Computing Eclipses, Occultations and Transits

Transits of Mercury and Venus

workingupdated 2026-10-17transitsmercuryvenusblack-dropsolar-radiusastronomical-unit
  • A transit is a solar eclipse by a planet, and the almanacs compute it that way. The 1961 Explanatory Supplement adapts Bessel's eclipse theory with the planet in place of the Moon, and Stellarium uses the solar-eclipse elements with the planet's radius as kk 1 2.
  • The NASA catalogues list 94 Mercury transits from 1601 to 2300 and 81 Venus transits from 2000 BCE to 4000 CE. Their elements and algorithms come from Meeus's Transits (1989), and the cycles are 13, 33 and 46 years for Mercury and 8, 105.5, 8 and 121.5 years for Venus 3 4.
  • Local contact times come from a three-constant formula per contact. The yearbook form T=T0+aρsinϕ′+bρcosϕ′cos(c+λ)T = T_0 + a\,\rho\sin\phi' + b\,\rho\cos\phi'\cos(c + \lambda) is still printed by the Institute of Applied Astronomy of the Russian Academy of Sciences (IAA RAS), and the longitude sign differs between sources 5 1.
  • The solar and planetary radii dominate the residual differences. One arcsecond of contact radius moves an interior contact by 10 to 21 seconds of time at the modern transits, computed from the Supplement's contact solution with the published elements 1 6 7 8 9 10. The 4 to 5 second differences between Espenak and the Institut de mécanique céleste et de calcul des éphémérides (IMCCE) for 2004 match a Venus semidiameter of 8.41 against 8.34 arcseconds 11 12 6.
  • The black drop is the telescope and atmosphere blurring a limb-darkened edge. The Transition Region and Coronal Explorer (TRACE) showed it from space at a Mercury transit, so it does not need a planetary atmosphere 13.
  • Transits measured the astronomical unit and now measure the solar radius. Newcomb's 1874 and 1882 reduction gave 8.79 arcseconds of solar parallax, the 2004 public campaign recovered the astronomical unit to 0.007 per cent after a selection, and space timings give radii from 959.57 to 960.12 arcseconds 14 15 16 17.

What this topic covers

This topic covers the computation of transits of Mercury and Venus across the Sun as seen from the Earth. It traces enumeration through node passages and recurrence cycles, the almanac elements in heliocentric and geocentric form, the contact-time solution and the local reduction formulas, the constants that matter, and the observations that test the predictions. It reuses the solar-eclipse machinery of Besselian elements, local circumstances, solar radius and Earth and time, and states only what differs for a planet. Transits seen from other planets are outside it, except Phobos and Deimos, which the Saturn and outer satellites topic covers briefly.

Notes in this topic

  • Enumerating transits and their elements: the node condition and limits, the Mercury and Venus cycles and series, grazing and topocentric-only transits, how the NASA, IMCCE, Stellarium and Swiss Ephemeris lists are computed, and the elements in the Supplement's heliocentric form, Stellarium's vector form, Jubier's polynomial form and the IAA yearbook form.
  • Transit contacts and local circumstances: the contact solution, least separation and position angles, the rigorous and linearised local formulas with worked examples, ΔT and the longitude sign, the radius constants each source uses, a sensitivity figure, and predictor-against-predictor comparisons for 2004, 2016 and 2019.
  • Observations, the black drop and the solar radius: the black drop and its four explanations, the Venus aureole, Halley's and Delisle's methods, the historical solar parallax values, the VT-2004 analysis, solar radius results from transits, and the observed-minus-predicted contacts that exist.

What this topic changes for the pipeline

The element and contact stages become body-agnostic. The radius of the occulting body, the exact rather than small-ratio point-Z formulas, and both semidiameters must be inputs, and a prediction must carry its planetary radius beside s0s_0, ΔT and the ephemeris. The catalogue schema needs room for missing geocentric contacts and for topocentric-only events. The validation suite gains space-based transit timings, which test the solar radius without a lunar limb profile.

References

  1. 1peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (1961), section 9F Transits of Mercury, pages 262 to 276 Read in full (OCR text). Node condition and the table of node limits, dates and frequencies (Mercury ascending node about 16 November 238 arcminutes, descending 8 May 108; Venus ascending 9 December 37, descending 7 June 41), Newcomb's heliocentric method and the transit fundamental plane, the light-time rule, the contact solution, least distance, position angles with V, Newcomb's q p w, the B C D reduction and Examples 9.26 to 9.32 for 1960 November 7. The node-limit formula for theta and the cone-angle equation are damaged in the OCR text and are not transcribed.
  2. 2company Stellarium, src/gui/AstroCalcDialog.cpp, functions generateTransits, localTransit and TransitBessel::computeElements, commit 69888f4f47af2aa4d1c9d14ec3853e874e253c73 Read lines 4523 to 4990 of a local clone at that commit. Besselian elements for transits adapted from the 1961 Supplement, synodic stepping from a reference conjunction, the gamma threshold 0.9972 + L1, the iteration to 0.1 s and the contact corrections.
  3. 3primary Espenak, F., NASA GSFC, Seven Century Catalog of Mercury Transits: 1601 CE to 2300 CE Read in full (curl). 94 transits, 31 in May at the descending node and 63 in November at the ascending node; the 13, 33 and 46 year commensurabilities with their residuals in days; the 46-year series; the full table of contacts, minimum separations and series; the 1937 May 11 row with no interior contacts; and the statement that Besselian elements come from Meeus's Transits (1989).
  4. 4primary Espenak, F., NASA GSFC, Six Millennium Catalog of Venus Transits: 2000 BCE to 4000 CE Read in full (curl). 81 transits, 44 in June and 37 in December; the 8-year resonance with the 2.45 day shortfall; the 243-year series of 88,756 days; node dates drifting from about May 21 and November 19 five thousand years ago; rows with missing contacts such as -1763 and -0540; elements from Meeus (1989).
  5. 5primary Institute of Applied Astronomy RAS, Astronomical Yearbook 2019: Transit of Mercury of 2019 November 11 Read in full (curl). Same layout as 2016, with TDT - UT1 = 70.0 s stated, geocentric contacts 12:35:26.3, 12:37:07.7, 15:19:47.3, 18:02:32.1, 18:04:13.5 UT and the four reduction formulas.
  6. 6primary IMCCE for ESO VT-2004, Extended InfoSheet B1: Predictions of the Venus transit of June 8, 2004 Read in full (curl). VSOP87, Lieske precession, Wahr nutation, Aoki sidereal time, IAU 1976 constants (solar semidiameter 15'59.63", Venus radius 6051.8 km, Earth radius 6378.140 km), flattening 1/298.257, TT - UTC 65.184 s, general and geocentric phases, the E, F, G coefficients and the Paris example. Formula (1) is an image; its form is reconstructed in the note from the Paris example.
  7. 7company Espenak, F., EclipseWise, 2016 Transit of Mercury (Observer's Handbook article) Read through a fetch summary with quoted table values. Geocentric contacts 11:12:19, 11:15:31, 14:57:26, 18:39:14, 18:42:26 UT1 with position angles 83.2, 83.5, 153.8, 224.1 and 224.4 degrees, minimum separation 318.5 arcseconds, JPL DE406.
  8. 8company Espenak, F., EclipseWise, 2019 Transit of Mercury (Observer's Handbook article) Read through a fetch summary with quoted table values. Geocentric contacts 12:35:27, 12:37:08, 15:19:48, 18:02:33, 18:04:14 UT1 with position angles, minimum separation 75.9 arcseconds, JPL DE406, local times differing by up to 2 minutes, and the remark that the thread-breaking time is delayed by some seconds, up to a minute.
  9. 9company Jubier, X., MercuryTransitCalculator.js (Mercury transit calculator source) Read in full (curl). Element arrays per transit with t0 and ΔT, Sun 959.62 arcseconds (comment 959.63), Mercury 3.3558 arcseconds (comment 3.3629), solar parallax 8.794142 arcseconds, the topocentric shift formulas, the contact iteration and the flattening factor 0.996647189335.
  10. 10company Jubier, X., VenusTransitCalculator.js (Venus transit calculator source) Read (curl). Same structure as the Mercury calculator, Venus 8.3442 arcseconds (comment 8.41), element arrays for 2004 (ΔT 64.184 and 64.635 s) and 2012 (66.76 s).
  11. 11trade Meeus J. (1991) Astronomical Algorithms, first edition, Willmann-Bell, Chapter 52 Eclipses Read from the Internet Archive OCR text (chapter 52 in the 1991 edition, chapter 54 in the 1998 edition). Source of the |sin F| > 0.36 rule, the corrections to the time of maximum, P, Q, W, gamma, u, the thresholds 0.9972, 1.0260, 1.5433, 0.0047, 0.00464, the partial-magnitude formula and the stated accuracy of 0.36 min mean and 1.1 min maximum for 1951 to 2050. Graded trade as a recognised practitioner's own algorithm text.
  12. 12primary Espenak, F., NASA GSFC, The 2004 Transit of Venus (Observer's Handbook 2004) Read (curl, text extracted). Geocentric contacts 05:13:29, 05:32:55, 08:19:44, 11:06:33, 11:25:59 UT with position angles, and the up to 30 arcsecond parallax shift of Venus.
  13. 13peer-reviewed Schneider, G., Pasachoff, J. M. and Golub, L. (2004) TRACE observations of the 15 November 1999 transit of Mercury and the Black Drop effect: considerations for the 2004 transit of Venus. Icarus 168, 249-256 Read in full from the uncorrected proof hosted at Williams College (web.williams.edu/Astronomy/eclipse/transits/Icarus_transit.pdf). Black drop seen from space, caused by the instrumental PSF convolved with solar limb darkening; 0.55 arcsecond radius narrowing after deconvolution; 0.56 arcsecond residual edge half-width; diffraction edge of about 43 km at Mercury.
  14. 14primary IMCCE for ESO VT-2004, Extended InfoSheet F5: The 1882 transit of Venus Read in full (curl). Expeditions of 1882 and the table of solar parallax determinations from 1761 to the 1990 radar value of 8.79415 arcseconds, including Encke 8.5776 and Newcomb (1890) 8.79 arcseconds.
  15. 15primary ESO and IMCCE, VT-2004 Observing Campaign Results Read in full (curl). 4550 contact timings from 1510 observers, the final AU of 149 608 708 plus or minus 11 835 km, and the alternative solution for the radii of the Sun (695 980.7 plus or minus 806 km) and Venus (6052.3 plus or minus 7.0 km).
  16. 16peer-reviewed Emilio, M., Kuhn, J. R., Bush, R. I. and Scholl, I. F. (2012) Measuring the solar radius from space during the 2003 and 2006 Mercury transits. ApJ 750, 135 (arXiv:1203.4898) Read in full from the arXiv PDF, with the radius equation checked against the ar5iv HTML rendering because the PDF text layer drops its symbols. SOHO/MDI contact fitting, the Shapiro (1980) duration relation, the adopted 959.645 arcsecond nominal radius, observed contacts offset 8 s (2003) and 5 s (2006) from the NASA ephemeris, and the 960.12 plus or minus 0.09 arcsecond result.
  17. 17peer-reviewed Emilio, M., Bush, R., Kuhn, J. and Scholl, I. (2020) Solar astrometry with planetary transits. Proceedings IAU Symposium 354, 481-490 Read in full (Cambridge PDF). Reviews transit radius work: MDI 2003 and 2006 at 960.12 arcseconds, HMI 2012 Venus at 959.57 plus or minus 0.02 arcseconds after PSF correction, a 0.23 arcsecond line-core minus continuum difference, ground scatter of 0.5 to 1 arcsecond per transit, and the claim that the IAU 2015 nominal radius is off by more than 300 km.

In this section

Enumerating transits and their elementsHow transits of Mercury and Venus are found and catalogued, their recurrence cycles, and the Besselian-style elements used to compute them.workingTransit contacts and local circumstancesThe contact-time solution for transits of Mercury and Venus, the almanac reduction formulas for any observer, and the constants that move the times.workingObservations, the black drop and the solar radiusWhat observed transit contacts show against predictions, why the black drop blurs them, the Venus aureole, the astronomical unit, and solar radius results.working