Alpha Centauri A and B Abundances and the Claimed Composition of the Trisolaran Homeworld: A Test Against the Sophon-Relayed Record

1. Introduction

It has been 49 years now since the Stellar Spectroscopy Group of the Yunnan Astronomical Observatory recorded its first clean echelle spectra of Alpha Centauri A and B, in Year 11 of the Deterrence Era. It was one of the earliest narrow windows in which the partial suspension of sophon interference let a ground-based spectrograph work at its design resolution. The suspension originates with Trisolaris, and the Observatory has logged every window and every lapse since (Vasquez-Hale and Tamsin DE 53). The first published abundances followed in Year 12 (Liao and Baptiste-Odum DE 12), and the Group has kept the pair under observation in each window it has been given.

A second record grew up beside the spectra. Through the sophon-mediated exchange that the Dark Forest Deterrence made possible, statements about the Trisolaran homeworld have reached the UN Planetary Defense Council (PDC) in small numbers, and a PDC compiler has gathered those that bear on composition into one digest (Okafor-Lindgren DE 49). The digest holds a pair of quantitative statements, one giving the homeworld about three Earth masses of refractory solids and the other giving its iron core half of its total mass. We treat both as claims to be tested and never as data, because testimony that passes through the sophons cannot be checked against its origin (Stannard DE 44).

Stellar abundances offer one of the few independent checks.

The physical argument is old and simple. When a rocky planet forms from the disc of a star, it locks refractory elements (those that condense above about 900 K) into solids, and the star is left slightly poorer in them, which in turn gives the abundance trend a downward tilt with rising condensation temperature. A planet that is rich in refractories and iron should therefore leave a deeper trace than an ordinary one. The three stars form a chaotic system, and nothing in the digest says which star the homeworld formed around, so we test both A and B. Proxima is too faint for the resolution we need and is not treated here.

Writing in the Deterrence Era, Year 60, we use only spectra and publications that date from Year 59 or earlier. In Section 2 we describe the spectra, the sophon hazard and the predictions we test; in Section 3 we set out the abundances and the tests; in Section 4 we compare our values with earlier work and state what follows; and in Section 5 we set out the limits of the result.

2. Methods

The spectra are held in the Observatory archive (Yunnan Astronomical Observatory DE 59). They comprise 41 exposures of Alpha Centauri A and 38 of Alpha Centauri B, all taken at a resolving power of 115,000 in 23 logged windows between Year 51 and Year 59. Co-added, they reach a signal-to-noise ratio of 1,150 per pixel for A and 980 for B near 600 nm. Twelve exposures of sunlight reflected from the Moon, taken with the same instrument, provide the solar reference, which makes every abundance a line-by-line difference from the Sun.

Sophon interference with observation is a known hazard, and we declare it here. The partial suspension that opens each window is announced by Trisolaris, we cannot verify it independently, and a sophon that wished to alter a line profile would not need to do so in all windows. We took exposures only inside logged windows. We also divided the windows into odd and even sets and compared the equivalent widths of 212 Fe I lines of star A between the two sets. That check can detect interference that differs between windows. Sophons coordinate without measurable delay, so the check cannot detect a distortion common to every window, and we do not claim that it does.

Stellar parameters were fixed by excitation and ionisation balance of Fe I and Fe II, giving an effective temperature of 5,810 K and a surface gravity of 4.31 for A, and 5,260 K and 4.37 for B. Arguably the most consequential choice in the analysis was the continuum, which we set for each echelle order by an iterative polynomial fit that rejects absorption points, with no placement by eye. We measured 18 elements, of which 15 have condensation temperatures of 900 K or above and 3 (carbon, sulphur and zinc) have lower ones. The temperatures are the 50 per cent values of the working compilation of Ferreira (DE 31). For each star we fitted [X/H] against condensation temperature by weighted least squares, with weights from the quoted abundance errors, and we inflated the slope error by the square root of the reduced chi-squared whenever that exceeded one. Intervals use Student's t with 16 degrees of freedom, and the one-sided limits below use t = 1.746.

The two statements of the digest were turned into predictions before any slope was fitted. For the first, we adopted convective envelope masses of 0.016 solar masses for A and 0.042 for B (Yarrow DE 38), each uncertain by 30 per cent. We took the refractory mass fraction of the envelope to be 0.0078. That is the 0.0045 of rock-forming material (iron, magnesium and silicon with the oxygen bound to them) in a solar mixture, scaled by the +0.24 dex iron abundance that both stars show. It is our own adopted value, and the limits below scale with it. The envelopes then hold 42 and 109 Earth masses of refractory solids. A planet of three Earth masses formed from either envelope would lower every refractory element by the same amount, 0.032 dex in A and 0.012 dex in B. We built the abundance pattern in which the 15 refractory elements are lowered by that amount and the 3 volatile ones are not, and we fitted it with the same weighted least squares and the same weights as the data. The fit returns 0.699 dex per 1000 K of slope for each dex of depletion in A and 0.708 in B, so the predicted slopes are -0.023 and -0.009 dex per 1000 K. The 30 per cent envelope uncertainty carries into the predicted slope in proportion, and we added it in quadrature to the slope error whenever we compared slope and prediction.

For the second statement we derived the core mass fraction of a rocky body made from the stellar material. We took number ratios Mg/Si and Fe/Si from the measured differences and the solar values (log ε of 7.55, 7.51 and 7.47 for Mg, Si and Fe), placed 90 per cent of the iron in the core and the remainder as FeO in a mantle of MgO and SiO₂ (Ilunga DE 20), and propagated the errors of the three element abundances numerically, with silicon entering both ratios. We varied the core share of iron between 80 and 100 per cent to set a systematic term. The claimed value is 0.50.

3. Results

Table 1 gives the abundances. The iron abundance is +0.24 ± 0.01 for A and +0.25 ± 0.01 for B, and neither star shows an abundance that departs from the common run by more than 0.05 dex in any element.

The trend with condensation temperature is flat in both stars. For A the slope is +0.011 ± 0.016 dex per 1000 K (95% CI -0.023 to 0.045, p = 0.50), and for B it is -0.019 ± 0.019 dex per 1000 K (95% CI -0.059 to 0.020, p = 0.31). The reduced chi-squared values are 1.04 and 1.14. Neither slope differs from zero, and the scatter about each fit is no larger than the quoted errors allow.

Three Earth masses of refractory solids predict slopes of -0.023 for A and -0.009 for B, which are small, and the test is correspondingly weak. For A the observed slope exceeds the prediction by 0.034 dex per 1000 K, which is 2.11 formal standard errors (one-sided p = 0.025). With the 30 per cent uncertainty on the predicted slope added in quadrature, the separation is 1.94 standard errors and the one-sided p value is 0.035. For B the observed slope lies 0.011 dex per 1000 K below the prediction, so the data do not disfavour it at all (-0.58 standard errors, one-sided p = 0.72).

The one-sided 95% lower bound of each slope (t = 1.746 on 16 degrees of freedom) converts, through the same relation, to an upper limit on the refractory mass locked away of 2.3 Earth masses for A and 17.0 for B, against a claimed 3.0. If the envelopes are 30 per cent more massive than we assume, the limits become 2.9 and 22.1 Earth masses, and if they are 30 per cent less massive they become 1.6 and 11.9. The limit for A reaches 3.0 only when the envelope refractory mass is 33 per cent larger than we adopt. The claim is therefore excluded for A, narrowly, and cannot be tested for B, where the slope error corresponds to a depletion that would remove about 17 Earth masses and a planet of three would shift the refractory elements by only 0.012 dex.

Our core fraction is also low, since we obtain Mg/Si of 1.07 and 1.12 and Fe/Si of 0.87 and 0.87 for A and B, which give core mass fractions of 0.286 ± 0.007 (statistical) for A and 0.282 ± 0.008 for B, each with a systematic term of 0.034. The claimed 0.50 is 0.21 above the value for A and 0.22 above that for B, or 6.1 and 6.3 times the combined statistical and systematic error. The core claim is excluded for both stars.

A window-split check found a mean difference in equivalent width of 0.03 mÅ between odd and even windows over the 212 lines (95% CI -0.07 to 0.13 mÅ), so no interference that varies from window to window is visible.

Our values agree with the first Yunnan analysis for iron and nickel to within 0.03 dex, and they differ for titanium, where the earlier work gave +0.17 and we obtain +0.25 for A. The earlier slope for B was -0.058 ± 0.031 dex per 1000 K from 11 elements, which Liao and Baptiste-Odum read as a refractory deficit. That is 0.039 dex per 1000 K below ours. A historical reanalysis of the Year 11 spectra with an automated continuum recovers -0.024 ± 0.033 (Prokhorenko DE 57), and so we attribute the difference to the hand placement of the continuum in the original reduction (albeit a careful one).

4. Discussion

The abundances give the composition that the digest attributes to the Trisolaran homeworld no support, but they do not exclude every part of it. A half-iron core is excluded for both stars, because the iron and magnesium to silicon ratios of either star give a core fraction near 0.29, more than six combined errors below the claim. Three Earth masses of refractory solids are excluded for A only, and narrowly, with an upper limit of 2.3 Earth masses that rises to 2.9 if the envelope is 30 per cent more massive. For B the limit of 17.0 Earth masses lies far above the claim, and a planet of three Earth masses would lower the refractory elements by 0.012 dex, inside the scatter of the abundances.

This is a negative result, and its extent matters as much as its sign. The relayed composition has no support in the abundances of either star. A refractory-rich planet around B is neither shown nor excluded.

Liao and Baptiste-Odum read their slope for B as a refractory deficit in the system. With the continuum set by a fixed procedure that deficit is absent, and the slope we measure for B lies 0.039 dex per 1000 K above theirs.

We write within the Luo-Ji / Ye framework, and within it the result carries a modest strategic weight. The relayed composition passed through an intermediary that had reason to shape it, so it cannot be taken at face value. The abundances contradict its core claim and give the rest no support, which in turn shows that the chain of suspicion extends even to physical statements about the Trisolaran homeworld.

For habitability the data say little beyond what they exclude. Both stars carry the iron-to-silicate ratio and magnesium-to-silicon ratio of an ordinary rocky inheritance, with a core fraction near 0.29 and a mantle of olivine and pyroxene. We find no abundance-based obstacle to a stable silicate planet. Water, atmosphere and the orbital history of a planet in a chaotic triple system lie outside what a photosphere can tell us.

5. Limitations

The sophon hazard is the first limit. The window-split check cannot detect interference that is common to all windows, and the suspension itself rests on Trisolaran announcements that we cannot verify. A coordinated distortion of equal size in every exposure would shift all abundances together and could leave a slope unchanged, though it could not restore the slope the claim requires without also altering the line-by-line pattern.

Further limits belong to the test itself. The exclusion of the refractory-mass claim for A is narrow and rests on two adopted inputs, the envelope mass and the rock fraction of 0.0078, and the limit rises in proportion to either. The volatile group contains only three elements. The envelope masses come from structure models and not from measurement. The core partition is a simple stoichiometric model, and the digest itself carries no formation history, so we have assumed formation from the material of one of the two stars. The digest reaches us through an intermediary that, by its nature, has every incentive to be selective, and we treat its statements only as claims.

How a planet survives the chaotic dynamics of this triple system is still not completely understood, but the present data favour an ordinary rocky inheritance for the stars' material, with no iron excess in either star and, for A, no refractory excess as large as the one claimed. This paper extends the abundance series that Liao and Baptiste-Odum began in Year 12, and the next set of windows will add the exposures needed to tighten the slope for B.