exoplanet-detection-period-transit-least-squares（系外行星探测周期凌日最小二乘）

Transit Least Squares (TLS)

Transit Least Squares is a specialized algorithm optimized for detecting exoplanet transits in light curves. It's more sensitive than Lomb-Scargle for transit-shaped signals because it fits actual transit models.

Overview

TLS searches for periodic transit-like dips in brightness by fitting transit models at different periods, durations, and epochs. It's the preferred method for exoplanet transit detection.

Installation

pip install transitleastsquares

Basic Usage

CRITICAL: Always include flux_err (flux uncertainties) for best results!

import transitleastsquares as tls
import lightkurve as lk
import numpy as np

# Example 1: Using Lightkurve (recommended)
lc = lk.LightCurve(time=time, flux=flux, flux_err=error)
lc_clean = lc.remove_outliers(sigma=3)
lc_flat = lc_clean.flatten()

# Create TLS object - MUST include flux_err!
pg_tls = tls.transitleastsquares(
    lc_flat.time.value,      # Time array
    lc_flat.flux.value,      # Flux array
    lc_flat.flux_err.value   # Flux uncertainties (REQUIRED!)
)

# Search for transits (uses default period range if not specified)
out_tls = pg_tls.power(
    show_progress_bar=False,  # Set True for progress tracking
    verbose=False
)

# Extract results
best_period = out_tls.period
period_uncertainty = out_tls.period_uncertainty
t0 = out_tls.T0                    # Transit epoch
depth = out_tls.depth               # Transit depth
snr = out_tls.snr                   # Signal-to-noise ratio
sde = out_tls.SDE                   # Signal Detection Efficiency

print(f"Best period: {best_period:.5f} ± {period_uncertainty:.5f} days")
print(f"Transit epoch (T0): {t0:.5f}")
print(f"Depth: {depth:.5f}")
print(f"SNR: {snr:.2f}")
print(f"SDE: {sde:.2f}")

Example 2: With explicit period range

# Search specific period range
out_tls = pg_tls.power(
    period_min=2.0,      # Minimum period (days)
    period_max=7.0,      # Maximum period (days)
    show_progress_bar=True,
    verbose=True
)

Period Refinement Strategy

Best Practice: Broad search first, then refine for precision.

Example workflow:

1. Initial search finds candidate at ~3.2 days
2. Refine by searching narrower range (e.g., 3.0-3.4 days)
3. Narrower range → finer grid → better precision

Why? Initial searches use coarse grids (fast). Refinement uses dense grid in small range (precise).

# After initial search finds a candidate, narrow the search:
results_refined = pg_tls.power(
    period_min=X,  # e.g., 90% of candidate
    period_max=Y   # e.g., 110% of candidate
)

Typical refinement window: ±2% to ±10% around candidate period.

Advanced Options

For very precise measurements, you can adjust:

• oversampling_factor: Finer period grid (default: 1, higher = slower but more precise)
• duration_grid_step: Transit duration sampling (default: 1.1)
• T0_fit_margin: Mid-transit time fitting margin (default: 5)

Advanced Parameters

• oversampling_factor: Higher values give finer period resolution (slower)
• duration_grid_step: Step size for transit duration grid (1.01 = 1% steps)
• T0_fit_margin: Margin for fitting transit epoch (0 = no margin, faster)

Phase-Folding

Once you have a period, TLS automatically computes phase-folded data:

# Phase-folded data is automatically computed
folded_phase = out_tls.folded_phase      # Phase (0-1)
folded_y = out_tls.folded_y              # Flux values
model_phase = out_tls.model_folded_phase # Model phase
model_flux = out_tls.model_folded_model # Model flux

# Plot phase-folded light curve
import matplotlib.pyplot as plt
plt.plot(folded_phase, folded_y, '.', label='Data')
plt.plot(model_phase, model_flux, '-', label='Model')
plt.xlabel('Phase')
plt.ylabel('Flux')
plt.legend()
plt.show()

Transit Masking

After finding a transit, mask it to search for additional planets:

from transitleastsquares import transit_mask

# Create transit mask
mask = transit_mask(time, period, duration, t0)
lc_masked = lc[~mask]  # Remove transit points

# Search for second planet
pg_tls2 = tls.transitleastsquares(
    lc_masked.time,
    lc_masked.flux,
    lc_masked.flux_err
)
out_tls2 = pg_tls2.power(period_min=2, period_max=7)

Interpreting Results

Signal Detection Efficiency (SDE)

SDE is TLS's measure of signal strength:

• SDE > 6: Strong candidate
• SDE > 9: Very strong candidate
• SDE < 6: Weak signal, may be false positive

Signal-to-Noise Ratio (SNR)

• SNR > 7: Generally considered reliable
• SNR < 7: May need additional validation

Common Warnings

TLS may warn: "X of Y transits without data. The true period may be twice the given period."

This suggests:

• Data gaps may cause period aliasing
• Check if period * 2 also shows a signal
• The true period might be longer

Model Light Curve

TLS provides the best-fit transit model:

# Model over full time range
model_time = out_tls.model_lightcurve_time
model_flux = out_tls.model_lightcurve_model

# Plot with data
import matplotlib.pyplot as plt
plt.plot(time, flux, '.', label='Data')
plt.plot(model_time, model_flux, '-', label='Model')
plt.xlabel('Time [days]')
plt.ylabel('Flux')
plt.legend()
plt.show()

Workflow Considerations

When designing a transit detection pipeline, consider:

1. Data quality: Filter by quality flags before analysis
2. Preprocessing order: Remove outliers → flatten/detrend → search for transits
3. Initial search: Use broad period range to find candidates
4. Refinement: Narrow search around candidates for better precision
5. Validation: Check SDE, SNR, and visual inspection of phase-folded data

Typical Parameter Ranges

• Outlier removal: sigma=3-5 (lower = more aggressive)
• Period search: Match expected orbital periods for your target
• Refinement window: ±2-10% around candidate period
• SDE threshold: >6 for candidates, >9 for strong detections

Multiple Planet Search Strategy

1. Initial broad search: Use TLS with default or wide period range
2. Identify candidate: Find period with highest SDE
3. Refine period: Narrow search around candidate period (±5%)
4. Mask transits: Remove data points during transits
5. Search for additional planets: Repeat TLS on masked data

Dependencies

pip install transitleastsquares lightkurve numpy matplotlib

References

• TLS GitHub Repository
• TLS Tutorials
• Lightkurve Tutorials
• Lightkurve Exoplanet Examples

When to Use TLS vs. Lomb-Scargle

• Use TLS: When specifically searching for exoplanet transits
• Use Lomb-Scargle: For general periodic signals (rotation, pulsation, eclipsing binaries)

TLS is optimized for transit-shaped signals and is typically more sensitive for exoplanet detection.

Common Issues

"flux_err is required"

Always pass flux uncertainties to TLS! Without them, TLS cannot properly weight data points.

Period is 2x or 0.5x expected

Check for period aliasing - the true period might be double or half of what TLS reports. Also check the SDE for both periods.

Low SDE (<6)

• Signal may be too weak
• Try different preprocessing (less aggressive flattening)
• Check if there's a data gap during transits