Earth, planets and orbits

1801

Finding a lost world

Giuseppe Piazzi; Carl Friedrich Gauss

How do you find a lost planet with a pencil?

On 1 January 1801 Piazzi spotted Ceres, then lost it behind the Sun’s glare. Using only a few weeks of observations, the young Gauss calculated its orbit. Astronomers pointed where he said and found it again that December.

How they figured it out

On 1 January 1801 Giuseppe Piazzi in Palermo spotted a faint moving object, Ceres, and followed it for 41 nights before it disappeared into the Sun’s glare. Astronomers had only a tiny arc of its path, about three degrees, and the standard methods assumed circular orbits.

The 24-year-old Gauss developed a new way to compute a full orbit from three observations, and used his method of least squares to balance out the measurement errors. On 31 December 1801 Franz von Zach found Ceres almost exactly where Gauss had said. Gauss became famous overnight.

The equation

min⁡∑i(observedi−predictedi)2\min \sum_i \left(\text{observed}_i - \text{predicted}_i\right)^2

Choose the orbit that makes the total squared error between prediction and observation as small as possible.

Everyday echoLine-of-best-fit in any spreadsheet is Gauss’s least squares.

See it in real life

The trend line on your fitness app

Weigh yourself every day and the numbers jump around. The smooth trend line your app draws through them is a least-squares fit: the line that makes the total squared error as small as possible. GPS uses the same idea to get one position from many noisy satellite signals.

Where engineers use it

  • Least squares

    Fitting a line to lab data, calibrating sensors and GPS positioning all use Gauss’s least squares.

  • Orbit determination

    Tracking stations work out satellite orbits from a handful of measurements, as Gauss did.

Where it came from

What it led to

  • A planet found on paper 1846

    Same technique

    Both turned scattered positions in the sky into a prediction of where to point a telescope.

    Where the comparison stops: Gauss fitted the orbit of an object people had already seen. Le Verrier inferred an unseen planet from its pull on another, a much harder problem.