A quantitative analysis of Galilei's observations of Jupiter satellites from the Sidereus Nuncius

TL;DR

This paper reanalyzes Galileo’s Sidereus Nuncius observations of Jupiter’s moons by digitizing sketches and using the text-based angular readings, then validating these against a modern sky simulator. Through Lomb-Scargle periodograms and sinusoidal fits, the authors extract orbital periods with 0.1–0.3% precision and semimajor axes with a few percent accuracy, demonstrating that Kepler’s third law and the 1:2:4 resonance are convincingly supported by Galileo’s data. A careful cross-calibration of the two datasets, accounting for Jupiter’s disk size and observational biases near the limb, yields a coherent orbital picture and reveals the limits of the original telescope’s separating power. The study extends the simulator's use to Moon and star-cluster observations, and a replication of the telescope underscores the remarkable quality and the practical difficulties of early 17th-century observations. Overall, the work highlights the exceptional value of Galileo’s data and provides a nuanced, quantitative bridge between early observational astronomy and modern celestial mechanics.

Abstract

We present a new careful and comprehensive analysis the observations of the satellites of Jupiter from the Sidereus Nuncius that extends and complements previous similar studies. Each observation is compared to the predictions obtained using a modern sky simulator, verifying and trying to understand them individually. The work considers both the information that can be extracted from the sketches and the angular measurements reported by Galilei. Angular measurements allow assessing the absolute accuracy in relation to modern ephemerides. We evaluate the performances of the telescope in terms of separation power of close-by satellites and the inefficiency in the detection connected to the proximity to the disk. A sinusoidal fit of the data, allows measuring the relative major semi-axes of the satellites' orbits and their periods with a statistical precision of 2-4\% and 0.1-0.3\% respectively. The posterior fit error is used to estimate the measurements precision. We show that with this data one can infer in a convincing way the third law of Kepler for the Jupiter system. The 1:2:4 orbital resonance between the periods of Io and Europa/Ganymede can be determined with \% precision. In order to obtain these results it is important to separate the four datasets. This operation was an extremely difficult task for Galilei. Nevertheless we show how some indication on the periods emerge from the using the modern Lomb-Scargle technique on the full data-set. We briefly extend the use of the simulator to verify the accuracy in the seven observations of the Moon and the performance in reproducing the Pleiades, the Orion belt, the Orion head and the Beehive cluster. Finally we present images obtained with a replica of the telescope that highlights the challenges of these observations thus confirming the excellence underlying this amazing set of early scientific data.

Figures (41)

• Figure 1: Motion of Jupiter (black) and of its satellites (Io in blue, Europa in red, Ganymede in green and Callisto in Orange) across the sky from Jan. 7 1610 to March 2 1610. The marker points the position of the star HD32811 that was recorded by Galileo in observations 61, 62 and 64 (see Tab. \ref{['fig:1']}.)
• Figure 2: Sidereus Nuncius sketches A, from January 7 to February 1st.
• Figure 3: Sidereus Nuncius sketches B, from February 2 to March 1st.
• Figure 4: Distribution of the observations in azimuth and altitude for the supposed location of the observations in Padua (45.4032, 11.8818, currently at via Galileo Galilei 18). The color code indicates CET time of the observation.
• Figure 5: Left: reconstructed positions of Jupiter satellites from the angular data of Galilei (in primes). Right: reconstructed positions from the digitization of the sketches in units of the planet's equatorial diameters in the sketches themselves. Times increases from top to bottom.