# Drake Equation Explorer (Monte Carlo)

> How many civilisations? The Drake equation with sourced ranges, Monte Carlo and sensitivity. Explore the Drake equation N = R* · fp · ne · fl · fi · fc · L with ranges instead of single guesses. The star formation rate, the fraction of stars with planets and the number of habitable-zone rocky planets default to published measurements; life, intelligence, detectability and lifetime are your assumptions. A seeded Monte Carlo gives the median N, the 90 % range, the share of draws with N < 1 and which factor drives the uncertainty. Part of the Proofline math notebook.

Part of usefulweb.app (https://usefulweb.app/). Part of [Proofline](https://proofline.usefulweb.app/) (ownerApp: proofline). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://prooflinedrake.usefulweb.app/ (interactive tool at https://prooflinedrake.usefulweb.app/tool)
- JSON API: POST https://prooflinedrake.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://prooflinedrake.usefulweb.app/api/v1/tool)
- OpenAPI: https://prooflinedrake.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://prooflinedrake.usefulweb.app/mcp - tool `proofline_drake`

## Inputs
- rStar (text): R*: Sun-like stars formed per year in the Milky Way - A value or a range “low - high” (per year). Default from Robitaille & Whitney 2010; Licquia & Newman 2015.
- fp (text): fp: Fraction of those stars with planets - A value or a range “low - high”. Default from Cassan et al. 2012 (microlensing).
- ne (text): ne: Habitable-zone rocky planets per planetary system - A value or a range “low - high”. Default from Bryson et al. 2021 (Kepler DR25).
- fl (text): fl: Fraction of those planets where life appears - A value or a range “low - high”. Default from assumption (no measurement exists).
- fi (text): fi: Fraction of life-bearing planets that develop intelligence - A value or a range “low - high”. Default from assumption (no measurement exists).
- fc (text): fc: Fraction of intelligent species that become detectable - A value or a range “low - high”. Default from assumption (no measurement exists).
- L (text): L: Years a civilisation stays detectable - A value or a range “low - high” (years). Default from assumption (no measurement exists).
- distribution (select): Sampling of the ranges
- samples (integer): Monte Carlo draws
- seed (integer): Random seed

## Method
- N = R* · fp · ne · fl · fi · fc · L (Drake, 1961). The lowest and highest possible N multiply the range ends; the central estimate multiplies the geometric middles (log-uniform) or the midpoints (uniform).
- Monte Carlo: each factor is drawn independently from its range with a seeded mulberry32 generator; the median, the 5 % and 95 % quantiles and the share of draws with N < 1 come from the sorted draws.
- Sensitivity: log₁₀ N is the sum of the factors’ logarithms, so its variance is the sum of their variances; each factor’s share is exact (log-uniform: (log₁₀ high − log₁₀ low)²/12).
- Default ranges: R* 0.68–1.65 Sun-like stars per year (Milky Way star formation of 0.68–1.45 and 1.65 ± 0.19 solar masses per year, read as one Sun-like star per solar mass), fp 0.5–1 (microlensing finds about one or more planets per star), ne 0.37–0.88 (Kepler DR25 η⊕ for Sun-like stars); fl, fi, fc and L are wide assumptions because nothing has measured them.

## Limits
- The Drake equation organises ignorance; it does not measure anything. With unmeasured factors spanning many orders of magnitude, every result mostly mirrors the assumptions.