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Magnetic Position Determination by Homing Pigeons?
J. theor. Biol. (2002) 218, 47–54 doi:10.1006/yjtbi.3058, available online at http://www.idealibrary.com on
Magnetic Position Determination by Homing Pigeons? W. Ian Reilly*w wSchool of Surveying, University of Otago, Dunedin, New Zealand (Received on 5 September 2001, Accepted in revised form on 15 April 2002)
How pigeons return home from unfamiliar release sites is a long-standing puzzle in animal behaviour. Walker (1998, 1999) has described a ‘‘vector summation model’’ which ‘‘identifies a novel coordinate that pigeons could use with magnetic total intensity to determine position’’. The model is not applicable in a magnetic field generated simply by a geocentric dipole, but requires a field perturbed by higher-order sources. Tests are devised to simulate the addition of both regional and local magnetic anomalies to a geocentric dipole field, and to calculate the directions of the home loft from a number of release sites. The results indicate that a pigeon would be unlikely to derive useful information from the model. r 2002 Elsevier Science Ltd. All rights reserved.
Introduction Recent work by Diebel et al. (2000) and Walker et al. (1997) has demonstrated the presence of magnetite crystals in trout, and Diebel et al. (2000, p. 301) have concluded, inter alia, that ‘‘ythe magnetite discovered in the trout would be suitable for detecting the small changes in magnetic intensity required by a new model of magnetic position determination by homing pigeons’’. The ‘‘new model’’ is that given by Walker (1998). This model has been criticized by Wallraff (1999), not only from a comparison with the observed behaviour of homing pigeons, but also on the grounds of the unreasonably high sensitivities with which pigeons are expected to detect very small changes in the intensity of the ambient magnetic field. In his reply, Walker fails to respond to the latter criticism, and restates his view that the direction *Corresponding author. University of Otago, Dunedin, New Zealand. Tel.: +64-3-3487-253; fax: +64-3-479-7586. E-mail address: [email protected] (W.I. Reilly). 0022-5193/02/$35.00/0
of the horizontal gradient of the intensity of the ambient magnetic field is a key element, or ‘‘coordinate’’, in position determination by homing pigeons (Walker, 1999). It is therefore apposite to note some flaws in Walker’s model that have not been specifically canvassed by Wallraff.
Magnetic Field Determination It is assumed that a pigeon can determine the direction of the magnetic field vector, in the sense of the ‘‘inclination compass’’ of Wiltschko & Wiltschko (1972); see also Wiltschko & Wiltschko (1995) for a comprehensive overview. At its most elementary, this might not include the determination of the polarity of the field, a circumstance that may explain the lack of response by some animals to reversal of the applied field under experimental conditions. However, since vertebrates seem to be able to determine the direction of the vertical, a pigeon would not only be able to determine the r 2002 Elsevier Science Ltd. All rights reserved.
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W. I. REILLY
direction of magnetic north with respect to its body axes, but also the angle of inclination of the magnetic field vector to the horizontal. Field polarity can be determined in any case if the pigeon knows that the field lines dip polewards in either (magnetic) hemisphere, i.e. towards the north pole in the northern hemisphere, and towards the south pole in the southern hemisphere. Most, if not all, examples of magnetic orientation (as distinct from position determination) could be explained without the need for the determination of the intensity of the ambient field, just its direction. The Walker Model Walker (1998) bases his model on the observation of ‘‘region-wide departure of directionsyfor pigeons homing to lofts in both the USA and Europe’’ (Walker, 1998, p. 342 and Fig. 2). He identifies a regional ‘‘axis of symmetry’’ at two sites, at Ithaca, New York, and at Frankfurt, Germany, and, from a comparison with the map of smoothed magnetic isointensity contours, concludes that these axes of symmetry lie ‘‘close to the lines of maximum slope in total intensity’’ [Walker, 1998, pp. 344–345 and Fig. 1(a)]. He consequently postulates a ‘‘vector coordinate system’’ to determine position as follows: ythe direction of steepest intensity slopey with total intensity, defines a vector coordinate system by which an animal able to measure these two dimensions of the Earth’s magnetic field could determine position.
In other words, Walker proposes that a pigeon can determine (a) the intensity of the ambient magnetic field at both home and release sites, and (b) the horizontal component of the direction of the gradient of the magnetic field intensity at these sites. Given that a ‘‘coordinate’’ is usually understood to be a scalar variable, (b) can be interpreted as the angle between this direction and some reference direction, also horizontal. It is implicit in Walker’s argument that the pigeon is to remember the field intensity at the home site, and to compare it with the intensity at the release site to an accuracy of 10 nT, or about 1 part in 5000 of the ambient field (Walker, 1998, p. 342). More-
over, as the argument is developed (e.g. Walker, 1999, p. 272), it is implicit, though nowhere stated, that a pigeon would need a sensitivity to changes in the field intensity of at least an order of magnitude greater than this, so as to be able to determine the direction of the field gradient at both home and release sites. Walker further postulates that a pigeon can combine these intensities and directions into an artificial vector (which we will call the ‘‘Walker vector’’) at each of the sites, and subtract one from the other to give a ‘‘homing vector’’, which specifies the direction from the release to the home site. A mathematical formulation of this hypothesis is given in Appendix A. Ambiguities in the Walker Model It is essential in the Walker model that a pigeon is to construct two vectors, and subtract one from the other. But a pigeon can not remember a ‘‘direction’’, unless it is quantified in terms of angles that refer it to a reference frame. Which reference frames are available to the pigeon? Either the geographic latitude or the magnetic latitude (via the magnetic inclination) may be available, but in neither case the longitudeFand that, of course, is the crux of the problem. A global geocentric reference frame is thus ruled out, and the only readily accessible ones are local: the combination of the vertical with either the geographic east and north directions, or with the magnetic east and north directions. But such frames are not the same at the home and release sites: the vertical at the home site is not parallel to the vertical at the release site; and neither the north nor the east directions are parallelFunless, of course, the home and the release sites are very close to each other, in which case the Walker model is unnecessary. Walker (1998) does not discuss the problem of reference directions, so we must make some reasonable assumptions. A pigeon lives in a three-dimensional world on the curved surface of the earth, but the Walker model presumes that the pigeon navigates in a horizontal ‘‘flat-earth’’ approximation to the real world. For distances of the order of 100 km or so, this is not unreasonable. The reference direction most
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accessible to a pigeon is probably magnetic north: the Earth’s magnetic field is omnipresent, and is available day and night, under clear skies and under heavy overcast. Moreover, a pigeon can construct this direction from a combination of its ‘‘inclination compass’’ and the local vertical. These reference directions at home and release sites are therefore site dependent, and in the absence of a common reference frame, the subtraction of two ‘‘Walker vectors’’ is not therefore a valid vector operation. However, a pigeon has no option but to use local reference directions, and is required by the Walker model to perform the subtractions. A further ambiguity in the Walker model is illustrated by reference to Fig. 3(b) in Walker (1998), reproduced here as Fig. 1. Though the map projection is not stated, we can imagine that Fig. 1 is drawn on a Mercator projection aligned with the axis of the geocentric inclined dipole that makes the main contribution to the Earth’s magnetic field. For such a projection, the isointensity (F) lines of this dipole field will be straight, horizontal and parallel. Any added regional magnetic anomaly will result in the isointensity contours becoming curved, as in the figure. But, under the Walker model, a pigeon has no way of determining the curvature of the isointensity contours: they could either be concave southwards as shown in Fig. 1, or curved in the opposite direction. This problem
has already been raised, in slightly different terms, by Wallraff [1999, p. 265, Section 1(ii)]. The Ambient Magnetic Field The simplest first approximation to the Earth’s magnetic field is that it arises from a geocentric inclined dipole, which constitutes about 90% of the observed field (see, for instance, Barton, 1997). The field of a geocentric dipole has rotational symmetry about the dipole axis: as a result, neither the field vector nor any higher derivative has any component in the magnetic east-west direction. In such a field, the horizontal component of the direction of the gradient of the magnetic field intensity will always coincide with magnetic north. Following the Walker model, the derived ‘‘homing vector’’ would lie precisely in either the magnetic north or the magnetic south direction. In this case, there would be no useful prediction of the homeward direction from a release site. It must be concluded, therefore, that the Walker model could only operate in a magnetic field perturbed by regional or local magnetic anomalies. A Test Case A simple test of the Walker model is to insert some representative values in the appropriate equations and to look at the results. We will assume a home site rather similar to that of the Ithaca, NY, case, at a magnetic latitude U of 601N (see Appendix A). For a geocentric dipole field, the corresponding field intensity FH would be about 55 000 nT, and the horizontal component of the gradient of the field intensity, about 0.0034 nT m1. We will assume that the azimuth aH of the horizontal component of the gradient of the field intensity, with respect to magnetic north, is zero. The components (east, north) of the ‘‘Walker vector’’ at the home site are therefore wH ¼ ðFH sinðaH Þ; FH cosðaH ÞÞ ¼ ð0; FH Þ;
Fig. 1. Example of a vector summation for a loft and release sites located within a relatively consistent change in magnetic total intensity (cf. the region around Ithaca, NY). (Fig. 3(b) and its caption from Walker, 1998).
ð1Þ
i.e. the vector is of magnitude 55 000 nT and direction due south.
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W. I. REILLY
We now take three release sites, each 100 km from the home site, with directions in terms of magnetic north, and with field intensities calculated using the above gradient value, which is equal to a decrease of 340 nT in a southerly direction, and assign the following values:
The Case of a Regional Magnetic Anomaly
since we have assumed that aH is zero. The magnitude jhj of the homing vector in this case is
For the chosen examples, with aH ¼ 0; then aR ¼ 0 corresponds to the field of a geocentric dipole. One effect of adding a regional magnetic anomaly to the dipole field will be to cause a small divergence between the direction of the horizontal component of the gradient of the field intensity, and the direction of magnetic north. We can therefore look at the effect of a small variation of aR about a zero value. The results of such a calculation are given in Table 1, showing both the magnitude (Magn) in nanotesla (nT), and the azimuth (b) of the homing vector hj : The expected value of the homing azimuth is shown in the last row. It is seen that the predicted values of the homing azimuth b fluctuate greatly within the small range of aR between 11 and +11. At Site A, the range of b is 1401; at Site B, it is 1511. The result for Site C, aR ¼ 0; is indeterminate, since the magnitude of the homing vector is zero. As is also seen from the table, the expected value of the homing azimuth b is in fact attained at some point, at least for Sites A and B. However, such points are particularly unstable. We can define a ‘‘sensitivity index’’, giving the change in b for a unit change in aR : in other words, the derivative db=daR : Since this quantity is the ratio of two angles, it is dimensionless. However, in terms of the Walker model, the results of Tables 1 and 2 apply only to the case of isointensity contours that are concave to the south, as shown in Fig. 1. If the curvature were reversed, the sign of the calculated homing azimuth b in Table 1 would be reversed. For example, the apparent conclusion for Site C is: for aR negative, fly east; for aR positive, fly west. But if the curvature of the contour is reversed, the contrary holds. If the pigeon cannot determine the curvature, then it has only a 50% chance of making the ‘‘correct’’ choice. The results shown in Tables 1 and 2 indicate that a pigeon would not derive any useful information from the Walker model.
jhj ¼ ½F2H þ F2R 2FH FR cosðaR Þ1=2 :
The Case of a Local Magnetic Anomaly
K Site AFdue south of the home site, with a field intensity of 54 660 nT; K Site BFsouth-west of the home site, with a field intensity of 54 760 nT; K Site CFdue west of the home site, and on the same isointensity contour, i.e. 55 000 nT.
Walker (1998, p. 346) comments that ‘‘homing should be more accurate from release sites along the axis of steepest intensity slope through the loft than from release sites along the isointensity contour through the loft’’; hence, in terms of his model, Site A should give a better result than Site C. We can now proceed to calculate the azimuth b of the homing vector hj as a function of the azimuth aR of the horizontal component of the gradient of the field intensity at each of the three release sites, using eqn (A.5) (Appendix A). The components of the ‘‘Walker vector’’ at a release site are thus wR ¼ ðFR sinðaR Þ; FR cosðaR ÞÞ:
ð2Þ
The homing vector is h ¼ wR wH ¼ ðFH sinðaH Þ FR sinðaR Þ; FH cosðaH Þ FR cosðaR ÞÞ: ð3Þ The azimuth b of the homing vector is given by tan b ¼ ðFH sinðaH Þ FR sinðaR ÞÞ= FH cosðaH Þ FR cosðaR ÞÞ ¼ ðFR sinðaR ÞÞ=ðFH FR cosðaR ÞÞ ð4Þ
ð5Þ
We also assume that in these examples the pigeon determines the azimuth aR with respect to the local direction of magnetic north.
We can now assume that each of the release sites A, B and C, lies on the flank of a sharp local magnetic anomaly, such that the direction of the
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Table 1 Homing azimuths b at release sites from the Walker model: regional magnetic anomalies Site A
Site B
Site C
Gradient azim aR
Magn (nT)
Azimuth (b)
Magn(nT)
Azimuth (b)
Magn (nT)
Azimuth (b)
1.01 0.81 0.61 0.41 0.21 0.01 0.21 0.41 0.61 0.81 1.01
1020 840 670 510 390 340 390 510 670 840 1020
69.91 65.71 59.11 48.21 29.31 0.01 29.31 48.21 59.11 65.71 69.91
990 800 20 450 310 240 310 450 620 800 990
75.41 72.21 67.01 57.71 38.51 0.01 38.51 57.71 67.01 72.21 75.41
960 770 580 380 190 0 190 380 580 770 960
89.51 89.61 89.71 89.81 89.91 B 89.91 89.81 89.71 89.61 89.51
Expected value
0.01
45.01
90.01
Table 2 Sensitivity of the homing azimuth b at release sites for a unit change in the azimuth aR of the Walker vector
directionFas found by the so-called ‘‘sun compass’’Fbut they have very similar results.
Site A
Site B
01
451
In support of his model, Walker states (1998, p. 344)
Expected homing azimuth b Corresponding Walker azimuth aR ‘‘Sensitivity index’’ db=daR
0.0001 161
0.2521 226
Site C 901 B N
horizontal component of the gradient of the field intensity could lie in any azimuth aR around the circle. To avoid moving the sites, we will imagine that the local anomaly moves around the site, and show, in Table 3, the resulting homing azimuth b as a function of aR in the range from 1801 to +1801. The predicted azimuths range over the semicircle from 901 to +901, and only by chance does any one coincide with the expected direction. Not surprisingly, the magnitudes of the homing vectors also range widely. These are estimates of the distance to the home site, though expressed in terms of the difference in field intensity between home and release site. Since the pigeon is presumed to remember the field intensity at the home site, the azimuth is obviously the most important parameter to come out of the model. Analogous tests can be made using geographic north as the local reference
Measurement of the Magnetic Field Gradient
It turns out that pigeons might be able to determine the direction of steepest intensity slope quite easily. A short term running record of intensity variations experienced while flying about the release site would provide the birds with the necessary ability to determine the slope in intensityySuch an ability to scan the field of intensity variations at a release site would facilitate recognition of the direction in which the axis of steepest intensity slope lies.
This supposition has already been strongly criticized by Wallraff (1999). His examples can be extended by examining the magnitudes of the quantities involved for the geocentric dipole model. The greatest magnitude of the horizontal component of the direction of the gradient of the magnetic field intensity for the geocentric dipole is about 0.005 nT m1 for a magnetic latitude of 301. If a pigeon were to fly around a horizontal circle of 200 m diameter, it would encounter a maximum change in field intensity of 200 0:005 ¼ 1 nT. Given the nature of the diurnal variation of magnetic intensity, a change of 1 nT could easily occur while the pigeon was flying around the circle (see Appendix B for an
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W. I. REILLY
Table 3 Homing azimuths b at release sites from the Walker model: local magnetic anomalies Site A
Site B
Site C
Gradient azim aR
Magn (nT)
Azimuth (b)
Magn (nT)
Azimuth (b)
Magn (nT)
Azimuth(b)
1801 1501 1201 901 601 301 01 301 601 901 1201 1501 1801
109 105 95 77 54 28
0.01 15.01 29.91 44.81 59.71 74.31 0.01 74.31 59.71 44.81 29.91 15.01 0.01
109 106 95 77 54 28
0.01 15.01 29.91 44.91 59.81 74.51 0.01 74.51 59.81 44.91 29.91 15.01 0.01
110 106 95 77 55 28
0.01 15.01 30.01 45.01 60.01 75.01 B 75.01 60.01 45.01 30.01 15.01 0.01
28 54 77 95 105 109
700 900 000 500 800 400 300 400 800 500 000 900 700
Expected value
28 54 77 95 106 109
0.01
example). Moreover, a sensitivity of 1 nT is an order of magnitude greater than that quoted by Walker of ‘‘10 nT sensitivity to intensity changes inferred for homing pigeons’’ (Walker, 1998, p. 342). The circle would need to be 2 km in diameter even to achieve Walker’s 10 nT value. Walker has therefore produced scant evidence that pigeons can achieve the sensitivity to intensity changes, in order to determine the direction of the horizontal component of the gradient vector to any reasonable accuracy. Conclusion The model of Walker (1998, 1999) seems incapable of providing a pigeon with a reliable direction in which to fly to reach its home loft. In consequence, it fails to explain the homing behaviour of pigeons. While there may be fairly general agreement that many animals, including pigeons, can determine magnetic north, and probably also magnetic inclination, the Walker model fails to establish any evidence for, or even the need for, a knowledge of magnetic field intensity. The understanding of the navigational capabilities of animals is, for us, a formidable problem. The range of speculation is almost boundless. There is a recognizable need to proceed on the basis of a principle of parsimony, and to give priority to explanations that are founded on well-established physical principles,
800 000 100 600 900 400 200 400 900 600 100 000 800
28 55 77 95 106 110
000 300 300 800 000 500 0 500 000 800 300 300 000
45.01
90.01
and lead to feasible models, as well as according with reliable observational evidence of animal behaviour. REFERENCES Barton, C. E. (1997). International geomagnetic reference field: the seventh generation. J. Geomag. Geoelectr. 49, 123–148. Diebel, C. E., Proksch, P., Green, C. R., Neilson, P. & Walker, M. M. (2000). Magnetite defines a vertebrate magnetoreceptor. Nature 406, 299–302. Walker, M. M. (1998). On a wing and a vector: a model for magnetic navigation by homing pigeons. J. theor. Biol. 192, 341–349. Walker, M. M. (1999). Magnetic position determination by homing pigeons. J. theor. Biol. 197, 271–276. Walker, M. M., Diebel, C. E., Haugh, C. V., Pankhurst, P. M., Montgomery, J. C. & Green, C. R. (1997). Structure and function of the vertebrate magnetic sense. Nature 390, 371–376. Wallraff, H. G. (1999). The magnetic map of homing pigeons: an evergreen phantom. J. theor. Biol. 197, 265–269. Wiltschko, W. & Wiltschko, R. (1972). Magnetic compass of European robins. Science 176, 62–64. Wiltschko, R. & Wiltschko, W. (1995). Magnetic Orientation in Animals. Berlin: Springer.
Appendix A The Ambient Magnetic Field and the Walker Vector We will develop Walker’s argument using a notation more apt for vector calculus, and
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MAGNETIC POSITION DETERMINATION
denote the ambient magnetic field vector by Bj : We can decompose this vector into a scalar magnitude b and a unit vector pj Bj ¼ b pj :
ðA:1Þ
The field intensity b is also commonly denoted, in geomagnetic usage, by the total force F; the angle that the direction pj makes with the horizontal by the inclination I, and the angle that its horizontal projection makes with geographic north by the declination D. A magnetic latitude U can be defined, as an analogue of geographic latitude, in terms of the inclination I by the relationship tan U ¼ 12 tan I: Walker’s ‘‘direction of steepest intensity slope’’ is the horizontal direction of the gradient vector of the intensity b. Differentiating eqn (A.1) we obtain Bj;k ¼ bk pj þ b pj;k
ðA:2Þ
Walker’s DIR(B0 OBS)MAX), and we will denote by a the angle that qj makes with some reference direction (e.g. magnetic north, or geographic north). We can now express the ‘‘Walker vector’’ wj ; as defined earlier, by wj ¼ b qj ;
ðA:4Þ
where the negative sign allows for the convention that a gradient vector points towards an increasing value of the scalar quantity. It can be seen that wj is an ‘‘artificial’’ vector, not a real physical quantity. The argument now runs that the pigeon can determine both b and qj at its home loft P; and can therefore construct and remember the Walker vector wj ðPÞ: At the release site Q; perhaps 200 km distant, the pigeon can similarly construct the vector wj ðQÞ: The pigeon thereupon determines its homeward direction, from Q to P; given by the ‘‘homing’’ vector hj by subtraction, viz.
and, contracting eqn (A.2) with the unit vector p j ; we obtain
hj ¼ wj ðQÞ wj ðPÞ:
bk ¼ Bj;k pj ¼ grad F;
We will denote by b the azimuth of the ‘‘homing’’ vector hj : Equation (A.5) is here presented as a vector equation, but, as discussed in the main text, Walker’s application of this equation is not a valid vector operation, in the
ðA:3Þ
where bk is the intensity gradient vector. We denote by the unit vector qj ; the direction of the projection of bk onto the horizontal plane (this is
ðA:5Þ
Fig. B1. Magnetic field intensity F recorded over 3 hr (bottom to top), beginning at 0h UT on 19 January 2002. The values of F are derived from observations of H and Z recorded at 20-s intervals at the Eyrewell Magnetic Observatory.
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W. I. REILLY
absence of a reference frame common to points P and Q: There is a corresponding ambiguity in the definition of b; so we will have to assume that it applies at point Q: Appendix B An Example of Quiet-day Variation in the Magnetic Field Intensity F An example of quiet-day variation in the magnetic field intensity F is given in Fig. B1. The Eyrewell Magnetic Observatory is situated in New Zealand at latitude 431 250 2600 S,
longitude 1721 210 1600 E, magnetic latitude 521 S. The data on which Fig. 2 is based were kindly supplied by Lester Tomlinson on behalf of the Institute of Geological & Nuclear Sciences. Comparable data at 1-min intervals, from Eyrewell and from other contributing observatories, can be obtained from the World Data Centre for Geomagnetism, Kyoto: http:// swdcdb.kugi.kyoto-u.ac.jp, which has links to other relevant sites. The magnetic field intensity values plotted in Fig. B1 were for the early afternoon, local time. Some hours later a disturbed period set in, with the amplitudes of fluctuations increasing ten-fold and more.
Magnetic Position Determination by Homing Pigeons? W. Ian Reilly*w wSchool of Surveying, University of Otago, Dunedin, New Zealand (Received on 5 September 2001, Accepted in revised form on 15 April 2002)
How pigeons return home from unfamiliar release sites is a long-standing puzzle in animal behaviour. Walker (1998, 1999) has described a ‘‘vector summation model’’ which ‘‘identifies a novel coordinate that pigeons could use with magnetic total intensity to determine position’’. The model is not applicable in a magnetic field generated simply by a geocentric dipole, but requires a field perturbed by higher-order sources. Tests are devised to simulate the addition of both regional and local magnetic anomalies to a geocentric dipole field, and to calculate the directions of the home loft from a number of release sites. The results indicate that a pigeon would be unlikely to derive useful information from the model. r 2002 Elsevier Science Ltd. All rights reserved.
Introduction Recent work by Diebel et al. (2000) and Walker et al. (1997) has demonstrated the presence of magnetite crystals in trout, and Diebel et al. (2000, p. 301) have concluded, inter alia, that ‘‘ythe magnetite discovered in the trout would be suitable for detecting the small changes in magnetic intensity required by a new model of magnetic position determination by homing pigeons’’. The ‘‘new model’’ is that given by Walker (1998). This model has been criticized by Wallraff (1999), not only from a comparison with the observed behaviour of homing pigeons, but also on the grounds of the unreasonably high sensitivities with which pigeons are expected to detect very small changes in the intensity of the ambient magnetic field. In his reply, Walker fails to respond to the latter criticism, and restates his view that the direction *Corresponding author. University of Otago, Dunedin, New Zealand. Tel.: +64-3-3487-253; fax: +64-3-479-7586. E-mail address: [email protected] (W.I. Reilly). 0022-5193/02/$35.00/0
of the horizontal gradient of the intensity of the ambient magnetic field is a key element, or ‘‘coordinate’’, in position determination by homing pigeons (Walker, 1999). It is therefore apposite to note some flaws in Walker’s model that have not been specifically canvassed by Wallraff.
Magnetic Field Determination It is assumed that a pigeon can determine the direction of the magnetic field vector, in the sense of the ‘‘inclination compass’’ of Wiltschko & Wiltschko (1972); see also Wiltschko & Wiltschko (1995) for a comprehensive overview. At its most elementary, this might not include the determination of the polarity of the field, a circumstance that may explain the lack of response by some animals to reversal of the applied field under experimental conditions. However, since vertebrates seem to be able to determine the direction of the vertical, a pigeon would not only be able to determine the r 2002 Elsevier Science Ltd. All rights reserved.
48
W. I. REILLY
direction of magnetic north with respect to its body axes, but also the angle of inclination of the magnetic field vector to the horizontal. Field polarity can be determined in any case if the pigeon knows that the field lines dip polewards in either (magnetic) hemisphere, i.e. towards the north pole in the northern hemisphere, and towards the south pole in the southern hemisphere. Most, if not all, examples of magnetic orientation (as distinct from position determination) could be explained without the need for the determination of the intensity of the ambient field, just its direction. The Walker Model Walker (1998) bases his model on the observation of ‘‘region-wide departure of directionsyfor pigeons homing to lofts in both the USA and Europe’’ (Walker, 1998, p. 342 and Fig. 2). He identifies a regional ‘‘axis of symmetry’’ at two sites, at Ithaca, New York, and at Frankfurt, Germany, and, from a comparison with the map of smoothed magnetic isointensity contours, concludes that these axes of symmetry lie ‘‘close to the lines of maximum slope in total intensity’’ [Walker, 1998, pp. 344–345 and Fig. 1(a)]. He consequently postulates a ‘‘vector coordinate system’’ to determine position as follows: ythe direction of steepest intensity slopey with total intensity, defines a vector coordinate system by which an animal able to measure these two dimensions of the Earth’s magnetic field could determine position.
In other words, Walker proposes that a pigeon can determine (a) the intensity of the ambient magnetic field at both home and release sites, and (b) the horizontal component of the direction of the gradient of the magnetic field intensity at these sites. Given that a ‘‘coordinate’’ is usually understood to be a scalar variable, (b) can be interpreted as the angle between this direction and some reference direction, also horizontal. It is implicit in Walker’s argument that the pigeon is to remember the field intensity at the home site, and to compare it with the intensity at the release site to an accuracy of 10 nT, or about 1 part in 5000 of the ambient field (Walker, 1998, p. 342). More-
over, as the argument is developed (e.g. Walker, 1999, p. 272), it is implicit, though nowhere stated, that a pigeon would need a sensitivity to changes in the field intensity of at least an order of magnitude greater than this, so as to be able to determine the direction of the field gradient at both home and release sites. Walker further postulates that a pigeon can combine these intensities and directions into an artificial vector (which we will call the ‘‘Walker vector’’) at each of the sites, and subtract one from the other to give a ‘‘homing vector’’, which specifies the direction from the release to the home site. A mathematical formulation of this hypothesis is given in Appendix A. Ambiguities in the Walker Model It is essential in the Walker model that a pigeon is to construct two vectors, and subtract one from the other. But a pigeon can not remember a ‘‘direction’’, unless it is quantified in terms of angles that refer it to a reference frame. Which reference frames are available to the pigeon? Either the geographic latitude or the magnetic latitude (via the magnetic inclination) may be available, but in neither case the longitudeFand that, of course, is the crux of the problem. A global geocentric reference frame is thus ruled out, and the only readily accessible ones are local: the combination of the vertical with either the geographic east and north directions, or with the magnetic east and north directions. But such frames are not the same at the home and release sites: the vertical at the home site is not parallel to the vertical at the release site; and neither the north nor the east directions are parallelFunless, of course, the home and the release sites are very close to each other, in which case the Walker model is unnecessary. Walker (1998) does not discuss the problem of reference directions, so we must make some reasonable assumptions. A pigeon lives in a three-dimensional world on the curved surface of the earth, but the Walker model presumes that the pigeon navigates in a horizontal ‘‘flat-earth’’ approximation to the real world. For distances of the order of 100 km or so, this is not unreasonable. The reference direction most
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MAGNETIC POSITION DETERMINATION
accessible to a pigeon is probably magnetic north: the Earth’s magnetic field is omnipresent, and is available day and night, under clear skies and under heavy overcast. Moreover, a pigeon can construct this direction from a combination of its ‘‘inclination compass’’ and the local vertical. These reference directions at home and release sites are therefore site dependent, and in the absence of a common reference frame, the subtraction of two ‘‘Walker vectors’’ is not therefore a valid vector operation. However, a pigeon has no option but to use local reference directions, and is required by the Walker model to perform the subtractions. A further ambiguity in the Walker model is illustrated by reference to Fig. 3(b) in Walker (1998), reproduced here as Fig. 1. Though the map projection is not stated, we can imagine that Fig. 1 is drawn on a Mercator projection aligned with the axis of the geocentric inclined dipole that makes the main contribution to the Earth’s magnetic field. For such a projection, the isointensity (F) lines of this dipole field will be straight, horizontal and parallel. Any added regional magnetic anomaly will result in the isointensity contours becoming curved, as in the figure. But, under the Walker model, a pigeon has no way of determining the curvature of the isointensity contours: they could either be concave southwards as shown in Fig. 1, or curved in the opposite direction. This problem
has already been raised, in slightly different terms, by Wallraff [1999, p. 265, Section 1(ii)]. The Ambient Magnetic Field The simplest first approximation to the Earth’s magnetic field is that it arises from a geocentric inclined dipole, which constitutes about 90% of the observed field (see, for instance, Barton, 1997). The field of a geocentric dipole has rotational symmetry about the dipole axis: as a result, neither the field vector nor any higher derivative has any component in the magnetic east-west direction. In such a field, the horizontal component of the direction of the gradient of the magnetic field intensity will always coincide with magnetic north. Following the Walker model, the derived ‘‘homing vector’’ would lie precisely in either the magnetic north or the magnetic south direction. In this case, there would be no useful prediction of the homeward direction from a release site. It must be concluded, therefore, that the Walker model could only operate in a magnetic field perturbed by regional or local magnetic anomalies. A Test Case A simple test of the Walker model is to insert some representative values in the appropriate equations and to look at the results. We will assume a home site rather similar to that of the Ithaca, NY, case, at a magnetic latitude U of 601N (see Appendix A). For a geocentric dipole field, the corresponding field intensity FH would be about 55 000 nT, and the horizontal component of the gradient of the field intensity, about 0.0034 nT m1. We will assume that the azimuth aH of the horizontal component of the gradient of the field intensity, with respect to magnetic north, is zero. The components (east, north) of the ‘‘Walker vector’’ at the home site are therefore wH ¼ ðFH sinðaH Þ; FH cosðaH ÞÞ ¼ ð0; FH Þ;
Fig. 1. Example of a vector summation for a loft and release sites located within a relatively consistent change in magnetic total intensity (cf. the region around Ithaca, NY). (Fig. 3(b) and its caption from Walker, 1998).
ð1Þ
i.e. the vector is of magnitude 55 000 nT and direction due south.
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W. I. REILLY
We now take three release sites, each 100 km from the home site, with directions in terms of magnetic north, and with field intensities calculated using the above gradient value, which is equal to a decrease of 340 nT in a southerly direction, and assign the following values:
The Case of a Regional Magnetic Anomaly
since we have assumed that aH is zero. The magnitude jhj of the homing vector in this case is
For the chosen examples, with aH ¼ 0; then aR ¼ 0 corresponds to the field of a geocentric dipole. One effect of adding a regional magnetic anomaly to the dipole field will be to cause a small divergence between the direction of the horizontal component of the gradient of the field intensity, and the direction of magnetic north. We can therefore look at the effect of a small variation of aR about a zero value. The results of such a calculation are given in Table 1, showing both the magnitude (Magn) in nanotesla (nT), and the azimuth (b) of the homing vector hj : The expected value of the homing azimuth is shown in the last row. It is seen that the predicted values of the homing azimuth b fluctuate greatly within the small range of aR between 11 and +11. At Site A, the range of b is 1401; at Site B, it is 1511. The result for Site C, aR ¼ 0; is indeterminate, since the magnitude of the homing vector is zero. As is also seen from the table, the expected value of the homing azimuth b is in fact attained at some point, at least for Sites A and B. However, such points are particularly unstable. We can define a ‘‘sensitivity index’’, giving the change in b for a unit change in aR : in other words, the derivative db=daR : Since this quantity is the ratio of two angles, it is dimensionless. However, in terms of the Walker model, the results of Tables 1 and 2 apply only to the case of isointensity contours that are concave to the south, as shown in Fig. 1. If the curvature were reversed, the sign of the calculated homing azimuth b in Table 1 would be reversed. For example, the apparent conclusion for Site C is: for aR negative, fly east; for aR positive, fly west. But if the curvature of the contour is reversed, the contrary holds. If the pigeon cannot determine the curvature, then it has only a 50% chance of making the ‘‘correct’’ choice. The results shown in Tables 1 and 2 indicate that a pigeon would not derive any useful information from the Walker model.
jhj ¼ ½F2H þ F2R 2FH FR cosðaR Þ1=2 :
The Case of a Local Magnetic Anomaly
K Site AFdue south of the home site, with a field intensity of 54 660 nT; K Site BFsouth-west of the home site, with a field intensity of 54 760 nT; K Site CFdue west of the home site, and on the same isointensity contour, i.e. 55 000 nT.
Walker (1998, p. 346) comments that ‘‘homing should be more accurate from release sites along the axis of steepest intensity slope through the loft than from release sites along the isointensity contour through the loft’’; hence, in terms of his model, Site A should give a better result than Site C. We can now proceed to calculate the azimuth b of the homing vector hj as a function of the azimuth aR of the horizontal component of the gradient of the field intensity at each of the three release sites, using eqn (A.5) (Appendix A). The components of the ‘‘Walker vector’’ at a release site are thus wR ¼ ðFR sinðaR Þ; FR cosðaR ÞÞ:
ð2Þ
The homing vector is h ¼ wR wH ¼ ðFH sinðaH Þ FR sinðaR Þ; FH cosðaH Þ FR cosðaR ÞÞ: ð3Þ The azimuth b of the homing vector is given by tan b ¼ ðFH sinðaH Þ FR sinðaR ÞÞ= FH cosðaH Þ FR cosðaR ÞÞ ¼ ðFR sinðaR ÞÞ=ðFH FR cosðaR ÞÞ ð4Þ
ð5Þ
We also assume that in these examples the pigeon determines the azimuth aR with respect to the local direction of magnetic north.
We can now assume that each of the release sites A, B and C, lies on the flank of a sharp local magnetic anomaly, such that the direction of the
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Table 1 Homing azimuths b at release sites from the Walker model: regional magnetic anomalies Site A
Site B
Site C
Gradient azim aR
Magn (nT)
Azimuth (b)
Magn(nT)
Azimuth (b)
Magn (nT)
Azimuth (b)
1.01 0.81 0.61 0.41 0.21 0.01 0.21 0.41 0.61 0.81 1.01
1020 840 670 510 390 340 390 510 670 840 1020
69.91 65.71 59.11 48.21 29.31 0.01 29.31 48.21 59.11 65.71 69.91
990 800 20 450 310 240 310 450 620 800 990
75.41 72.21 67.01 57.71 38.51 0.01 38.51 57.71 67.01 72.21 75.41
960 770 580 380 190 0 190 380 580 770 960
89.51 89.61 89.71 89.81 89.91 B 89.91 89.81 89.71 89.61 89.51
Expected value
0.01
45.01
90.01
Table 2 Sensitivity of the homing azimuth b at release sites for a unit change in the azimuth aR of the Walker vector
directionFas found by the so-called ‘‘sun compass’’Fbut they have very similar results.
Site A
Site B
01
451
In support of his model, Walker states (1998, p. 344)
Expected homing azimuth b Corresponding Walker azimuth aR ‘‘Sensitivity index’’ db=daR
0.0001 161
0.2521 226
Site C 901 B N
horizontal component of the gradient of the field intensity could lie in any azimuth aR around the circle. To avoid moving the sites, we will imagine that the local anomaly moves around the site, and show, in Table 3, the resulting homing azimuth b as a function of aR in the range from 1801 to +1801. The predicted azimuths range over the semicircle from 901 to +901, and only by chance does any one coincide with the expected direction. Not surprisingly, the magnitudes of the homing vectors also range widely. These are estimates of the distance to the home site, though expressed in terms of the difference in field intensity between home and release site. Since the pigeon is presumed to remember the field intensity at the home site, the azimuth is obviously the most important parameter to come out of the model. Analogous tests can be made using geographic north as the local reference
Measurement of the Magnetic Field Gradient
It turns out that pigeons might be able to determine the direction of steepest intensity slope quite easily. A short term running record of intensity variations experienced while flying about the release site would provide the birds with the necessary ability to determine the slope in intensityySuch an ability to scan the field of intensity variations at a release site would facilitate recognition of the direction in which the axis of steepest intensity slope lies.
This supposition has already been strongly criticized by Wallraff (1999). His examples can be extended by examining the magnitudes of the quantities involved for the geocentric dipole model. The greatest magnitude of the horizontal component of the direction of the gradient of the magnetic field intensity for the geocentric dipole is about 0.005 nT m1 for a magnetic latitude of 301. If a pigeon were to fly around a horizontal circle of 200 m diameter, it would encounter a maximum change in field intensity of 200 0:005 ¼ 1 nT. Given the nature of the diurnal variation of magnetic intensity, a change of 1 nT could easily occur while the pigeon was flying around the circle (see Appendix B for an
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W. I. REILLY
Table 3 Homing azimuths b at release sites from the Walker model: local magnetic anomalies Site A
Site B
Site C
Gradient azim aR
Magn (nT)
Azimuth (b)
Magn (nT)
Azimuth (b)
Magn (nT)
Azimuth(b)
1801 1501 1201 901 601 301 01 301 601 901 1201 1501 1801
109 105 95 77 54 28
0.01 15.01 29.91 44.81 59.71 74.31 0.01 74.31 59.71 44.81 29.91 15.01 0.01
109 106 95 77 54 28
0.01 15.01 29.91 44.91 59.81 74.51 0.01 74.51 59.81 44.91 29.91 15.01 0.01
110 106 95 77 55 28
0.01 15.01 30.01 45.01 60.01 75.01 B 75.01 60.01 45.01 30.01 15.01 0.01
28 54 77 95 105 109
700 900 000 500 800 400 300 400 800 500 000 900 700
Expected value
28 54 77 95 106 109
0.01
example). Moreover, a sensitivity of 1 nT is an order of magnitude greater than that quoted by Walker of ‘‘10 nT sensitivity to intensity changes inferred for homing pigeons’’ (Walker, 1998, p. 342). The circle would need to be 2 km in diameter even to achieve Walker’s 10 nT value. Walker has therefore produced scant evidence that pigeons can achieve the sensitivity to intensity changes, in order to determine the direction of the horizontal component of the gradient vector to any reasonable accuracy. Conclusion The model of Walker (1998, 1999) seems incapable of providing a pigeon with a reliable direction in which to fly to reach its home loft. In consequence, it fails to explain the homing behaviour of pigeons. While there may be fairly general agreement that many animals, including pigeons, can determine magnetic north, and probably also magnetic inclination, the Walker model fails to establish any evidence for, or even the need for, a knowledge of magnetic field intensity. The understanding of the navigational capabilities of animals is, for us, a formidable problem. The range of speculation is almost boundless. There is a recognizable need to proceed on the basis of a principle of parsimony, and to give priority to explanations that are founded on well-established physical principles,
800 000 100 600 900 400 200 400 900 600 100 000 800
28 55 77 95 106 110
000 300 300 800 000 500 0 500 000 800 300 300 000
45.01
90.01
and lead to feasible models, as well as according with reliable observational evidence of animal behaviour. REFERENCES Barton, C. E. (1997). International geomagnetic reference field: the seventh generation. J. Geomag. Geoelectr. 49, 123–148. Diebel, C. E., Proksch, P., Green, C. R., Neilson, P. & Walker, M. M. (2000). Magnetite defines a vertebrate magnetoreceptor. Nature 406, 299–302. Walker, M. M. (1998). On a wing and a vector: a model for magnetic navigation by homing pigeons. J. theor. Biol. 192, 341–349. Walker, M. M. (1999). Magnetic position determination by homing pigeons. J. theor. Biol. 197, 271–276. Walker, M. M., Diebel, C. E., Haugh, C. V., Pankhurst, P. M., Montgomery, J. C. & Green, C. R. (1997). Structure and function of the vertebrate magnetic sense. Nature 390, 371–376. Wallraff, H. G. (1999). The magnetic map of homing pigeons: an evergreen phantom. J. theor. Biol. 197, 265–269. Wiltschko, W. & Wiltschko, R. (1972). Magnetic compass of European robins. Science 176, 62–64. Wiltschko, R. & Wiltschko, W. (1995). Magnetic Orientation in Animals. Berlin: Springer.
Appendix A The Ambient Magnetic Field and the Walker Vector We will develop Walker’s argument using a notation more apt for vector calculus, and
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MAGNETIC POSITION DETERMINATION
denote the ambient magnetic field vector by Bj : We can decompose this vector into a scalar magnitude b and a unit vector pj Bj ¼ b pj :
ðA:1Þ
The field intensity b is also commonly denoted, in geomagnetic usage, by the total force F; the angle that the direction pj makes with the horizontal by the inclination I, and the angle that its horizontal projection makes with geographic north by the declination D. A magnetic latitude U can be defined, as an analogue of geographic latitude, in terms of the inclination I by the relationship tan U ¼ 12 tan I: Walker’s ‘‘direction of steepest intensity slope’’ is the horizontal direction of the gradient vector of the intensity b. Differentiating eqn (A.1) we obtain Bj;k ¼ bk pj þ b pj;k
ðA:2Þ
Walker’s DIR(B0 OBS)MAX), and we will denote by a the angle that qj makes with some reference direction (e.g. magnetic north, or geographic north). We can now express the ‘‘Walker vector’’ wj ; as defined earlier, by wj ¼ b qj ;
ðA:4Þ
where the negative sign allows for the convention that a gradient vector points towards an increasing value of the scalar quantity. It can be seen that wj is an ‘‘artificial’’ vector, not a real physical quantity. The argument now runs that the pigeon can determine both b and qj at its home loft P; and can therefore construct and remember the Walker vector wj ðPÞ: At the release site Q; perhaps 200 km distant, the pigeon can similarly construct the vector wj ðQÞ: The pigeon thereupon determines its homeward direction, from Q to P; given by the ‘‘homing’’ vector hj by subtraction, viz.
and, contracting eqn (A.2) with the unit vector p j ; we obtain
hj ¼ wj ðQÞ wj ðPÞ:
bk ¼ Bj;k pj ¼ grad F;
We will denote by b the azimuth of the ‘‘homing’’ vector hj : Equation (A.5) is here presented as a vector equation, but, as discussed in the main text, Walker’s application of this equation is not a valid vector operation, in the
ðA:3Þ
where bk is the intensity gradient vector. We denote by the unit vector qj ; the direction of the projection of bk onto the horizontal plane (this is
ðA:5Þ
Fig. B1. Magnetic field intensity F recorded over 3 hr (bottom to top), beginning at 0h UT on 19 January 2002. The values of F are derived from observations of H and Z recorded at 20-s intervals at the Eyrewell Magnetic Observatory.
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W. I. REILLY
absence of a reference frame common to points P and Q: There is a corresponding ambiguity in the definition of b; so we will have to assume that it applies at point Q: Appendix B An Example of Quiet-day Variation in the Magnetic Field Intensity F An example of quiet-day variation in the magnetic field intensity F is given in Fig. B1. The Eyrewell Magnetic Observatory is situated in New Zealand at latitude 431 250 2600 S,
longitude 1721 210 1600 E, magnetic latitude 521 S. The data on which Fig. 2 is based were kindly supplied by Lester Tomlinson on behalf of the Institute of Geological & Nuclear Sciences. Comparable data at 1-min intervals, from Eyrewell and from other contributing observatories, can be obtained from the World Data Centre for Geomagnetism, Kyoto: http:// swdcdb.kugi.kyoto-u.ac.jp, which has links to other relevant sites. The magnetic field intensity values plotted in Fig. B1 were for the early afternoon, local time. Some hours later a disturbed period set in, with the amplitudes of fluctuations increasing ten-fold and more.