exoplanet-detection-period-box-least-squares（系外行星探测周期盒最小二乘法）

Box Least Squares (BLS) Periodogram

The Box Least Squares (BLS) periodogram is a statistical tool for detecting transiting exoplanets and eclipsing binaries in photometric time series data. BLS models a transit as a periodic upside-down top hat (box shape) and finds the period, duration, depth, and reference time that best fit the data.

Overview

BLS is built into Astropy and provides an alternative to Transit Least Squares (TLS). Both search for transits, but with different implementations and performance characteristics.

Key parameters BLS searches for:

• Period (orbital period)
• Duration (transit duration)
• Depth (how much flux drops during transit)
• Reference time (mid-transit time of first transit)

Installation

BLS is part of Astropy:

pip install astropy

Basic Usage

import numpy as np
import astropy.units as u
from astropy.timeseries import BoxLeastSquares

# Prepare data
# time, flux, and flux_err should be numpy arrays or Quantities
t = time * u.day  # Add units if not already present
y = flux
dy = flux_err  # Optional but recommended

# Create BLS object
model = BoxLeastSquares(t, y, dy=dy)

# Automatic period search with specified duration
duration = 0.2 * u.day  # Expected transit duration
periodogram = model.autopower(duration)

# Extract results
best_period = periodogram.period[np.argmax(periodogram.power)]
print(f"Best period: {best_period:.5f}")

Using autopower vs power

autopower: Automatic Period Grid

Recommended for initial searches. Automatically determines appropriate period grid:

# Specify duration (or multiple durations)
duration = 0.2 * u.day
periodogram = model.autopower(duration)

# Or search multiple durations
durations = [0.1, 0.15, 0.2, 0.25] * u.day
periodogram = model.autopower(durations)

power: Custom Period Grid

For more control over the search:

# Define custom period grid
periods = np.linspace(2.0, 10.0, 1000) * u.day
duration = 0.2 * u.day

periodogram = model.power(periods, duration)

Warning: Period grid quality matters! Too coarse and you'll miss the true period.

Objective Functions

BLS supports two objective functions:

1. Log Likelihood (default)

Maximizes the statistical likelihood of the model fit:

periodogram = model.autopower(0.2 * u.day, objective='likelihood')

2. Signal-to-Noise Ratio (SNR)

Uses the SNR with which the transit depth is measured:

periodogram = model.autopower(0.2 * u.day, objective='snr')

The SNR objective can improve reliability in the presence of correlated noise.

Complete Example

import numpy as np
import matplotlib.pyplot as plt
import astropy.units as u
from astropy.timeseries import BoxLeastSquares

# Load and prepare data
data = np.loadtxt('light_curve.txt')
time = data[:, 0] * u.day
flux = data[:, 1]
flux_err = data[:, 3]

# Create BLS model
model = BoxLeastSquares(time, flux, dy=flux_err)

# Run BLS with automatic period grid
# Try multiple durations to find best fit
durations = np.linspace(0.05, 0.3, 10) * u.day
periodogram = model.autopower(durations, objective='likelihood')

# Find peak
max_power_idx = np.argmax(periodogram.power)
best_period = periodogram.period[max_power_idx]
best_duration = periodogram.duration[max_power_idx]
best_t0 = periodogram.transit_time[max_power_idx]
max_power = periodogram.power[max_power_idx]

print(f"Period: {best_period:.5f}")
print(f"Duration: {best_duration:.5f}")
print(f"T0: {best_t0:.5f}")
print(f"Power: {max_power:.2f}")

# Plot periodogram
import matplotlib.pyplot as plt
plt.plot(periodogram.period, periodogram.power)
plt.xlabel('Period [days]')
plt.ylabel('BLS Power')
plt.show()

Peak Statistics for Validation

Use compute_stats() to calculate detailed statistics about a candidate transit:

# Get statistics for the best period
stats = model.compute_stats(
    periodogram.period[max_power_idx],
    periodogram.duration[max_power_idx],
    periodogram.transit_time[max_power_idx]
)

# Key statistics for validation
print(f"Depth: {stats['depth']:.6f}")
print(f"Depth uncertainty: {stats['depth_err']:.6f}")
print(f"SNR: {stats['depth_snr']:.2f}")
print(f"Odd/Even mismatch: {stats['depth_odd'] - stats['depth_even']:.6f}")
print(f"Number of transits: {stats['transit_count']}")

# Check for false positives
if abs(stats['depth_odd'] - stats['depth_even']) > 3 * stats['depth_err']:
    print("Warning: Significant odd-even mismatch - may not be planetary")

Validation criteria:

• High depth SNR (>7): Strong signal
• Low odd-even mismatch: Consistent transit depth
• Multiple transits observed: More reliable
• Reasonable duration: Not too long or too short for orbit

Period Grid Sensitivity

The BLS periodogram is sensitive to period grid spacing. The autoperiod() method provides a conservative grid:

# Get automatic period grid
periods = model.autoperiod(durations, minimum_period=1*u.day, maximum_period=10*u.day)
print(f"Period grid has {len(periods)} points")

# Use this grid with power()
periodogram = model.power(periods, durations)

Tips:

• Use autopower() for initial searches
• Use finer grids around promising candidates
• Period grid quality matters more for BLS than for Lomb-Scargle

Comparing BLS Results

To compare multiple peaks:

# Find top 5 peaks
sorted_idx = np.argsort(periodogram.power)[::-1]
top_5 = sorted_idx[:5]

print("Top 5 candidates:")
for i, idx in enumerate(top_5):
    period = periodogram.period[idx]
    power = periodogram.power[idx]
    duration = periodogram.duration[idx]

    stats = model.compute_stats(period, duration, periodogram.transit_time[idx])

    print(f"\
{i+1}. Period: {period:.5f}")
    print(f"   Power: {power:.2f}")
    print(f"   Duration: {duration:.5f}")
    print(f"   SNR: {stats['depth_snr']:.2f}")
    print(f"   Transits: {stats['transit_count']}")

Phase-Folded Light Curve

After finding a candidate, phase-fold to visualize the transit:

# Fold the light curve at the best period
phase = ((time.value - best_t0.value) % best_period.value) / best_period.value

# Plot to verify transit shape
import matplotlib.pyplot as plt
plt.plot(phase, flux, '.')
plt.xlabel('Phase')
plt.ylabel('Flux')
plt.show()

BLS vs Transit Least Squares (TLS)

Both methods search for transits, but differ in implementation:

Box Least Squares (BLS)

Pros:

• Built into Astropy (no extra install)
• Fast for targeted searches
• Good statistical framework
• compute_stats() provides detailed validation

Cons:

• Simpler transit model (box shape)
• Requires careful period grid setup
• May be less sensitive to grazing transits

Transit Least Squares (TLS)

Pros:

• More sophisticated transit models
• Generally more sensitive
• Better handles grazing transits
• Automatic period grid is more robust

Cons:

• Requires separate package
• Slower for very long time series
• Less control over transit shape

Recommendation: Try both! TLS is often more sensitive, but BLS is faster and built-in.

Integration with Preprocessing

BLS works best with preprocessed data. Consider this pipeline:

1. Quality filtering: Remove flagged data points
2. Outlier removal: Clean obvious artifacts
3. Detrending: Remove stellar variability (rotation, trends)
4. BLS search: Run period search on cleaned data
5. Validation: Use compute_stats() to check candidate quality

Key Considerations

• Preprocessing should preserve transit shapes (use gentle methods like flatten())
• Don't over-process - too aggressive cleaning removes real signals
• BLS needs reasonable period and duration ranges
• Always validate with multiple metrics (power, SNR, odd-even)

Common Issues

Issue: No clear peak

Causes:

• Transits too shallow
• Wrong duration range
• Period outside search range
• Over-aggressive preprocessing

Solutions:

• Try wider duration range
• Extend period search range
• Use less aggressive flatten() window
• Check raw data for transits

Issue: Period is 2× or 0.5× expected

Causes:

• Missing alternating transits
• Data gaps
• Period aliasing

Solutions:

• Check both periods manually
• Examine odd-even statistics
• Look at phase-folded plots for both periods

Issue: High odd-even mismatch

Cause:

• Not a planetary transit
• Eclipsing binary
• Instrumental artifact

Solution:

• Check stats['depth_odd'] vs stats['depth_even']
• May not be a transiting planet

Dependencies

pip install astropy numpy matplotlib
# Optional: lightkurve for preprocessing
pip install lightkurve

References

Official Documentation

• Astropy BLS Documentation
• Astropy Time Series Guide

Key Papers

• Kovács, Zucker, & Mazeh (2002): Original BLS paper - A&A 391, 369
• Hartman & Bakos (2016): VARTOOLS implementation - A&C 17, 1

Related Resources

• Lightkurve Tutorials
• TLS GitHub - Alternative transit detection method

When to Use BLS

Use BLS when:

• You want a fast, built-in solution
• You need detailed validation statistics (compute_stats)
• Working within the Astropy ecosystem
• You want fine control over period grid

Use TLS when:

• Maximum sensitivity is critical
• Dealing with grazing or partial transits
• Want automatic robust period grid
• Prefer more sophisticated transit models

Use Lomb-Scargle when:

• Searching for general periodic signals (not specifically transits)
• Detecting stellar rotation, pulsation
• Initial exploration of periodicity

For exoplanet detection, both BLS and TLS are valid choices. Try both and compare results!