There’s no elementary formula for the circumference of an ellipse, but there is an elementary approximation that is extremely accurate.

An ellipse has equation (x/a)² + (y/b)² = 1. If a = b, the ellipse reduces to a circle and the circumference is simply 2πa. But the general formula for circumference requires the hypergeometric function ₂F₁:

where λ = (a – b)/(a + b).

However, if the ellipse is anywhere near circular, the following approximation due to Ramanujan is extremely good:

To quantify what we mean by extremely good, the error is O(λ¹⁰). When an ellipse is roughly circular, λ is fairly small, and the error is on the order of λ to the 10th power.

To illustrate the accuracy of the approximation, I tried the formula out on some planets. The error increases with ellipticity, so I took the most most elliptical orbit of a planet or object formerly known as a planet. That distinction belongs to Pluto, in which case λ = 0.016. If Pluto’s orbit were exactly elliptical, you could use Ramanujan’s approximation to find the circumference of its orbit with an error less than one micrometer.

Next I tried it on something with a much more elliptical orbit: Halley’s comet. Its orbit is nearly four times longer than it is wide. For Halley’s comet, λ = 0.59 and Ramanujan’s approximation agrees with the exact result to seven significant figures. The exact result is 11,464,319,022 km and the approximation is 11,464,316,437 km.

Here’s a video showing how elliptical the comet’s orbit is.

If you’d like to experiment with the approximation, here’s some Python code:

from scipy import pi, sqrt
from scipy.special import hyp2f1

def exact(a, b):
    t = ((a-b)/(a+b))**2
    return pi*(a+b)*hyp2f1(-0.5, -0.5, 1, t)

def approx(a, b):
    t = ((a-b)/(a+b))**2
    return pi*(a+b)*(1 + 3*t/(10 + sqrt(4 - 3*t)))

# Semimajor and semiminor axes for Halley's comet orbit
a = 2.667950e9 # km
b = 6.782819e8 # km

print exact(a, b)
print approx(a, b)

Modern radio pulsar surveys produce a large volume of prospective candidates, the majority of which are polluted by human-created radio frequency interference or other forms of noise. Typically, large numbers of candidates need to be visually inspected in order to determine if they are real pulsars. This process can be labor intensive. In this paper, we introduce an algorithm called PEACE (Pulsar Evaluation Algorithm for Candidate Extraction) which improves the efficiency of identifying pulsar signals. The algorithm ranks the candidates based on a score function. Unlike popular machine-learning based algorithms, no prior training data sets are required. This algorithm has been applied to data from several large-scale radio pulsar surveys. Using the human-based ranking results generated by students in the Arecibo Remote Command enter programme, the statistical performance of PEACE was evaluated. It was found that PEACE ranked 68% of the student-identified pulsars within the top 0.17% of sorted candidates, 95% within the top 0.34%, and 100% within the top 3.7%. This clearly demonstrates that PEACE significantly increases the pulsar identification rate by a factor of about 50 to 1000. To date, PEACE has been directly responsible for the discovery of 47 new pulsars, 5 of which are millisecond pulsars that may be useful for pulsar timing based gravitational-wave detection projects.