Integration error over very long time spans

The main limit to the time span of a numerical integration of the planetary orbits is no longer set by the availability of computer resources, but rather by the accumulation of the integration error. By the latter we mean the difference between the computed orbit and the dynamical behaviour of the real physical system, whatever the causes. The analysis of these causes requires an interdisciplinary effort: there are physical model and parameters errors, algorithm and discretisation errors, rounding off errors and reliability problems in the computer hardware and system software, as well as instabilities in the dynamical system. We list all the sources of integration error we are aware of and discuss their relevance in determining the present limit to the time span of a meaningful integration of the orbit of the planets. At present this limit is of the order of 10⁸ years for the outer planets. We discuss in more detail the truncation error of multistep algorithms (when applied to eccentric orbits), the coefficient error, the method of Encke and the associated coordinate change error, the procedures used to test the numerical integration software and their limitations. Many problems remain open, including the one of a realistic statistical model of the rounding off error; at present, the latter can only be described by a semiempirical model based upon the simpleN ² formula (N=number of steps, =machine accuracy), with an unknown numerical coefficient which is determined only a posteriori.     

Stable planetary orbits around one component in nearby binary stars

Numerical simulations are made within the frame of the elliptic planar restricted three-body problem, in order to search if stable orbits exist for planets around one of the two components in double stars. The values taken for the binary's mass ratio and eccentricity correspond to real nearby double and multiple systems; the Centauri and Sirius systems are investigated here. Large stable planetary orbits, already known to exist through a systematic exploration of the circular model and for the Sun-Jupiter elliptic case, are found to exist around Centauri A and Centauri B, as well as around Sirius A, up to distances from each star of the order of more than half the binary's periastron separation. Similar studies for other nearby well-known double and multiple systems will follow.     

Reviews

OFFICE DEVELOPMENT: A Geographical Analysis by M. Bateman. 14 × 22 cm, xvi and 175 pages. Croom Helm: London 1985 (ISBN 0 7099 0697 8) $A35.95 (cloth).

ATLAS OF YOUTH UNEMPLOYMENT, 1981: The Geographic Distribution of Youth Unemployment in Australian Cities from the 1981 Census According to Birthplace and Gender by P. Matwijiw with E. Bamford and C. Maher. 21 × 29 cm, 127 pages. Australian Institute of Multicultural Affairs: Melbourne 1985 (ISBN 949890 33 2) $A12.00 (limp).

ENVIRONMENT AND HEALTH by A. J. Rowland and P. Cooper. 23 × 16 cm, vi and 205 pages. Edward Arnold: London 1983 (ISBN 0 7131 2855 0) $A27.95 (limp).

FAMINE AS A GEOGRAPHICAL PHENOMENON edited by B. Currey and G. Hugo. 17 × 24 cm, vi and 202 pages. Reidel: Dordrecht 1984 (ISBN 90 277 1762 1) Dfl 95.00, $US37.00 (cloth).

SOUTH AUSTRALIA'S CHANGING POPULATION (South Australian Geographical Papers No. 1) by G. J. Hugo. 25 × 18 cm, 71 pages. Royal Geographical Society of Australasia (SA Branch): Adelaide 1983 (ISBN 0 909112 05 03) $A4.00 (limp).

DEVELOPMENTS IN POLITICAL GEOGRAPHY edited by M. A. Busteed. 23 × 15 cm, 340 pages. Academic Press: London 1983 (ISBN 0 12 148420 3) $US42.00 (cloth).

NEITHER JUSTICE NOR REASON: A Legal and Anthropological Analysis of Aboriginal Land Rights by M. Gumbert. 22 × 14 cm, xv and 215 pages. University of Queensland Press: St Lucia 1984 (ISBN 0 7022 1746 8) $A20.00 (cloth).

RURAL RESOURCE MANAGEMENT by P. J. Cloke and C. C. Park. 14 × 22 cm, xiii and 473 pages. Croom Helm: London 1985 (ISBN 0 7099 2037 7) $A44.95 (cloth).

FARM POLICY IN AUSTRALIA by R. K. Hefford. 14 × 22 cm, xviii and 415 pages. University of Queensland Press: St Lucia 1985 (ISBN 0 7022 1698 4) $A40.00 (cloth).

TRANSPORT AND COMMUNICATIONS FOR PACIFIC MICROSTATES: Issues in Organisation and Management edited by C. C. Kissling. 14 × 21 cm, 192 pages. Institute of Pacific Studies, University of the South Pacific: Suva 1984. $A10.00 (cloth), $A8.00 (limp).

TIMES OF CRISIS: Epidemics in Sydney 1788–1900 by P. H. Curson. 15 × 25 cm, xii and 195 pages. Sydney University Press: Sydney 1985 (ISBN 0 424 00112 8). $A35.00 (cloth).

PEASANTS, SUBSISTENCE, ECOLOGY AND DEVELOPMENT IN THE HIGHLANDS OF PAPUA NEW GUINEA by L. S. Grossman. 24 × 16 cm, xxi and 302 pages. Princeton University Press: Princeton, NJ 1984 (ISBN 0 691 09406 3) $US45.50.

THE WORLD FOOD PROBLEM 1950–1980 by D. Grigg. 15 × 24 cm, 276 pages. Basil Blackwell: Oxford 1985 (ISBN 0 631 13481 6) $A58.50 (cloth).

POPULATION REDISTRIBUTION AND DEVELOPMENT IN SOUTH ASIA edited by L. A. Kosinski and K. M. Elahi. 17 × 24 cm, 243 pages. Reichel: Dordrecht 1985 (ISBN 90 277 1938 1) Dfl 120.00, $US44.00, £30.50 (cloth).

GEOGRAPHY AND ETHNIC PLURALISM edited by C. Clarke, D. Ley and C. Peach. 15 × 23 cm, xvii and 294 pages. Allen & Unwin: London 1984 (ISBN 0 04 309107 5) $A45.00 (cloth); (ISBN 0 04 309108 3) $A24.95 (limp).

SOCIAL GEOGRAPHY IN INTERNATIONAL PERSPECTIVE edited by J. Eyles. 14 × 22 cm, 295 pages. Croom Helm: London 1986 (ISBN 0 7099 0944 6) $A49.95 (cloth).

THE GENTRIFICATION OF INNER MELBOURNE: A Political Geography of Inner City Housing by W. S. Logan. 14 × 22 cm, xvi and 328 pages. University of Queensland Press: St. Lucia 1985 (ISBN 0 7022 1729 8) $A50.00 (cloth).

YELLOWCAKE AND CROCODILES: Town Planning, Government and Society in Northern Australia by J. Lea and R. Zehner. 14 × 22 cm, xxvi and 200 pages. Allen & Unwin: Sydney 1986 (ISBN 0 86861 875 6) $A24.95 (cloth); (ISBN 0 86861 843 8) $A 14.95 (limp).

THE BUNGALOW: The Production of a Global Culture by A. D. King. 20 × 25 cm, xii and 310 pages. Routledge & Kegan Paul: London 1984 (ISBN 0 7100 9538 4) $A62.50 (cloth).

THE WEST EUROPEAN CITY: A Social Geography by P. White. 15 × 23 cm, xviii and 269 pages. Longman: London 1984 (ISBN 0 582 30047 9) $A22.95 (limp).

THE SHAPING OF AMERICA: A Geographical Perspective on 500 years of History. Volume 1. Atlantic America, 1492–1800 by D. W. Meinig. 18 × 25 cm, xii and 500 pages. Yale University Press: New Haven 1986 (ISBN 0 300 03548 9) $US35.00 (cloth).

CANBERRA: MYTHS AND MODELS. Forces at Work in the Formation of the Australian Capital by K. F. Fischer. 21 × 30 cm, 166 pages. Institute of Asian Studies: Hamburg 1984 (ISBN 3 88910 009 0).

THE GEOGRAPHY OF PEACE AND WAR edited by D. Pepper and A. Jenkins. 15 × 23 cm, vi and 222 pages. Basil Blackwell: Oxford 1985 (ISBN 0 631 13559 6) $A75.00 (cloth); (ISBN 0 631 14069 7) $A23.95 (limp).

POLITICAL GEOGRAPHY: World‐economy, Nation‐state and Locality by P. J. Taylor. 15 × 23 cm, × and 238 pages. Longman: London 1985 (ISBN 0 582 30088 6) $A24.95 (limp).

THE INDUSTRIAL GEOGRAPHY OF CANADA by A. Blackbourn and R. G. Putnam 14 × 22 cm, 201 pages. Croom Helm: London 1984 (ISBN 0 7099 0622 6) $A42.50 (cloth).

THE FOODMAKERS by S. Sargent. 13 × 20 cm, 296 pages. Penguin: Ringwood 1985 (ISBN 0 14 007359 0) $A8.95 (limp).

LIMITS TO PREDICTION edited by R. B. McKern and G. C. Lowenthal. 13 × 19 cm, xiii and 163 pages. Australian Professional Publications: Sydney 1985 (ISBN 0 949416 029) $A14.95 (limp).

THE KANGAROO KEEPERS edited by H. J. Lavery. 18 × 25 cm, xxvi and 211 pages. University of Queensland Press: St Lucia 1985 (ISBN 0 7022 1875 8).

PRACTICAL ECOLOGY by D. D. Gilbertson, M. Kent and F. B. Pyatt. 15 × 23 cm, 320 pages, Hutchinson: London 1985 (ISBN 0 09 162651 X) £8.95.

BASIC BIOGEOGRAPHY (Second edition) by N. Pears. 15 × 24 cm, × and 358 pages. Longman: London 1985 (ISBN 0 582 30120 3) $A24.95 (limp).

ENVIRONMENTAL CHANGE (Second edition) by A. Goudie. 23 × 15 cm, xi and 258 pages. Oxford University Press: Oxford 1983 (ISBN 0 19 874135 9) A16.00 (limp).

MEGA‐GEOMORPHOLOGY edited by R. Gardner and H. Scoging. 16 × 24 cm, xiii and 240 pages. Oxford University Press: Oxford (ISBN 0 19 823244 6) $A48.00.

GRANITE LANDFORMS by C. R. Twidale. 25 × 17 cm, xxiii and 372 pages. Elsevier: Amsterdam 1982 (ISBN 0 444 42116 5) $A148.25.     

The Soviet settlement system--current issues and future prospects

"A survey of current urban issues and trends in the Soviet Union sets the stage for an analysis of the problems of the urban system of the USSR and discussion of recent efforts at planning and policy development for a national settlement system. The General Scheme of Settlement, issued in 1975 and revised in 1982 to guide the development of such a national system, is examined as is the ensuing debate between two opposing schools of thought as to the best approach for restructuring the urban system."     

Education as a determinant of fertility: a case study of Kullu Town

Using a sample of 885 females (1/2 of all married women of reproductive age), this study examines the role of education as a determinant of fertility among couples in Kullu town of Himachol Pradesh, India. Of the 885 respondents, only 149 were illiterate. The average family size was 2.88. Findings reveal that 1) the average family size was largest for illiterate respondents (3.57) and for illiterate males (3.76); 2) average family size declined consistently after the middle level of education to 1.29 for post graduate respondents and 2.33 for post graduate males; 3) a negative correlation exists between fertility and education of both husband and wife, with the wife's education having a stronger negative correlation with fertility; and 4) couples with an educational level of matriculation and above have a distinctly smaller family size than those less educated.     

Multiple windbreaks: An aeolean ensemble

Near-neutral measurements of the turbulent wind field within and above a sequence of 15 parallel windbreaks on a flat pastoral site are presented. The windbreak fences each had a porosity of 60% and were equally-spaced at 6 times their height (h = 2 m). The following conclusions seem justified for wind directions within 10 ° of the normal to the array:

1. Above the windbreaks (2h), mean windspeeds first decreased and then increased asymptotically to a value in equilibrium with the new surface roughness. At 0.5h, windspeeds exhibited a slow increase down the entire array.
2. Reflecting differences in approach flows, the drag on the initial fence was almost twice that on barriers farther downstream. This reduction in momentum extraction per windbreak was associated with an elevation in the zero-plane displacement to a level equal to 0.8h.
3. At positions well-removed from the initial fences, mean windspeeds were reduced throughout the entire region below shelter height. In this region, the flow became increasingly dominated by downward moving air with velocities much greater than the local average. The zone of reduced turbulence was small, extending only 2h downstream of a barrier at a height of 0.25h. This corresponded with the region excluded from smoke trails released at the top of windbreaks.
4. An approximate TKE budget mid-way between windbreaks 7 and 8 suggests that shear and wake production peak near z = h and that production is balanced by dissipation and vertical transport components. Advective and inertial interaction terms are negligible at this midway position but are likely to be major sources of TKE closer to the windbreak. Local equilibrium is attained above z = 1.5h implying the existence of a constant-stress layer.

The measurements show the practical difficulty of simultaneously reducing both mean windspeeds and turbulence levels with repeated windbreaks at conventional spacings for horticultural applications.     

The nature and significance of the feedback of changes in terrestrial vegetation on global atmospheric and climatic change

The potential feedback on global atmospheric and climate change of climate-driven changes in terrestrial vegetation is examined by systematically relating the surface exchanges of energy, mass and momentum to two dimensions of vegetation, structure and taxonomy, such that the significance of climate driven changes in these characteristics can be assessed. A detailed quantitative understanding of this feedback is an important prerequisite to realistic and dynamic representations of the Earth's surface within general circulation and biological models (GCMs and GBMs). Without realistic representations of terrestrial vegetation within these models, any forecasts of future climates by these models must be suspect.Several general conclusions are drawn. The first is that the indirect feedbacks, those associated with the clouds and aerosols of the planetary boundary layer, appear to be very powerful but as yet their behaviour and connections with the underlying surface are both poorly understood and captured within GCMs.The physical structure of vegetation, the disposition of biomass in 3-D, is the characteristic that most strongly influences the exchange of momentum (via aerodynamic roughness) and solar radiation (via albedo). Vegetation structure and species composition determine the most important of the mass exchanges, evapotranspiration. Of all of the surface exchanges, the parameterization of evapotranspiration (E ) and the simulation of the water balance over time is the most critical.Lastly, the problems of scaling and spatial heterogeneity, the sub-grid variability of the modellers, looms as a difficult, but not insoluble, problem. It remains a critical problem however, and the detailed parameterization of the various big leaf models stands in absurd contrast to the simplistic generalization of the spatial heterogeneity of terrestrial landscapes.Plant ecologists can contribute to the task of improving the representation of vegetated landscapes within GCMs. There is need to simply and unify the way in which vegetation can be grouped at landscape scales. A classification that is based on function rather than phylogeny is required. The definition of Vegetation Functional Types (VFTs) would expedite research on both the impact of, and feedback on, climate change.     

Surface drag and turbulence over an inhomogeneous land surface

Data collected over an inhomogeneous semi-rural area are presented. The data are compared with previous surface-layer data to determine how representative the fixed-point flux measurements are of area averages. Departures from the standard surface-layer results are found to be relatively small (~10–20%), which supports the concept of a blending height above which the flow ceases to respond to variations in the underlying surface and becomes horizontally homogeneous.Effective roughness lengths are derived for different wind directions and the relationship between the effective roughness length and upwind surface is examined in the light of recent ideas on averaging surface roughness lengths. It is found that by averaging drag coefficients, realistic values of the effective roughness length can be calculated which are not very sensitive to the precise choice of the component roughness lengths.     

Diffuse radiation,twilight, and photochemistry — I

A photochemical scheme which includes a detailed treatment of multiple scattering up to solar zenith angles of 96° (developed for use in a GCM) has been used to study partitioning within chemical families. Attention is drawn to the different zenith angle dependence of diffuse radiation for the two spectral regions <310 nm and >310 nm. The effect that this has on the so-called 40 km ozone problem is discussed. The importance of correctly including multiple scattering for polar ozone studies is emphasised.