The Astronomy Thread

Guess we know where Gozer is being summoned at least

Thought the relatively close approaches of 3I/Atlas by Mars and Jupiter both was interesting and asked Grok what the odds are of that... C3P0 gave better odds of successfully navigating an asteroid field. Grok's estimate for this two-planet pass trajectory was 114,000 to 1:

3. Geometric Probability of Passing Close to One Planet The solar system’s ecliptic plane, where 3I/ATLAS crosses, is where planets are most likely to be encountered. Let’s estimate the probability of a random interstellar object passing within a given distance of a planet:

• Solar System’s Cross-Section: Assume the comet crosses the ecliptic within 5.2 AU (Jupiter’s orbital radius), as 3I/ATLAS’s perihelion is 1.357 AU and its Jupiter encounter occurs near 5.2 AU. The cross-sectional area of the ecliptic plane out to 5.2 AU is:
  Asolar=π×(5.2 AU)2≈85 AU2≈1.9×1022 m2.A_{\text{solar}} = \pi \times (5.2 \, \text{AU})^2 \approx 85 \, \text{AU}^2 \approx 1.9 \times 10^{22} \, \text{m}^2.A_{\text{solar}} = \pi \times (5.2 \, \text{AU})^2 \approx 85 \, \text{AU}^2 \approx 1.9 \times 10^{22} \, \text{m}^2.
• Mars’s Target Area: For a close approach within 0.2 AU of Mars (radius of Mars’ orbit ~1.52 AU), the target area is:
  AMars=π×(0.2 AU)2≈0.1256 AU2.A_{\text{Mars}} = \pi \times (0.2 \, \text{AU})^2 \approx 0.1256 \, \text{AU}^2.A_{\text{Mars}} = \pi \times (0.2 \, \text{AU})^2 \approx 0.1256 \, \text{AU}^2.
  Probability of passing within 0.2 AU of Mars:
  PMars=0.125685≈0.00148 or 0.148% (1 in 676).P_{\text{Mars}} = \frac{0.1256}{85} \approx 0.00148 \text{ or } 0.148\% \text{ (1 in 676)}.P_{\text{Mars}} = \frac{0.1256}{85} \approx 0.00148 \text{ or } 0.148\% \text{ (1 in 676)}.
• Jupiter’s Target Area: For a close approach within 0.4 AU of Jupiter (orbital radius ~5.2 AU):
  AJupiter=π×(0.4 AU)2≈0.5024 AU2.A_{\text{Jupiter}} = \pi \times (0.4 \, \text{AU})^2 \approx 0.5024 \, \text{AU}^2.A_{\text{Jupiter}} = \pi \times (0.4 \, \text{AU})^2 \approx 0.5024 \, \text{AU}^2.
  Probability of passing within 0.4 AU of Jupiter:
  PJupiter=0.502485≈0.00591 or 0.591% (1 in 169).P_{\text{Jupiter}} = \frac{0.5024}{85} \approx 0.00591 \text{ or } 0.591\% \text{ (1 in 169)}.P_{\text{Jupiter}} = \frac{0.5024}{85} \approx 0.00591 \text{ or } 0.591\% \text{ (1 in 169)}.

These probabilities assume the comet’s entry point into the ecliptic is random, which is reasonable for interstellar objects like 3I/ATLAS, whose trajectories are determined by their origins in the galactic disk or other star systems and are not influenced by the solar system’s structure until they approach closely.4. Odds of Passing Close to Both Mars and Jupiter Since Mars and Jupiter are at different heliocentric distances (1.52 AU vs. 5.2 AU), their positions are independent in the context of a fast-moving interstellar object crossing the ecliptic. The comet’s trajectory is nearly straight (hyperbolic, eccentricity 6.14), and its high speed (58–68 km/s) means it crosses the solar system in months, so we can treat the Mars and Jupiter encounters as independent events for a first-order approximation:

• Joint Probability: The probability of passing within 0.2 AU of Mars and 0.4 AU of Jupiter is the product of the individual probabilities:
  Pboth=PMars×PJupiter=0.00148×0.00591≈8.75×10−6 or 0.000875% (1 in 114,286).P_{\text{both}} = P_{\text{Mars}} \times P_{\text{Jupiter}} = 0.00148 \times 0.00591 \approx 8.75 \times 10^{-6} \text{ or } 0.000875\% \text{ (1 in 114,286)}.P_{\text{both}} = P_{\text{Mars}} \times P_{\text{Jupiter}} = 0.00148 \times 0.00591 \approx 8.75 \times 10^{-6} \text{ or } 0.000875\% \text{ (1 in 114,286)}.
  This suggests that for a random interstellar object, the odds of passing within 0.2 AU of Mars and 0.4 AU of Jupiter are extremely low, about 1 in 114,000.

Only the third interstellar object ever witnessed, has some early on strange characteristics (bright, massive?), just happening to pass in on an extremely unlikely trajectory that gets it relatively close to a couple planets. Just saying...