4. Noise, Probability, and Exoplanet Transits

Space question

How can a tiny dip in a star’s brightness reveal a planet we cannot see?

When a planet crosses its star from our viewpoint, it blocks a fraction of the stellar disk. A time series of relative brightness is called a light curve.

4.1 Geometry predicts the signal

Ignoring limb darkening and assuming a small planet, transit depth is

$$\delta\approx\left(\frac{R_p}{R_\star}\right)^2.$$

An Earth-sized planet crossing a Sun-sized star blocks only about $(0.00916)^2\approx84$ parts per million. This is why calibration and statistics are central, not optional.

If dips repeat every $P$, that suggests the orbital period. With an estimate of stellar mass $M_\star$, Kepler’s law gives orbital scale:

$$a^3\approx\frac{GM_\star P^2}{4\pi^2}.$$

The inference chain already joins Chapters 1–3: flux, orbital dynamics, stellar spectroscopy, and now probability.

4.2 A measurement is signal plus noise

Write

$$y_i=m(t_i;\boldsymbol\theta)+\epsilon_i.$$

The model $m$ might contain period, transit time, duration, and depth. If independent noise is approximately Gaussian with standard deviation $\sigma_i$, the log likelihood is, up to a constant,

$$\log p(\mathbf y|\boldsymbol\theta)= -\frac12\sum_i \left[\frac{y_i-m(t_i;\boldsymbol\theta)}{\sigma_i}\right]^2.$$

Maximizing this likelihood is equivalent to minimizing weighted $\chi^2$. Probability states what assumptions connect residuals to evidence.

4.3 Fold the light curve

For a trial period $P$ and reference time $t_0$, phase can be written

$$\phi=\left[\frac{t-t_0+P/2}{P}\bmod 1\right]-\frac12.$$

If $P$ is right, separate transits line up near phase zero. Averaging many events increases signal-to-noise. If the noise is independent, stacking $N$ similar events improves signal-to-noise roughly as $\sqrt N$.

4.4 Detection is not confirmation

A periodic dip can also arise from:

• an eclipsing binary blended with another star;
• starspots and rotation;
• pointing drift or detector systematics;
• a period-search coincidence;
• an incorrect stellar radius.

Confirmation uses additional information: dip shape and colour, odd/even event comparison, centroid motion, spectroscopy, higher-resolution imaging, and sometimes radial velocities.

4.5 Uncertainty and selection

A fitted depth gives a radius ratio only within the transit model. Stellar radius uncertainty propagates directly to planet radius. Detection pipelines also find large, short-period planets more easily. Therefore the observed catalogue is not automatically the underlying planet population.

This is a selection effect: the probability of entering the sample depends on the thing being measured.

Lab

Open 03_exoplanet_transit_lab.ipynb. You will generate an irregular, noisy light curve, search trial periods, recover radius ratio, and test the danger of choosing the tallest periodogram peak.

Used in space science

• detecting and characterizing exoplanets;
• finding periodic and transient stars;
• estimating parameters of gravitational-wave sources;
• discovering faint sources in images;
• inferring populations while correcting survey completeness.

Journey to the frontier

The frontier includes characterizing atmospheres of small planets and deciding which combinations of gases, climate, and stellar environment could support a claim about habitability or life. More data do not eliminate the need for good models: correlated stellar variability, clouds, atmospheric chemistry, and selection effects can dominate the interpretation.

The frontier skill introduced here is calibrated skepticism: quantify how surprising a signal is under explicit assumptions, then actively test plausible alternatives.

Connections

• The period links back to gravitational dynamics.
• Stellar spectra determine the host star’s properties.
• Linear algebra and optimization fit richer transit models.
• Thermodynamics and chemistry later interpret atmospheric spectra.