Detectability of Stars Using the ESPRESSO Spectrograph at VLT

Assessing Stellar Observability Based on Signal-to-Noise Ratio with ESPRESSO

Key Findings

• Four out of six stars are detectable: Canopus, Polaris, and two stars at 5 pc and 10 pc distances are within ESPRESSO's detection capabilities.
• Apparent magnitude is crucial: Stars with apparent magnitude up to 15 can achieve S/N ≥ 10 with ESPRESSO during a 1-hour exposure.
• Distance affects detectability: Stars at greater distances with the same absolute magnitude become fainter and may fall below ESPRESSO's sensitivity.

Introduction

The ESPRESSO spectrograph, mounted on one of the 8-meter Very Large Telescope (VLT) units at the Paranal Observatory, is designed for high-precision spectroscopy. One of its key performance metrics is the ability to achieve a signal-to-noise ratio (S/N) of at least 10 per binned pixel during a 1-hour exposure. This analysis assesses the detectability of six specific stars using ESPRESSO under these conditions.

Understanding ESPRESSO's Sensitivity

ESPRESSO (Echelle SPectrograph for Rocky Exoplanet and Stable Spectroscopic Observations) is capable of observing stars within a certain range of apparent magnitudes, depending on the desired S/N and exposure time. Factors influencing detectability include the star's apparent magnitude, exposure duration, atmospheric conditions, and the instrument's efficiency.

Apparent Magnitude and Detectability

Apparent magnitude (m) describes how bright a star appears from Earth. It is calculated using the absolute magnitude (M) and the distance (d) in parsecs, based on the distance modulus formula:

\[ m = M + 5 \log_{10}(d) - 5 \]

The higher the apparent magnitude, the fainter the star appears. ESPRESSO can typically achieve S/N ≥ 10 in a 1-hour exposure for stars with apparent magnitudes up to around 15–16, depending on observing conditions and instrument settings.

Star Analysis Summary

The following table summarizes the key parameters for each star and their detectability status:

Detailed Analysis of Each Star

a) Canopus

Parameters:

• Absolute Magnitude (V): -5.71
• Distance: 95 pc

Apparent Magnitude Calculation:

\[ m = M + 5 \log_{10}(d) - 5 \]

\[ m = -5.71 + 5 \log_{10}(95) - 5 \]

\[ m \approx -5.71 + 5 \times 1.9777 - 5 \]

\[ m \approx -5.71 + 9.8885 - 5 \]

\[ m \approx -0.8215 \]

Canopus has an apparent magnitude of approximately -0.82, making it one of the brightest stars in the sky.

Detectability: Due to its high brightness, Canopus is easily detectable by ESPRESSO, achieving S/N ≥ 10 well within a 1-hour exposure. Care must be taken to avoid detector saturation.

b) Polaris

Parameters:

• Absolute Magnitude (V): -3.64
• Distance: 133 pc

Apparent Magnitude Calculation:

\[ m = M + 5 \log_{10}(d) - 5 \]

\[ m = -3.64 + 5 \log_{10}(133) - 5 \]

\[ m \approx -3.64 + 5 \times 2.1239 - 5 \]

\[ m \approx -3.64 + 10.6195 - 5 \]

\[ m \approx 1.9795 \]

Polaris has an apparent magnitude of approximately 1.98.

Detectability: Polaris is bright and well within ESPRESSO's detection capabilities.

c) Star at 10 pc (Absolute V magnitude 15)

Parameters:

• Absolute Magnitude (V): 15
• Distance: 10 pc

Apparent Magnitude Calculation:

\[ m = M + 5 \log_{10}(d) - 5 \]

\[ m = 15 + 5 \log_{10}(10) - 5 \]

\[ m = 15 + 5 \times 1 - 5 \]

\[ m = 15 + 5 - 5 \]

\[ m = 15 \]

Detectability: An apparent magnitude of 15 places this star at the limit of ESPRESSO's sensitivity for achieving S/N ≥ 10 in a 1-hour exposure. Under optimal observing conditions and instrument settings, this star is detectable, though it is near the faint-end limit.

d) Star at 200 pc (Absolute V magnitude 15)

Parameters:

• Absolute Magnitude (V): 15
• Distance: 200 pc

Apparent Magnitude Calculation:

\[ m = M + 5 \log_{10}(d) - 5 \]

\[ m = 15 + 5 \log_{10}(200) - 5 \]

\[ m = 15 + 5 \times 2.3010 - 5 \]

\[ m = 15 + 11.505 - 5 \]

\[ m \approx 21.505 \]

Detectability: With an apparent magnitude of approximately 21.51, this star is too faint for ESPRESSO to achieve S/N ≥ 10 in a 1-hour exposure. It falls well below the instrument's sensitivity threshold for such an exposure time.

e) Star at 5 pc (Absolute V magnitude 15)

Parameters:

• Absolute Magnitude (V): 15
• Distance: 5 pc

Apparent Magnitude Calculation:

\[ m = M + 5 \log_{10}(d) - 5 \]

\[ m = 15 + 5 \log_{10}(5) - 5 \]

\[ m = 15 + 5 \times 0.6990 - 5 \]

\[ m = 15 + 3.495 - 5 \]

\[ m \approx 13.495 \]

Detectability: With an apparent magnitude of approximately 13.49, this star is well within ESPRESSO's detection capabilities for achieving S/N ≥ 10 in a 1-hour exposure.

f) Star at 50 pc (Absolute V magnitude 15)

Parameters:

• Absolute Magnitude (V): 15
• Distance: 50 pc

Apparent Magnitude Calculation:

\[ m = M + 5 \log_{10}(d) - 5 \]

\[ m = 15 + 5 \log_{10}(50) - 5 \]

\[ m = 15 + 5 \times 1.6989 - 5 \]

\[ m = 15 + 8.4945 - 5 \]

\[ m \approx 18.4945 \]

Detectability: With an apparent magnitude of approximately 18.49, this star is too faint to achieve S/N ≥ 10 during a 1-hour exposure with ESPRESSO.

Factors Affecting Detectability

Instrument Sensitivity

ESPRESSO's sensitivity enables it to observe stars with apparent magnitudes up to approximately 15–16 under optimal conditions in a 1-hour exposure. For fainter stars, achieving the required S/N ratio would require longer exposures or might not be feasible.

Atmospheric Conditions

The Earth's atmosphere can affect observations through absorption, scattering, and turbulence. Observing conditions such as sky brightness, seeing, and atmospheric transparency play significant roles in the achievable S/N ratio.

Exposure Time

Longer exposure times increase the number of photons collected from the target, improving the S/N ratio. However, practical limitations such as telescope scheduling and tracking accuracy limit exposure durations.

Conclusion

Based on the analysis, four of the six listed stars are detectable using the ESPRESSO spectrograph coupled with one of the 8m VLT telescopes during a 1-hour exposure:

• Canopus (a)
• Polaris (b)
• Star at 10 pc with absolute magnitude 15 (c)
• Star at 5 pc with absolute magnitude 15 (e)

The stars at 200 pc (d) and 50 pc (f) are too faint to achieve the required S/N ratio in a 1-hour exposure with ESPRESSO.

Therefore, the correct answer is Answer C: 4.

References

eso.org

ESPRESSO Overview - ESO

eso.org

ESPRESSO User Manual