Module 1: Celestial Compasses — Sun, Stars and Polarised Light
Migrating animals can reach a specific wintering ground 10 000 km away without any instrument more sophisticated than their own eyes, an internal clock and a cerebellum. This module builds the physics and neuroscience of the three principal celestial compass systems: the Kramer time-compensated sun compass (1950), the Emlen star-pattern compass (1967, 1975) and the polarised-light compass operating on the Rayleigh sky at sunset. We derive the Mueller calculus of skylight polarisation, the 15°/h deflection law, and the role of internal clocks in compass operation.
1. The Emlen funnel and Kramer cage
Quantitative ornithological work on orientation begins with the two instruments that turned captive nocturnal restlessness into directional data. Gustav Kramer’s orientation cage (1949, 1950) was a circular arena with radially arranged perches that allowed a European starling to indicate its preferred heading by the directional distribution of hopping. Stephen Emlen’s funnel (1967, Auk) replaced perches with an ink-pad floor and paper-lined funnel walls, so that every foot-print left a visible record of the bird’s attempts to escape in its preferred direction. Both cages permit statistical tests of mean vector length and angular concentration against the null hypothesis of random orientation (Rayleigh’s circular uniformity test).
\[ \bar{\mathbf r} = \frac{1}{n}\sum_{k=1}^{n} e^{i\theta_k}\,,\quad R = |\bar{\mathbf r}|\,,\quad Z = nR^2 \]
Rayleigh resultant length \(R\) and test statistic \(Z\). Under the null, \(2Z\sim\chi^2_2\).
Kramer’s first striking observation was that caged starlings orient relative to the position of the sun, and that if the sun is replaced by a stationary mirror image, the birds orient relative to the reflected sun. When the stationary sun is displaced in azimuth by a fixed angle \(\Delta\), the birds also rotate their preferred heading by \(\Delta\). The sun compass is genuine, not merely a response to phototactic bias. Emlen (1967) extended the method to nocturnal migrants using planetarium stars and, in his classic 1975 paper on indigo buntings, mapped the rotational axis of the northern sky onto the bird’s learned stellar template.
What a compass is not
A compass supplies heading only, not position. Compasses cannot distinguish an eastern from a western displacement. To perform true navigation—returning to a known point after displacement—birds must also have access to a map signal such as magnetic inclination, infrasound gradients, olfactory plumes, or celestial rotational poles. Our subject in this module is exclusively the compass layer.
2. Kramer’s time-compensated sun compass
The apparent azimuth of the sun changes with time of day, local latitude and season. For a bird to use the sun as a compass, it must correct for the sun’s apparent motion using an internal clock. Kramer (1950, 1952) demonstrated this with the now-classical clock-shift experiment: starlings held in a constant-photoperiod room with lights turned on and off 6 hours earlier than local time shifted their heading by 90° when released under the true sun, because their internal clock believed noon had already passed.
Solar azimuth formulae
Solar azimuth \(A_\odot(t)\) and altitude \(h_\odot(t)\) depend on latitude \(\varphi\), declination \(\delta(t)\) and hour angle \(H = 15^\circ(t_{\text{sol}} - 12)\):
\[ \sin h_\odot = \sin\varphi\sin\delta + \cos\varphi\cos\delta\cos H \]
\[ \tan A_\odot = \frac{-\cos\delta\sin H}{\cos\varphi\sin\delta - \sin\varphi\cos\delta\cos H} \]
At local solar noon \(H = 0\) and \(A_\odot\) points due south in the northern hemisphere (and due north in the southern). Between sunrise and sunset the azimuth sweeps clockwise over \(\approx 180^\circ\) at 45° latitude.
The 15°/hour deflection law
If the bird’s internal clock is phase-shifted by \(\Delta\varphi\) (hours), it expects the sun at azimuth \(A_\odot(t + \Delta\varphi)\). Near solar noon \(dA_\odot/dt \approx 15^\circ/\text{h}\)(exactly at the equator, more at high latitudes), so the heading error is
\[ \Delta\theta = A_\odot(t) - A_\odot(t + \Delta\varphi) \approx -15^\circ\,\Delta\varphi \]
the signature linear signature of Kramer’s clock-shift experiments.
The deflection is not constant; it depends on solar altitude. At high latitudes near the summer solstice \(dA_\odot/dt \) can exceed 30°/hour around solar noon, and a bird that uses a linear clock correction introduces a systematic error. Experiments show that birds use a curvilinear internal representation of azimuth, encoded via pineal melatonin and suprachiasmatic neuronal rhythms; see Gwinner (2003) and Cassone & Menaker (1984).
Sun azimuth and clock-shifted heading
3. Emlen’s star-pattern compass
Unlike the sun, stars do not rotate in azimuth uniformly: the celestial pole is fixed in the sky (to within precession), and the whole celestial sphere appears to rotate around it at 15°/hour. Emlen (1967, 1975) showed that indigo buntings (Passerina cyanea) identify the northern celestial pole by learning the centre of rotation of the visible star pattern during their first summer. The compass does not depend on the identity of any single constellation; any planetarium pattern that rotates about a given point is accepted as “north”.
The rotating planetarium experiment
Emlen (1970) raised buntings under a planetarium rotating around Betelgeuse (in Orion) rather than Polaris. The birds treated Betelgeuse as “north” and oriented southward by reference to the visual sky pattern around it. In a separate experiment he raised buntings under a fixed, non-rotating planetarium; these birds failed to orient at all, demonstrating that rotation itself is the cue.
No time-compensation needed
A key advantage of the star compass is that it requires no time-compensation. Since the identified celestial pole is fixed, a bird reading its heading from the pole is insensitive to clock drift. This is why birds shifted in photoperiod but allowed to see the natural stars show no heading deflection, while those shown only the sun do (Emlen 1967).
\[ \omega_{\text{sky}} = 2\pi/T_{\text{sidereal}} = 7.292\times 10^{-5}\,\text{rad/s} \]
Earth’s sidereal rotation rate. The celestial pole is the unique point with \(\omega = 0\).
To learn the pole, a young bird needs only to identify the locus of minimum angular velocity in the night sky. Neuro-computationally, this can be done by integrating retinal flow across multiple nights with an attention bias toward the brightest near-polar stars. Emlen’s work laid out this operational model long before computational ethology formalised it.
Celestial rotation around the pole
4. The polarised-light compass
Skylight is partially linearly polarised through Rayleigh scattering of sunlight by atmospheric molecules. Lord Rayleigh (J. W. Strutt, 1871) showed that the degree of polarisation \(P(\gamma)\) at scattering angle \(\gamma\) from the sun is
\[ P(\gamma) = P_{\max}\,\frac{\sin^2\gamma}{1 + \cos^2\gamma}\,,\quad P_{\max}\approx 0.85 \]
The degree of polarization is maximal along the great-circle band perpendicular to the sun.
At sunset this maximum-polarisation band passes through the zenith and the antisolar horizon. Its orientation defines a geographic axis independent of any magnetic or stellar reference. Able & Able (1993) and Muheim, Moore & Phillips (2006) showed that both Savannah sparrows and European robins use this band to calibrate the magnetic and stellar compasses at twilight.
Stokes vectors and Mueller calculus
The full state of partial polarisation is encoded in the four-component Stokes vector \(\mathbf S = (I, Q, U, V)^\top\). For skylight \(V\approx 0\) (no circular polarisation) and the degree of polarisation is \(P = \sqrt{Q^2+U^2}/I\). The E-vector angle is
\[ \chi = \tfrac{1}{2}\arctan(U/Q) \]
A biological analyser—an ommatidial rhabdom in a dragonfly, or an ordered cryptochrome layer in a vertebrate retina—acts as an imperfect linear polaroid whose transmission axis is at angle \(\phi_a\). Mueller calculus gives the transmitted Stokes vector as \(\mathbf S' = M_{\text{pol}}(\phi_a)\,\mathbf S\) with transmitted intensity (Malus’s law for partially polarised light)
\[ I' = \tfrac{1}{2}\,I\bigl[1 + P\cos 2(\phi_a - \chi)\bigr] \]
Rotation of the analyser modulates intensity by a factor \(P\), and a bird rotating its head scans this modulation to extract \(\chi\).
Cryptoflavin, peropsin and the retinal polarisation receptor
The identity of the avian polarisation analyser remains debated. Muheim (2011) and Möller et al. (2004) argue that specialised double-cone photoreceptors in the ultraviolet-blue range transduce polarised light via the oriented dipole of an opsin pigment. Candidate molecules include cryptoflavin (a flavin-based blue-sensitive pigment distinct from the cryptochromes of the magnetic compass) and peropsin, which is expressed in the retinal pigment epithelium and has been linked to non-visual photoentrainment. In dragonflies and bees the polarisation compass is solidly known to involve the dorsal rim area (DRA) of the compound eye.
Sunset calibration
The polarisation compass is most reliable in the 30 minutes around sunset, when the maximum-polarisation band traverses the zenith. The bird rotates its head so that the polaroid transmission axis is aligned with the band (maximising intensity), and this fixes the east-west axis to within a few degrees. Field experiments with depolarising filters over the sunset sky disrupt compass orientation in both European and Nearctic passerines, confirming the causal role of the E-vector cue.
5. Polarisation compasses across taxa
Polarisation vision is ancient. Insects exploit it via the dorsal rim area, a dedicated patch of ommatidia at the top of the compound eye whose rhabdomeres contain highly aligned rhodopsin microvilli acting as narrow-bandwidth linear polaroids. In desert ants (Cataglyphis, Wehner 2003) the polarisation compass is the principal navigational input for path integration. In migratory dragonflies (Anax junius, Pantala flavescens, May et al. 2017) it orients flights across open water where no other cue is available.
Monarch and desert locust comparisons
The monarch butterfly (Danaus plexippus) possesses both a time-compensated sun compass (Reppert 2006, 2010) and a polarisation compass via its DRA. Module 6 develops the monarch biochemistry in detail. The desert locust uses its DRA to stabilise swarm direction during mass migratory flights—a neural circuit now mapped to the central complex, the insect analogue of the mammalian hippocampus.
Multimodal integration
Real migratory animals rarely use a single compass. Experimental work (reviewed by Muheim 2011; Chernetsov 2015) indicates a hierarchy: in most passerines studied, the sunset polarisation compass calibrates the magnetic compass, which in turn provides direction through the night until dawn recalibrates against the sun. The star compass is used on clear nights by some species but not others. Cue conflicts are resolved in species- specific ways: European robins follow magnetic cues in dim light, while Savannah sparrows follow polarisation.
Cue hierarchy through the diel cycle
6. The circadian foundation of the sun compass
The sun compass is only as accurate as the clock that corrects it. In birds, the suprachiasmatic nucleus and the pineal gland together generate a circadian rhythm in melatonin that entrains to the local photoperiod. Clock-shift experiments (e.g. Schmidt-Koenig 1958 for pigeons) produce the Kramer 15°/h deflection with remarkable fidelity, proving that the correction is computed from an internal time signal rather than read instantaneously from the sun’s altitude.
Melatonin and compass
Lesion and transplant studies in pigeons and starlings place the sun-compass clock at least partly in the pineal: replacing a resident pigeon’s pineal with one from a clock-shifted donor transfers the compass deflection to the host (Cassone & Menaker 1984). Cryptochromes are also present in the avian retina and pineal, and may subserve both magnetoreception (M2) and photoentrainment of the circadian oscillator.
Latitudinal azimuth rate
The instantaneous solar azimuth rate \(dA_\odot/dt\) varies with latitude. At the equator it is \(15^\circ/\text{h}\) year-round; at 60°N in midsummer it exceeds \(30^\circ/\text{h}\) around noon. A bird using a fixed-slope clock correction accumulates a systematic error that is only avoided by a nonlinear internal representation of azimuth. Empirically, pigeons trained in the temperate zone maintain accurate homing as far north as 65°N, implying some form of latitudinal calibration.
\[ \frac{dA_\odot}{dt} = \frac{15^\circ\cos\delta\cos h_\odot}{\cos^2 h_\odot - \sin^2\varphi\cos^2\delta}\quad\text{[deg/h]} \]
Instantaneous solar azimuth rate; reduces to 15°/h at the equator.
7. Ontogeny of compass templates
Each compass in the avian toolbox has a different developmental origin. The magnetic compass (M2) appears to be innate: hand-raised buntings and blackcaps orient by magnetic cues without prior exposure. The sun compass requires learned linking of the sun’s apparent motion to the circadian clock during the first weeks post-fledging. The star compass requires actual rotational experience of the night sky during a sensitive period before the first autumn migration (Emlen 1970). And the polarisation compass is learned at sunset by viewing the natural E-vector field.
This sensitive-period scaffolding implies that compasses are not a single innate module but a layered, partly learned, partly innate inventory that the bird assembles during its first year. It also explains why young birds often follow an inappropriate heading on their first migration when raised in aberrant light environments, while experienced adults maintain fidelity to familiar stopover sites year after year.
Simulation 1: Rayleigh sky polarization E-vector pattern at sunset
We implement the single-scattering Rayleigh theory, compute the degree of polarization \(P(\gamma)\) and the E-vector angle \(\chi\) over the upper hemisphere with the sun at 93° zenith (just below the horizon), and display the classical figure-eight band of high polarization perpendicular to the solar meridian. We then apply Mueller calculus to an imperfect polaroid analyser and recover the Malus’s-law rotation curve that a bird’s head-tilt must maximise.
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Code will be executed with Python 3 on the server
Simulation 2: Time-compensated sun compass error propagation
A bird at 45°N targets magnetic south using the sun compass. We compute actual solar azimuth across the day, then propagate clock-drift offsets of 0, +1, +3 and +6 hours and recover heading deflections that obey the Kramer 15°/h law. A radial plot shows the distribution of compass directions reported by clock-shifted individuals, matching the classical pigeon-release results of Schmidt-Koenig (1958).
Click Run to execute the Python code
Code will be executed with Python 3 on the server
8. Synthesis
Every migratory animal studied to date employs multiple compasses. The three celestial compasses—sun, stars, polarisation—constitute a redundant system whose members cross-calibrate each other during the sensitive twilight window. None of them alone would be sufficient across the full diel cycle, under variable cloud cover, or across the full latitudinal range of the annual journey. Combined with the geomagnetic compass (M2), this network enables reliable great-circle headings under a wide range of environmental conditions.
Unresolved questions include: (i) the precise identity of the avian polarisation receptor and its wavelength tuning, (ii) the neural representation of the learned star-pattern centre of rotation, and (iii) the degree to which cue-conflict hierarchies differ between Nearctic-Neotropical and Palearctic-African migrants. Module 2 turns to the quantum biology of the most mysterious compass of all—the magnetic sense.
Key references
• Able, K. P. & Able, M. A. (1993). Daytime calibration of magnetic orientation in a migratory bird requires a view of skylight polarization. Nature, 364, 523–525.
• Cassone, V. M. & Menaker, M. (1984). Is the avian circadian system a neuroendocrine loop? J. Exp. Zool., 232, 539–549.
• Chernetsov, N. (2015). Compass systems of songbirds: a review. J. Comp. Physiol. A, 201, 681–691.
• Emlen, S. T. (1967). Migratory orientation in the Indigo Bunting, Passerina cyanea. I. Evidence for use of celestial cues. Auk, 84, 309–342.
• Emlen, S. T. (1970). Celestial rotation: its importance in the development of migratory orientation. Science, 170, 1198–1201.
• Emlen, S. T. (1975). The stellar-orientation system of a migratory bird. Scientific American, 233(2), 102–111.
• Gwinner, E. (2003). Circannual rhythms in birds. Current Opinion in Neurobiology, 13, 770–778.
• Hoffmann, K. (1954). Versuche zu der im Richtungsfinden der Vögel enthaltenen Zeitschätzung. Z. Tierpsychol., 11, 453–475.
• Kramer, G. (1950). Weitere Analyse der Faktoren, welche die Zugaktivität des gekäfigten Vogels orientieren. Naturwissenschaften, 37, 377–378.
• May, M. L. (2017). Dragonfly migration. Annu. Rev. Entomol., 58, 47–68.
• Möller, A., Sagasser, S., Wiltschko, W. & Schierwater, B. (2004). Retinal cryptochrome in a migratory passerine bird: a possible transducer for the avian magnetic compass. Naturwissenschaften, 91, 585–588.
• Muheim, R., Moore, F. R. & Phillips, J. B. (2006). Calibration of magnetic and celestial compass cues in migratory birds—a review of cue-conflict experiments. J. Exp. Biol., 209, 2–17.
• Muheim, R. (2011). Behavioural and physiological mechanisms of polarized light sensitivity in birds. Phil. Trans. R. Soc. B, 366, 763–771.
• Reppert, S. M. (2006). A colorful model of the circadian clock. Cell, 124, 233–236.
• Schmidt-Koenig, K. (1958). Experimentelle Einflussnahme auf die 24-Stunden-Periodik bei Brieftauben und deren Auswirkungen. Z. Tierpsychol., 15, 301–331.
• Strutt, J. W. (Lord Rayleigh) (1871). On the scattering of light by small particles. Phil. Mag., 41, 447–454.
• Wehner, R. (2003). Desert ant navigation: how miniature brains solve complex tasks. J. Comp. Physiol. A, 189, 579–588.