercury has the strongest coupling. Within our minimal model, in which 1:1 is the only asymptotic attractor, such a 3:2 resonance is metastable, and Mercury would eventually decay into 1:1 spin-orbit resonance. On the other hand, it may be possible to stabilize 3:2 resonance in our model by adding to it interactions with further bodies (the other planets); three-dimensionality (noncoplanarity of spin and orbit); and post-Newtonian effects.

Isolating the dominant interactions between the degrees of freedom in terms of a Taylor expansion allows us to to study them separately. In general, the second-order approximation is sufficient to reproduce the typical behaviour of the complete model, as long as we are in the 1:1 resonance. This has been studied for apapsis precession, chaotic regimes and eccentricity evolution.

Finally, we summarize features of our model which are expected to remain valid in more general cases as well. Even if the expression for < Ė > q can, in general, be quite complex, the energy condition [START_REF] Ott | Chaos in Dynamical Systems[END_REF] for resonant states holds in any case. The mechanism of the response of the secondary to the change of the orbital elements should remain similar to in our model. Of course, we have to use the general formula [START_REF] Tél | Chaotic Dynamics, An introduction based on classical mechanics[END_REF] where q (2)

1 and q (2)
2 can be rather complicated.

Quasistationary behaviour of l

We introduce ∆ l = ll 0 . Its equation of motion in the 2nd order approximation is

∆ l = l φ 2 -ω 2 ∆ l -2γ ∆ l - l r 3 1 -3 cos 2 α . (41)
According to numerical results ∆ l/l ≪ 1 which is why the left side and the third term of the right side are neglectable. The second term is generally not negligible. After writing ∆ l = ll 0 we obtain 0 = l φ 2 -ω 2 (ll 0 ) -l r 3 1 -3 cos 2 α .

This means that l ≈ 0, and the spring is almost in equilibrium at each moment. On expanding l we obtain l equ = ω 2 l 0 ω 2 -φ 2 + 1 r 3 (1 -3 cos 2 α)

= l 0 1 1 - 1 ω 2 φ 2 + 1 r 3 3 cos 2 α -1 (43)
where we added the subscript equ. A comparison of l(t) and l equ (t) is shown in Fig. 13.

Fig. 1

 1 Fig. 1 (a) Instantaneous configuration of the system given by the generalized coordinates r, β , l, φ . The relative angle α = φβ is indicated. (b) Schematic diagram illustrating β apapsis(t)

Fig. 2 (

 2 Fig. 2 (a) Resonance function R(t i ), ε(t i ) (upper continuous line). Parameters and initial conditions: ω = 10, γ = 1, l 0 = 10 -3 ; ε 0 = 0.4, φ (0) = 0, φ (0) = 5. The series of resonances is 5:1, 4:2, 4:1, 2:2. (b) Resonance function R(t i ), E c (t i ) (upper continuous line). Parameters and initial conditions: ω = 10, γ = 5, l 0 = 10 -3 ; ε 0 = 0.4, φ (0) = 0, φ (0) = 5. The series of resonances is 2:1, 1:1, 2:2, 1:1.

Fig. 4 Fig. 5

 45 Fig.4(a) Energy transfer from the orbit (centre-of-mass kinetic energy, dark, red) to the secondary (rotation and vibration, grey, yellow). After some orbital periods the energy is transferred back to the centre-of-mass motion. ε 0 = 0.50, γ = 0.0, l 0 = 1.0 -2 , ω = 10. (b) Longer runs show that this behaviour is periodic. ε 0 = 0.30, γ = 0.0, l 0 = 1.0 -2 , ω = 10.

Fig. 6 (

 6 Fig.6(a) Temporal change of the factors of the first term in the integral[START_REF] Tél | Chaotic Dynamics, An introduction based on classical mechanics[END_REF] over two orbital periods at different eccentricities. The secondary is at apapsis at t = 0. The line styles correspond to the eccentricities: 0.9 (heavy continuous), 0.8 (thin continuous), 0.7 (thin dashed), 0.65 (thin dotted) and 0.63 (heavy dashed). (b) Temporal change of the factors of the second term in the integral[START_REF] Tél | Chaotic Dynamics, An introduction based on classical mechanics[END_REF].

6 Fig. 9 Fig. 10 (

 6910 Fig. 9 Numerical data (dots) and fits (continuous lines) for the apapsis precession as a function of spring frequency (a) and eccentricity (b). (a) Parameters are γ = 1, l 0 = 10 -3 ; ε 0 = 0.05, and the fitted parameters of (34) are c 1 = 9.85, c 2 = 8.11. (b) Parameters are ω = 10, γ = 1, l 0 = 10 -3 . The form of the fitted function is aε 2 + b. The result of the fit is a = 3.78 • 10 -5 , b = 1.13 • 10 -5

Fig. 11

 11 Fig. 11 Comparison of the average behaviour of the different approximate equations. Initial eccentricity is (a) ε 0 = 0.1; (b) ε 0 = 0.2; (c) ε 0 = 0.3.

Fig. 12 (

 12 Fig. 12 (a) Comparison of eccentricities with the different approximate equations. Parameters and initial conditions: ω = 10, γ = 1, l 0 = 10 -3 ; ε 0 = 0.1, φ (0) = 0, φ (0) = 5. (b) Comparison of the averaged eccentricities with the different approximate equations. Parameters and initial conditions: ω = 10, γ = 1, l 0 = 10 -3 ; ε 0 = 0.1.

Fig. 13

 13 Fig.[START_REF] Levin | Dynamic Analysis of Space Tether Missions[END_REF] Comparison of the measured l (continuous line) and l equ calculated by Eq. (43) (dashed line). Parameters and initial conditions: ω = 10, γ = 1, l 0 = 10 -3 ; ε 0 = 0.1, φ (0) = 0, φ (0) = 5.

  Comparison of the apapsis precession obtained by means of the different approximate equations. The measured exact and the exact ∆ β (t) averaged coincide with the measured ∆ β in 2nd order (heavy continuous line). The exact ∆ β (t) coincides with 2nd order ∆ β (t) (heavy dashed line). Parameters and initial conditions:

  1,

						exact								exact
						4th order								4th order
						3rd order							4th order (only 1:1 resonances)
	0.10000					2nd order 2nd a order 1st order		0.10000					3rd order 3rd order (only 1:1 resonances) 2nd order
						1st a order								2dn a order
	ε							ε
	0.09999
	0	1000	2000	3000 t	4000	5000	6000	0.09999	0	1000	2000	3000 t	4000	5000	6000

Acknowledgements

We acknowledge projects OTKA NK72037 (Hungary), Hielocris (Spain), CGL2008-06245-CO2 (Spain), and the Hispano-Hungarian Binational project TeT ESP-34/2006.

Appendix

Temporal change of the eccentricity

As we saw there are two typical temporal behaviours of E c ; linear and exponential. From Eq. [START_REF] Landau | Course of Theoretical Physics -Mechanics[END_REF] and considering that the approximate equation N 1 ≈ N is valid we can estimate the temporal change of eccentricity.

a. Exponential temporal behaviour of E c Assuming the function of E c (t i ) to be in the form E c (t i ) = E c,∞ + A Ec e -λ Ec t i we obtain an exponential function for the eccentricity

where we used that 1

The eccentricity thus also decays exponentially and its exponent is the half of the exponent of the function

b. Linear temporal behaviour of E c In p : q resonant states the temporal behaviour of E c is linear with the negative slope C p:q so E c (t i ) = C p q t i + E c,0 where E c,0 = E c (t 0 ). The eccentricity at the instant t 0 is

Using this we obtain the temporal change of the eccentricity

which is always a concave function.

Radius of the circularized orbit

Consider a circular orbit in a 1:1 resonance: a ∞ = r = const, β∞ = const, φ∞ = const, l ∞ = const, α = 0 and l = 0. The parameters of the circularized orbit are determined from the condition φ∞ = β∞ .

Because l ∞ ≪ a ∞ from Eq. ( 12), a good approximation can be obtained for a ∞ if we consider that N 1 ≈ N 0 . By writing down Eq. ( 12) initially we obtain

In the final state ε ∞ = 0, 0 = 1 -N 2 0 /a ∞ and thus

The larger the initial eccentricity (ε 0 ), the smaller the final radius of the orbit (a ∞ ). It is important to note that a ∞ is independent of any other parameters i.e., γ, ω or l 0 .