# Exoplanet Detection Simulator - Transit Light Curves, Radial Velocity, Habitable Zone

> Transit light curves with limb darkening and noise, radial-velocity curves and Doppler shifts, and Kopparapu habitable zones with planet temperatures.

- URL: https://qilingtools.com/en/physics/exoplanet
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

This tool shows how astronomers find planets around other stars. The transit page integrates a quadratically limb-darkened stellar disc numerically to draw a realistic light curve from the star and planet sizes, period and impact parameter, then adds noise so you can see whether the planet would be detected. The radial velocity page solves Kepler's equation for the star's line-of-sight velocity and spectral line shift, and explains the m sin i degeneracy. The habitable zone page uses the Kopparapu 2013 fits for conservative and optimistic zones and places the planets of the Solar System, TRAPPIST-1, Kepler-186 and others with their equilibrium temperatures.

## How to use

1. On the transit tab, pick a preset (Earth, hot Jupiter HD 209458 b, TRAPPIST-1e, Kepler-186f) or set the star and planet
2. Press Play to watch the planet cross the star with a cursor on the light curve; change the noise, cadence and number of transits to see if SNR passes 7.1
3. On the radial velocity tab, change mass, period, eccentricity and inclination to see K, the RV curve and the exaggerated line shift
4. On the habitable zone tab, choose a star or set its temperature and luminosity to see the zone edges and planet temperatures

## Features

- Transit light curves with quadratic limb darkening by numerical disc integration, including grazing transits
- Depth, T₁₄/T₂₃, ingress time, inclination, transit probability and signal-to-noise
- Exact radial-velocity semi-amplitude, eccentric RV curves, top-down orbit and Doppler line animation
- m sin i explained: change the inclination to compare measured and true mass
- Kopparapu 2013 habitable zone edges (runaway/maximum greenhouse, recent Venus, early Mars) and equilibrium temperatures
- Comparison of transit, radial velocity, direct imaging, microlensing and astrometry

## FAQ

### What does the transit method measure?

The depth is roughly (Rp/R*)², so a known stellar radius gives the planet's radius: Earth dims the Sun by only about 84 ppm, a hot Jupiter by 1–2%. The time between transits is the period, and Kepler's third law then gives the orbital distance; the duration and ingress shape give the inclination and impact parameter. The orbit must be almost edge-on for a transit, which happens with probability about R*/a: only 0.47% for an Earth–Sun system.

### What is limb darkening, and why is the bottom of the dip curved?

A star's disc is brighter at the centre than at the edge, because near the limb we see cooler, higher layers. A common description is the quadratic law I(μ) = 1 − u₁(1 − μ) − u₂(1 − μ)², where μ is the cosine of the angle between the line of sight and the surface normal. At mid-transit the planet blocks the brightest part, so the dip is a little deeper than (Rp/R*)² and gets shallower towards the limb, giving a rounded bottom. The tool integrates numerically over every stellar annulus the planet covers.

### Why does radial velocity only give m sin i?

Doppler shifts only reveal the star's velocity along our line of sight. In the semi-amplitude K = (2πG/P)^⅓ · m sin i / (M* + m)^⅔ / √(1 − e²), the planet mass and inclination always appear together as m sin i, so the same curve could come from an ordinary planet at i = 90° or a much heavier body seen nearly face-on. Jupiter moves the Sun at about 12.5 m/s, Earth only about 0.09 m/s, which is why Earth twins are so hard to find this way.

### How is the habitable zone defined?

It is the range of orbital distances where a planet could keep liquid water on its surface. Kopparapu et al. (2013) used 1-D climate models: the inner edge is the runaway greenhouse (oceans boil away) and the outer edge the maximum greenhouse (more CO₂ no longer helps). The effective stellar flux at each edge, S_eff, is a polynomial in stellar temperature, and the distance is d = √(L/S_eff). For the Sun the conservative zone is about 0.98–1.69 AU, the optimistic one (recent Venus to early Mars) about 0.75–1.77 AU.

### Why is the equilibrium temperature different from the real surface temperature?

Equilibrium temperature balances absorbed starlight against thermal emission, T_eq = [L(1 − A)/(16πσa²)]^¼, with no greenhouse effect. Earth's is about 255 K (−18 °C) while its mean surface temperature is about 15 °C; the 33 °C difference is the atmosphere's greenhouse effect, and on Venus the gap is over 400 °C. So it is a way to compare how much energy planets receive, not a verdict on habitability.

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