Closeby Habitable Exoplanet Survey (CHES). V. Planetary Parameters Derived from Angular Separation Variations

This paper proposes a novel relative astrometric measurement model for the Closeby Habitable Exoplanet Survey (CHES) that relies solely on variations in the angular separation length between target and reference stars, thereby overcoming the limitations of Gaia catalog-based absolute astrometry and traditional direction-dependent methods to achieve the microarcsecond precision required for detecting Earth-like planets around nearby FGK stars.

The Gist

Imagine you are trying to spot a tiny, invisible fly buzzing around a giant, bright lighthouse in the middle of a foggy ocean. You can't see the fly directly, but you know that if the fly is heavy enough, it will make the lighthouse wobble just a tiny bit as it orbits.

This is the challenge astronomers face when trying to find Earth-like planets orbiting other stars. The stars are the lighthouses, and the planets are the flies. The "wobble" is so incredibly small that it's like trying to measure the width of a human hair from 400 kilometers away.

This paper introduces a new, clever way to measure that wobble for the Closeby Habitable Exoplanet Survey (CHES), a future space mission. Here is the breakdown of their new method using simple analogies.

The Old Problem: The Shaky Ruler

Traditionally, to measure a star's wobble, astronomers used a "ruler" made of other stars in the sky (a reference frame). They would measure the target star's position relative to these background stars.

However, this method has two big flaws:

1. The Ruler is Moving: The background stars aren't actually fixed; they are drifting through space. Over time, your "ruler" gets blurry and inaccurate.
2. The Camera is Rotating: The space telescope has to turn to look at different stars. This rotation makes it incredibly hard to know exactly which way is "up" or "down" in the sky, introducing errors. It's like trying to measure the distance between two people while standing on a spinning merry-go-round.

The New Solution: Measuring the "Stretch"

Instead of trying to measure where the star is on a map (which requires knowing exactly which way the camera is pointing), the authors propose measuring only the distance between the target star and a nearby reference star.

The Analogy: The Rubber Band
Imagine the target star and a reference star are two people holding opposite ends of a rubber band.

• You don't care about the direction they are facing or where they are standing on a map.
• You only care about how long the rubber band gets or shrinks over time.

Even if the whole group is spinning (the telescope rotating) or the people are walking in different directions (stars moving through space), the change in the length of the rubber band tells you exactly what is happening.

What Makes the Rubber Band Stretch?

The paper explains that the length of this "rubber band" changes due to several cosmic forces, and their new mathematical model accounts for all of them:

1. The Drift (Proper Motion): The stars are naturally walking through space. This slowly changes the distance between them.
2. The Perspective Shift (Parallax): As the Earth (and the telescope) orbits the Sun, our viewpoint changes. It's like holding your finger up and closing one eye, then the other; your finger seems to jump back and forth. This makes the "rubber band" stretch and shrink once a year.
3. The Wobble (Planets): If an Earth-like planet is orbiting the target star, it pulls the star back and forth. This causes a tiny, rhythmic change in the rubber band's length.
4. The "Glitch" Effects: The paper also corrects for weird physics, like light bending around the Sun (gravitational lensing) and the telescope moving so fast that light seems to come from a slightly different angle (aberration).

Why This is a Game-Changer

By focusing only on the length of the connection between stars, the scientists remove the need for a perfect, unchanging map of the universe.

• No more "Up" or "Down": They don't need to know if the telescope is tilted.
• No more "Drifting Rulers": They don't need to rely on old catalogs that get outdated.
• Super Precision: They can measure these length changes down to microarcseconds. To visualize this: it's like measuring the thickness of a coin from the distance of the Moon.

What They Tested

The team ran computer simulations to see if their "rubber band" math works for different scenarios:

• Earth-like Planets: They simulated a tiny wobble caused by an Earth twin. The model successfully found it.
• Jupiter-like Planets: They simulated massive planets. The model found these easily because they make the "rubber band" stretch a lot.
• Black Holes: They even tested it on invisible black holes tugging on stars. The model could detect the gravitational tug of these dark giants.

The Bottom Line

This paper proposes a smarter way to hunt for new worlds. Instead of trying to draw a perfect map of the universe while standing on a spinning, shaky platform, they suggest simply measuring the distance between two stars. If that distance wiggles in a specific rhythm, it means there is a planet (or a black hole) dancing around the star, invisible to the naked eye but revealed by the stretch of the cosmic rubber band.

This method could help the CHES mission find Earth 2.0, and it might even be used by other telescopes that aren't even designed for this kind of work, turning them into planet hunters.

Technical Summary

1. Problem Statement

The detection of Earth-like exoplanets in the habitable zones of nearby stars requires microarcsecond ($\mu$as) level astrometric precision. Current and traditional methods face two primary limitations:

• Absolute Astrometry Limitations: Reliance on global catalogs like Gaia is insufficient for long-term, high-precision detection. While Gaia provides $\sim20$ $\mu$as precision, the temporal propagation of errors in proper motion and parallax leads to increasing positional uncertainties over time, degrading the ability to detect subtle Earth-like signals.
• Traditional Relative Astrometry Limitations: Conventional relative astrometry measures positional shifts in Right Ascension (RA) and Declination (Dec). This requires precise knowledge of the field rotation (satellite attitude changes) and a fixed reference direction. Determining these angles introduces significant systematic errors from catalog uncertainties, spacecraft attitude control, and instrumental effects, often dominating the error budget.

The Closeby Habitable Exoplanet Survey (CHES) mission aims to overcome these hurdles but needs a measurement model that does not rely on a global reference frame or precise attitude determination.

2. Methodology

The authors propose a novel one-dimensional relative astrometric model based solely on the temporal variation of the angular separation length between a target star and multiple reference stars. This approach is independent of the orientation (direction) of the separation vector.

Key Components of the Model:

The model mathematically reconstructs the variation in angular separation ($l$) by integrating several physical effects:

1. Proper Motion & Perspective Acceleration: Instead of fitting RA/Dec components, the model treats stars as moving along great circles on the celestial sphere defined by their total proper motion. It incorporates perspective acceleration (a geometric effect caused by radial velocity changing the apparent transverse velocity over time).
2. Parallax: The model accounts for the annual apparent shift of stars due to the observer's orbit around the Sun (specifically from the Sun-Earth L2 point). It calculates the displacement vector toward the Sun and its projection onto the angular separation line.
3. Relativistic Corrections:
   • Stellar Aberration: Caused by the satellite's velocity relative to incoming starlight.
   • Gravitational Lensing: Light deflection by solar system bodies (dominated by the Sun).
   • Note: These effects are nearly identical for stars within the same narrow field of view, allowing them to be modeled as differential corrections.
4. Planetary Perturbations:
   • The model uses the Thiele-Innes (TI) formalism to describe planetary-induced stellar reflex motion.
   • Since only the length of the separation is measured, the planetary signal is treated as a projection of the star's 3D orbital motion onto the 1D line connecting the target and reference stars.
   • By observing the target against multiple reference stars (typically $\ge 6$) in different directions, the 1D projections can be combined to reconstruct the full two-dimensional orbital motion of the star.

Fitting Strategy:

• Parameters: The model fits 18 parameters, including initial separations, proper motion magnitudes, radial velocities, parallaxes, geometric angles, and planetary orbital elements (eccentricity $e$, period $P$, mean anomaly $M_0$, and projection coefficients $D_1, D_2$).
• Algorithm: Parameter estimation is performed using the dynesty nested sampling algorithm to explore the posterior distribution and evaluate Bayesian evidence.
• Data Input: The method requires only time-series measurements of angular separation lengths, eliminating the need for RA/Dec coordinates or global reference frames.

3. Key Contributions

• Decoupling from Reference Frames: The method removes the dependency on external catalogs (like Gaia) for defining the reference frame and eliminates the need to determine field rotation angles, significantly reducing systematic errors.
• 1D to 2D Reconstruction: It demonstrates how one-dimensional angular separation variations, when observed against multiple reference stars, can mathematically reconstruct the full two-dimensional astrometric orbit of a star.
• Unified Physical Model: It provides a comprehensive framework that simultaneously models proper motion, perspective acceleration, parallax, aberration, lensing, and planetary perturbations within a single fitting routine.
• Applicability to Non-Astrometric Missions: The model is robust enough to be applied to photometric missions (e.g., Kepler) if they can achieve high-precision relative angular measurements, expanding the scope of exoplanet detection.

4. Results (Numerical Simulations)

The authors validated the model through simulations of three distinct scenarios, assuming a measurement precision of 1 $\mu$as (CHES design goal):

• Earth-like Planets:
  • Target: HD 88230 with a hypothetical Earth-like planet (1 $M_{\oplus}$, 1 AU).
  • Result: The model successfully recovered the planetary signal ($\sim1$ $\mu$as) and reconstructed the 2D orbit. While the Signal-to-Noise Ratio (SNR) is low ($\sim1$), the residual-based approach effectively isolated the planetary signal from proper motion and parallax.
• Jovian Planets:
  • Targets: HD 88230 (Warm Jupiter) and HD 147513 (Classical Jovian).
  • Result: For high-SNR cases (signals of tens to hundreds of $\mu$as), the model robustly fitted all five planetary parameters ($e, P, M_0, D_1, D_2$) and accurately reconstructed the orbital motion. The fitted parameters matched the input "true" values with high precision.
• Black Holes:
  • Targets: Gaia BH1, BH2, and BH3.
  • Result: The model successfully detected the gravitational influence of stellar-mass black holes on their companion stars.
  • Low-Precision Test: The model was also tested using Kepler-level precision ($\sim4$ mas). While less precise than dedicated astrometry, it still detected the orbital perturbations, proving the method's versatility for identifying dark compact objects in photometric data.

5. Significance

• Enabling CHES: This methodology provides the theoretical foundation for the CHES mission to achieve its goal of detecting Earth-like planets, overcoming the specific limitations of field rotation and catalog errors inherent in narrow-field observations.
• Beyond Exoplanets: The framework is not limited to exoplanets; it is applicable to binary stars, compact objects (black holes, neutron stars), and potentially dark matter sub-halos, provided they induce Keplerian perturbations.
• Future Missions: By relying on relative angular separation rather than absolute coordinates, this approach offers a pathway for future space missions (both astrometric and photometric) to achieve microarcsecond-level science without the massive computational and calibration overhead of global reference frame construction.
• Data Synergy: Since the data consists of 1D separation variations, observations from different missions or epochs can be combined, allowing for the joint analysis of stellar motion signals that transcend individual mission limitations.

In conclusion, the paper presents a robust, mathematically rigorous solution to the "reference frame problem" in high-precision astrometry, enabling the detection of Earth-analogs and other compact objects through a novel, direction-independent measurement strategy.