Testing Axion-Like Particles with X-Ray Emission from Magnetic White Dwarf Stars

Home , Horizontal branch, List of white dwarfs, Red giant, White dwarf

with Chris Dessert & Ben Safdi (1903.05088)

Testing axion-like particles with X-ray emission from magnetic white dwarf stars

Andrew J. Long! University of Michigan! @ Pheno 2019! May 6, 2019! Andrew J. Long (UMich)! Axion-like particles

Interactions with Standard Model particles è Leading-order interactions are dimension-5 operators

1 2 1 1 gaee L = @ a m2a2 g aF F˜ + e¯µ e @ a ALP µ a a 5 µ 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 2 2 4 2me ···

Connection to new physics scale è Couplings arise from new physics at the scale fa è Very weakly interacting!

↵ me ga = C gaee = Ce

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 2⇡fa 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 fa

2 Andrew J. Long (UMich)! Testing ALPs with white dwarf cooling [Raﬀelt (1986)] [Nakagawa, Adachi, Kohyama, & Itoh (1987,88)] Axions are produced via bremstrahlung è axion luminosity:

2 4 4 gaee MWD Tc LLAAACV3iclVDLTsJAFJ3WF+ILdOmmkZi4oi2ayMqQ6MKFJhgtkAAh02EKE+bRzExV0vAJbvXb+BodoAsBN95kkpPzyL1zwpgSpT1vatkbm1vbO7nd/N7+weFRoXjcUCKRCAdIUCFbIVSYEo4DTTTFrVhiyEKKm+HodqY3X7FURPAXPY5xl8EBJxFBUBvq+aEHe4WSV/bm46wDPwMlkE29V7RuOn2BEoa5RhQq1fa9WHdTKDVBFE/ynUThGKIRHOC2gRwyrLrp/NaJc26YvhMJaR7Xzpz9nUghU2rMQuNkUA/VqjYj/9LaiY6q3ZTwONGYo8WiKKGOFs7s406fSIw0HRsAkSTmVgcNoYRIm3qWtriBMpQLeV/iNyr4wL2TIg7Fu/tIFHLZODOqyf9iZhEzmbxp3F/tdx00KmX/slx5uirVqln3OXAKzsAF8ME1qIF7UAcBQGAAPsAn+LKm1re9becWVtvKMidgaeziDxh4tY4= a a 1.6 10 L ' ⇥ 10 13 1 M 107 K 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 ✓ ◆✓ ◆ ⇣ ⌘

aAAACkHiclVFNSxxBEO2daGI2mqg55tJkEXJyZkSItxjiIQQCCllX2FmkpqdmbeyPobsm6zDML8g1+XH+G3vXOUTNJQUNj1fvUdX18kpJT0lyO4iera0/f7Hxcvhqc+v1m+2d3XNvaydwLKyy7iIHj0oaHJMkhReVQ9C5wkl+/WXZn/xE56U1P6ipcKZhbmQpBVCgzuBye5TsJ6viT0HagxHr6/RyZ9BkhRW1RkNCgffTNKlo1oIjKRR2w6z2WIG4hjlOAzSg0c/a1aYd3wtMwUvrwjPEV+zfjha0943Og1IDXfnHvSX5r960pvJo1kpT1YRG3A8qa8XJ8uW3eSEdClJNACCcDLtycQUOBIXjPJgSj32gYjCFw4WyZh6fOFvl9ib+Lr2IddMLffd/tjBIB88wM7gQVuugbLPJSddmhDfUZl7wbHmUErRUTVuR7rKQa9h6dapF0XUhrPRxNE/B+cF+GvDZ4eg46WPbYO/Ye/aBpewjO2Zf2SkbM8GQ/WK/2Z9oNzqKPkWf76XRoPe8ZQ8q+nYHhq/Mag==

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 Tc Axion emission causes WDs to cool è constrains the axion-electron coupling: 13 g < 3 10 (3) 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 aee ⇥ [PDG & Miller Bertolami et al (2014)]

3 Andrew J. Long (UMich)! A hint of new physics!

Evidence of anomalous cooling? è Seen in white dwarf, red giant branch, and horizontal branch stars

[Gianno et. al. (2017)] Evidence for ALPs? 0.8 3σ WD+RGB+HB è favored couplings are … 0.6 2σ ]

1 11 1 - ga 1.4 10 GeV

GeV

' ⇥ 10 13 - 0.4 1σ

10 [

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 gaee 1.5 10 γ a ' ⇥ g ALPS II 0.2

IAXO 0.0 A target for future observations! 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -13 gae [10 ]

4 Andrew J. Long (UMich)! Detecting astrophysical ALPs at Earth

Can we detect the radiated ALPs when they reach Earth? è Axion energy flux density at Earth: F = L / (4⇡d2 ) 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 a a WD g

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 a Axions are converted into photons in a B-field è Conversion probability is given by

2 2 2 pa g B L /4 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 T 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 a T ! ⇡

Predicted signal (is very weak): 2 4 31 2 gaee MWD Tc F 4 10 erg/cm /sec 13 7 ' ⇥ 10 1 M 10 Kel ✓ ◆✓ ◆ 2 ⇣2 ⌘ 2 2 ga BT L dWD 10 11/GeV 5T 100 cm 10 pc

AAAEK3iclVJbaxNBFN4mXmq8tfroy8GitGCb3STSgiBFRQUtVEjaQiddZmdPtmNnL52ZtQ3D/CR/jb4ovvo/nE3SS3p58MAuZ875vnP5OFEhuNK+/2umVr9x89bt2TuNu/fuP3g4N/9oS+WlZNhjucjlTkQVCp5hT3MtcKeQSNNI4HZ08LbKb39DqXiedfWwwH5Kk4wPOKPahcL5mZ/vQ5LQNKXwHIjiKR5Cg0Q8EYvQAaJdQEHg75nldmCBoEygCSSlel+mhqV2r3XurZA5jCPLJSAvgAgc6EUgA0mZMWQ0rIlEiRaS0FBEa60Z1Q7a1hElT/b1kqs4RdwIicZjbYhicBRXDCAbqsxOCdPwbshGRVeBnM71CcW5+h23YJZV6x6WND7ZsXHltBLj8bBjjcb/s7GdJGfbf8Ct67d4E3ateQmki0rQa1Gfq8I+ECfsdZD4khwOXzB7AjfLLQvh3IK/4o8MLjvBxFnwJrbp7uA1iXNWpphpJqhSu4Ff6L6hUnMm0DZIqbCg7IAmuOvcjDrN+mYkkoVnLhLDIJfuyzSMoucZhqZKDdPIISut1MVcFbwqt1vqwVrf8KwoNWZs3GhQCtA5VMcMMZfItBg6hzLJ3azA9qmTSbuTn+rS7CkXatIslngk8ixpvpN5EeXHzQ2uWDMdToDK/h/NNUodp+EUDy7qe9nZaq0E7ZXWl87C+tpE+1nviffUW/QCb9Vb9z56m17PY7VXNVr7Wjuof6//qP+u/xlDazMTzmNvyup//wEbgFwi ⇥ ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ 5 Andrew J. Long (UMich)! A stronger B-ﬁeld at the source!

Often compact stars already sustain strong B-fields è Neutron stars (magnetars): ~1012 – 1015 Gauss è Magnetic white dwarfs: ~106 – 109 Gauss

The conversion of axions into photons will occur near the star!

è Previous talk: Kuver Sinha … axions from magnetars produce an X-ray signal

è This talk: Andrew Long … axions from magnetic white dwarfs produce an X-ray signal

è Next talk: Josh Foster … dropping axion dark matter onto a magnetic neutron star produces a radio signal

6 Andrew J. Long (UMich)! Axion conversion in MWD magnetosphere [D. E. Morris (1986); Raﬀelt & Stodolsky (1987)] [Forn & Sinha (2018)] [Dessert, Long, & Safdi (2019)] Focus on ALP production in magnetic white dwarfs (MWDs) è ALPs convert to X-ray photons as they exit the star

L AAACm3iclVFNSxxBEO2dfOnmS5OjBJosgZycGRGSo6CHIAYMZFxhe1lqenrWxv6iu8d1aOZveI1/K/8mPescouaSgoLHq3pU8V5pBHc+y36PkidPnz1/sbE5fvnq9Zu3W9vvzpxuLGUF1ULb8xIcE1yxwnMv2LmxDGQp2LS8POzn0ytmHdfqp28Nm0tYKl5zCj5S5GQRyBKkhAV0i61JtputCz8G+QAmaKjTxfaoJZWmjWTKUwHOzfLM+HkA6zkVrBuTxjED9BKWbBahAsncPKyf7vCnyFS41ja28njN/q0IIJ1rZRk3JfgL93DWk/+azRpff50HrkzjmaJ3h+pGYK9x7wCuuGXUizYCoJbHXzG9AAvUR5/uXUkLF6kUVGXZSmi1TI+sNqW+Tr9zR1PZDouu+z9ZPCSjZkwUW1Ed3VdVINOjLhDPrn0gjmLSm1KD5KINxsuOxIjj12urVlXXh5U/jOYxONvbzSP+sT852Bti20A76CP6jHL0BR2gb+gUFYgig27QL3SbfEgOk+Pk5G41GQ2a9+heJcUfwWfROw== a

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 Te↵ ~ 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 ga

B 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

7 signal = thermal X-ray emission (Tc ~ 10 K ~ keV) 4 background = surface emission negligible (Teff ~ 10 K)

7 Andrew J. Long (UMich)! Conversion probability (reﬁned calculation)

Axion-photon conversion in a weak & homogenous B-field 2 2 2 pa g B RWD 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 a T ! ⇠

But more generally … è The B-field falls off moving away from the star

B = gaBT /2 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 B A = m2/(2E) i@r + E + k k =0 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 a a B a a 2 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 7 ↵ BT  ✓ ◆ ✓ ◆ = E k 2 45⇡ B2 (Euler-Heisenberg boundary condition: A (r = 0) = 0 AAACq3iclVDLbhMxFHWGVwmvtCzZWERIbMhM0qJ2Q1VBkdggFdFpKzJhdMe5k1j1S7YHiEbzZXwJS7bwE3jSWdCWDVeyfHzuObq+pzCCO58kP3rRjZu3bt/ZuNu/d//Bw0eDza0TpyvLMGVaaHtWgEPBFaaee4FnxiLIQuBpcf6m7Z9+Qeu4Vsd+ZXAmYaF4yRn4QOWDNDtE4SHPDFgQAgV9RbPSAqt3m3rS0LfdKwNhltDUOy8zw5uOfJ0ff5404cok+KWVNbPcN4HKB8NklKyLXgfjDgxJV0f5Zm8/m2tWSVSeCXBuOk6Mn9VgPWcCm35WOTTAzmGB0wAVSHSzer1/Q58FZk5LbcNRnq7Zvx01SOdWsgjK9qPuaq8l/9WbVr7cm9VcmcqjYheDykpQr2kbJp1zi8yLVQDQ7s4ZZcsQJPMh8ktT4tQFKgY1t/hVaLWID602hf4Wv+eOxXLVCV3zf7YwSAZPPyQ+vprvdXAyGY23R5MPO8ODvS77DfKEPCXPyZjskgPyjhyRlDDynfwkv8jv6EX0MfoUZRfSqNd5HpNLFeEfUoHVvw== crit photon self-int’n) 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 k

è Axion-to-photon conversion probability 2 pa A (r = ) 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 ! / k 1 8 Andrew J. Long (UMich)! Conversion probability (reﬁned calculation)

For a dipole profile B-field profile 3 r B (r)=B T T,0 ⇥ R 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 ✓ WD ◆ 2 2 2 i 2 (B,0 RWD) ( 5 ) ( 5 , 5 ,0RWD) pa 4 k ! 3/5 ⇡ ( ,0 RWD) 5 5 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 k

10 2/5 ∝ BT,0

5 4 10

× 1 2 →γ ∝ B

a T,0 p 0.5 photon self-interacon (Euler-Heisenberg) suppresses mixing

0.1

5 10 50 100 500 1000 9 BT,0 [MG] A quasi-thermal X-ray spectrum

The spectrum of bremm’d axions è Very close to thermal @ core temperature Tc

dLa 3 E/Tc E / (e 1) 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 dE /

The spectrum of secondary (X-ray) photons è Very slight energy modulation by axion-photon conversion

dF dLa pa (E) a = ! 10-10 2 axion spectrum

dE dE 4⇡d ] photon spectrum 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 WD sec / 10-12 2 Suzaku (95% CL) cm / erg [ 10-14 dE / dF 10-16 × E

10-18 0.1 0.5 1 5 10 50 10 E [keV] Andrew J. Long (UMich)! How many MWDs are there?

The Gaia survey maps out nearby WDs è Expects to see 100% of WDs with 100 pc of Earth. [Torres et. al. (2015) è Current catalog contains ~70,000 WDs.

The number of known MWDs is far fewer è Roughly 300-550 have been identified. [Ferrario, deMarno, & Gansike (2015)]

B-field measured from spectra ① Measure the spectrum well ② MWD contains hydrogen (Balmer) ③ B-field induces Zeeman effect ④ Spectra splitting ó B-field

11 1

X-ray signatures of axion conversion in magnetic white dwarf stars Supplementary Material Christopher Dessert, Andrew J. Long, and Benjamin R. Safdi

This Supplementary Material is organized as follows. Section I includes a list of several MWD stars that are expected to be promising candidates for observations of axion-induced X-ray ﬂux. In Sec. II we present a more general formalism for calculating the axion-to-photon conversion probability to account for the fact that the axions are emitted isotropically and homogeneously throughout the WD core. In Sec. III we perform a more detailed study of the X-ray emission from RE J0317-853, including a more accurate modeling of its magnetic ﬁeld structure, an assessment of uncertainties in its temperature measure- ments, and an evaluation of its X-ray spectrum. Finally Sec. IV presents the radiatively-induced axion-electron coupling that arises from the axion-photon coupling, which allows us to recast our limit on g g in terms of the axion-photon coupling | a aee| alone.

I. ADDITIONAL MWD CANDIDATES FOR X-RAY OBSERVATION

In the main text we have focused our analysis on the MWD star RE J0317-853, since it is expected to have a particularly strong X-ray flux, and because X-ray data is already available from Suzaku. However, there are over 200 MWD stars with well-measured field strengths, temperatures, and distances. For each of these stars, we calculate the expected axion-induced 9 24 Andrew1 J. Long (UMich)! X-ray flux in the 2 10 keV energy window, denoted by F2 10, assuming ma = 10 eV and gagaee = 10 GeV . The flux is insensitive to the axion mass in the limit ma 0, and more generally the flux has an overall scaling with the couplings, 2 ! F2 10 (gagaee) . Magnetic white / dwarf candidates Our results are summarized in Table I, which shows the ten MWDs with the largest predicted X-ray flux, F2 10. We con- structed a full list by merging the SDSS DR7 magnetic WD catalog [73], a review MWD catalog [16], and Gaia DR2 WD catalog [58]. The former two provide the magnetic field strengths and temperatures of the WDs, while the latter provides distances, luminosities, masses, and radii, if known. Since the mass and radius of WD 2010+310 are not known, we take its mass to be MWD =1M , and we infer its radius from the Stefan-Boltzmann law. Top 10 MWD candidates

TABLE I. MWD stars that make good candidates for measurement of their secondary, axion-induced X-ray flux. The columns correspond to the star’s mass in solar masses, radius in solar radii, luminosity in solar luminosities, effective temperature in Kelvin, magnetic field strength in 2 9 mega-Gauss, distance from Earth in parsecs, and predicted X-ray flux from 2 10 keV in erg/cm /s, calculated assuming ma =10 eV 24 1 and gagaee =10 GeV . The parameters were obtained by merging the catalogs in Refs. [16, 58, 73]. We infer the mass and radius of WD 2010+310 as discussed in the text.

2 MWD [M ] RWD [R ] L [L ] Te↵ [K] B [MG] dWD [pc] F2 10 [erg/cm /s] 14 RE J0317-853 1.32 0.00405 0.0120 30000 200 29.54 6.8 10 ⇥ 14 WD 2010+310 1⇤ 0.00643⇤ 0.00566 19750 520 30.77 4.4 10 ⇥ 14 WD 0041-102 (Feige 7) 1.05 0.00756 0.00635 18750 35 31.09 3.0 10 ⇥ 14 WD 1031+234 0.937 0.00872 0.0109 20000 200 64.09 2.3 10 ⇥ 14 WD 1533-057 0.717 0.0114 0.0121 18000 31 68.96 1.3 10 ⇥ 15 WD 1017+367 0.730 0.0111 0.0082 16500 65 79.24 7.1 10 ⇥ 15 WD 1043-050 1.02 0.00787 0.00388 16250 820 83.33 5.4 10 ⇥ 15 WD 1211-171 1.06 0.00754 0.00992 21000 50 92.61 5.4 10 ⇥ 15 SDSS 131508.97+093713.87 0.848 0.00968 0.01347 20000 14 101.73.5 10 ⇥ 15 WD 1743-520 1.13 0.00681 0.00184 14500 36 38.93 2.9 10 ⇥

RE J0317-853II. THE PROBABILITY FOR AXION-PHOTON CONVERSION IN A GENERAL MAGNETIC FIELD BACKGROUND In this section we present a more general formalism to calculate the axion-photon conversion probability, and in the following è MWD section= 1.32 we apply M thissun calculation to study the X-ray emission from RE J0317-853 in more detail. Interactions between the axion è RWD = 0.00405 Rsun è Lgam = 0.0120 Lsun è Teff = 30,000 Kelvin è B = 200 MG è dWD = 29.54 pc (Gaia) [Barstow et. al. (1995)] 2 [Burleigh et. al. (1999)] è F2-10 < 1.7e-13 erg/cm /sec @ 95% CL (Suzaku, 60 ks) [Kulebi et. al. (2010)] [Suzaku (2013)] Andrew J. Long (UMich)! Constraining ALPs [Dessert, Long, & Safdi (2019)]

10-21 RE J0317-853

10-22 CAST gaγγ × gaee ] 1 - SN1987a gaγγ + CAST gaγγ + WD gaee -23 GeV 10 |[ cooling hints aee g 10-24 Suzaku (this work) × γγ a axion g | -25 10 axion XMM-Newton (projected) DFSZ

KSVZ 10-26 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

ma [eV] More than 1 order of magnitude improvement over previous limits (CAST). -5 Fiducial model for cooling hints is excluded up to ma ~ 10 eV @ 95% CL 13 Andrew J. Long (UMich)! Constraining ALPs [Dessert, Long, & Safdi (2019)]

10-9 RE J0317-853 γ-ray transparency hint -10 10 CAST ] 1 -

GeV -11 10 SN1987a |[

γγ Suzaku (this work) a g |

-12 axion 10 axion

KSVZDFSZ XMM-Newton (projected) 10-13 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

ma [eV]

14 Andrew J. Long (UMich)! Summary with Chris Dessert & Ben Safdi (1903.05088)

AAACkHiclVFNS+RAEO2JruuOq+vH0UvjIHgyiQh6W5f1IIKi4DjCZJBOpzI29kforuwYQn6BV/fH7b+xZ8xh/bhY0PB49R5VXS8tpHAYRf86wdz8l4Wvi9+6S9+XV36srq1fO1NaDn1upLE3KXMghYY+CpRwU1hgKpUwSO9/T/uDP2CdMPoKqwJGio21yAVn6KnL89vVXrQbzYq+B3ELeqSti9u1TpVkhpcKNHLJnBvGUYGjmlkUXELTTUoHBeP3bAxDDzVT4Eb1bNOGbnsmo7mx/mmkM/Z/R82Uc5VKvVIxvHNve1Pyo96wxPxwVAtdlAiavwzKS0nR0Om3aSYscJSVB4xb4Xel/I5ZxtEf59WUsO88FTKdWZhIo8fhsTVFah7CM+F4qKpW6JrP2fwg5T3dRMOEG6W8sk4Gx02dIDxgnThOk+lRcqaErOoCVZP4XP3Ws1NNsqbxYcVvo3kPrvd2Y48v93tHURvbItkkW2SHxOSAHJETckH6hBMgj+SJ/A3Wg8PgZ/DrRRp0Ws8GeVXB6TNda8xX handle on axion-like particles. 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 N 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 N è Complementary to probes of 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 Tc cooling (WD luminosity funcion)

10-21 RE J0317-853

10-22 CAST gaγγ × gaee ] 1 - SN1987a gaγγ + CAST gaγγ + WD gaee This technique yields the strongest -23 GeV 10 |[ cooling hints

aee constraints to date on light ALPs. g 10-24 Suzaku (this work) × γγ

a axion g | -25 10 axion XMM-Newton (projected) DFSZ Lots of room to improve the limits

KSVZ 10-26 with more data! 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

ma [eV]

15 BACKUP SLIDES

16 Estimating the core temperature [Chabrier, Brassard, Fontaine, & Saumon (2000)]

There is no direct measurement of the WD core temperature è E.g., photons produced inside do not escape

However there220 is extensive literature CHABRIER to model ET AL. WD interiors Vol. 543

è A highSJ99). electron Indeed, an important thermal feature in conductivity these cool and dense atmospheres is the e†ect of the surrounding particles on the impliespartition a nearly function of anisothermal atom, which eventually core. leads to the pressure ionization of hydrogen. These modiÐed internal è Core temperaturepartition functions imply is a di†erent related (nonideal) to ionization surface equilibrium, in particular for the abundances of H ,H`, andH`. This in turn modiÐes the abundance of free2 elec-2 brightness3 (or effective temperature) trons, and thus of H~ ions, a dominant source of opacity in the optical. The atmosphere models of SJ99 were used in the range1500 ¹ T ¹ 4000 K and 7.5 ¹ log g ¹ 9.0, com- plemented by BSW95eff for4000 ¹ T ¹ 10,000 K, with a pure hydrogen composition.0.4 Theseeff synthetic spectra and atmosphereL models have been successfully confronted with T 3 106 Kavailable spectroscopic and photometric observations of c cool H-rich (DA)4 WDs above 4000 K (Bergeron et al. 1997). ' ⇥ 10 L Optical✓ colors for cool◆ WDs with a small but nonzero N(He)/N(H) composition ratio1.6 will be only slightly di†erent0.8 ([0.2 mag) from the colors of pure hydrogen atmospheres Te↵ RWD 4 106 Kpresented here (see BSW95; Bergeron et al. 1997). The color indices20 of Table,000 2 ofK BSW95 can then0 be.01 used aboveR 4000 K 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 ' ⇥ for mixed composition atmospheres. ✓ ◆ ✓ ◆ 3. RESULTS FIG. 2.ÈL -Tc relations from our calculations (solid line) and those of 3.1. Cooling Curves Hansen (1999) (short-dashed line), Wood (1995) (dotted line), and Montgo- For the purposes of comparisons, we deÐne a set of mery et al. (1999) (long-dashed line) for a 0.6M_ WD with hydrogen and reference models that include the physics described in helium mass fractions q(H) \ 10~4, q(He) \ 10~2 and pure H atmosphere.17 the previous section, including crystallization-induced fragmentation (see below), with the Salaris et al. (1997) high- is rapidly followed by a phase whereby convection speeds rate stratiÐed initial C/O proÐles. TheL -Tc relations that we used for these reference models have been computed as up the cooling process as compared to the case of purely described above, from full static stellar models with chemi- radiative models (see Fig. 3 of Tassoul et al. 1990 and related discussion). For luminosities lower than that of the cal layering deÐned by log q(H) \[4.0 and log q(He) \ breaks in slope shown in Figure 2, the- relation [2.0. The photometric colors are taken from pure H atmo- L Tc sphere calculations. becomes sensitive, among other things, to the details of the atmosphere. From this point of view, the Hansen (1999) 3.1.1. Comparison with Existing Calculations calculations and our own improve upon those of Wood We Ðrst compare, in Figure 2, ourL -Tc relation for a 0.6 (1995) or Montgomery et al. (1999) because the latter ones M_ WD with the one obtained by Wood (1995), the one by rely on a standard gray atmosphere strategy that overesti- Montgomery et al. (1999, Table 1), and another one kindly mates the penetration of the superÐcial convection zone and provided by Hansen (1999) for the same values of q(H) and leads to a value ofTc slightly lower than it should be, as q(He). The four curves agree reasonably well, but the rela- discussed previously. tively small di†erences shown here translate into signiÐcant On the other hand, for luminosities larger than those of di†erences on the cooling time (see below). Unfortunately, it the changes of slope, it is well established that the relation is is currently impossible to account in detail for the di†er- completely insensitive to the stratiÐcation of the upper ences in theL -Tc relations illustrated in Figure 2; di†erences envelope and, in particular, to the details of the atmosphere in the constitutive physics used by the various groups are (see Tassoul et al. 1990 for a complete discussion). The dis- probably at the origin of most of the deviations. The L -Tc crepancies shown in Figure 2 between the three curves for relation is particularly sensitive to the opacity proÐle these higher luminosities are then the explicit proof that throughout the star, so slightly di†erent implementations of there are indeed signiÐcant di†erences in the the conductive and radiative opacities, as well as the use of implementation/calculation of the constitutive physics di†erent generations of these data, could very well account between the three groups, notably at the level of the con- for most of the discrepancies. As expected, the Wood (1995) ductive opacities. So, to a certain extent, the di†erences and the Montgomery et al. (1999) relations are very similar, found in Figure 2 result from the fact that we are comparing since they rely on the same input physics, yielding similar models with di†erent physical inputs. cooling sequences. Figure 3 illustrates the consequences of these di†erences As mentioned in ° 2.2, the change of slope observed in all on the cooling time of the star. For eachL -Tc relation, we three curves is a well-known phenomenon and is due to show two cases: (1) taking into account the chemical frac- convection breaking into the degenerate thermal reservoir tionation of C and O at crystallization (rightmost curves) at sufficiently low luminosities. When that occurs, there is a and (2) ignoring this process (leftmost curves). Interestingly Ñattening of the temperature gradient between the core and enough, we Ðnd that in the phases of interest for the present the surface as a result of the larger efficiency of convection study[log (L /L _) \ [4.5], our results (solid curves) agree as compared to radiation. Initially, this leads to a slowdown reasonably well with those obtained when using the L -Tc of the cooling process as an excess energy is liberated relation provided by Hansen (long-dashed curves). The through the more transparent convective envelope, but this results derived on the basis of the Wood (1995) (or Montgo- 5

EstimatingFIG. S2. Theuncertainties limit on ga gaee , calculated using two different models for the background magnetic field. The alternate B-field model, with the displaced dipole,| gives a comparable| result at low axion masses and slightly improved sensitivity at high axion masses, due to the increased magnetic field strength. [Kulebi et. al. (2010)]

TABLE II. The X-ray ﬂux of RE J0317-853 is calculated for different values of MWD, RWD, and Te↵ , which were determined by [60]. The 9 24 1 luminosities, L , are inferred from the Stefan-Boltzmann law. We calculate F2 10 for ma =10 eV and gagaee =10 GeV . 2 MWD [M ] RWD [0.01 R ] L [L ] Te↵ [K] B [MG] dWD [pc] F2 10 [erg/cm /sec] 14 CO-low-T 1.32 0.020 0.405 0.011 0.0120 30000 200 29.54 6.8 10 ± ± ⇥ 13 CO-high-T > 1.46 0.299 0.008 0.0503 50000 200 29.54 2.4 10 ± ⇥ 14 ONe-low-T 1.28 0.015 0.416 0.011 0.0126 30000 200 29.54 7.2 10 ± ± ⇥ 13 ONe-high-T 1.38 0.020 0.293 0.008 0.0483 50000 200 29.54 2.2 10 ± ± ⇥

-21 2 parameters used to10 describe RE J0317-853. Since the ﬂux depends on the axion’s couplings through F2 10 (gagaee) , the / effect on our limit is a factorRE of J03172. The- ﬂux853 uncertainty follows primarily from the uncertainty in T . In Fig. S3 we show ⇠ e↵ how these uncertainties translate into the uncertainty in the 95% limit from Suzaku observations and (projected) XMM-Newton observations. The bands encompass the range× of limits obtained by cycling through the parameters presented in Tab. II. 10-22 CAST gaγγ gaee ] 1 - SN1987a gaγγ + CASTC.ga Spectrumγγ + WD gaee -23 GeV 10 Fig. S4 shows the predicted spectra of axion and photon emission from our candidate MWD star, RE J0317-853. The shape

of the axion spectrum|[ (blue curve) is very well approximated by a blackbody at temperature T 2 107 K 1.7 keV whereas cooling hints c ' ⇥ ' the amplitude ofaee the spectrum is set by the magnitude of the axion-electron coupling according to (1). To draw the blue curve g 13-24 we take gaee = 1010 . The spectrum of secondary, axion-induced photons (red curve) tracks the thermal spectral shape up to an additional× energy dependence comingSuzaku from( thethis axion-photon work) conversion probability (5). To draw the red curve we take 9 γγ 11 1 24 a ma = 10 eV and ga = 10 GeV , such that the product ga gaee = 10 is marginallyaxion consistent with the Suzaku g

| | | limit. We assume that all axions are emitted isotropically and homogeneously from throughout the interior of the WD core, and -25 the axion-photon conversion10 probability is calculated using (S5). axion DFSZ XMM-Newton (projected) KSVZ IV. THE RUNNING OF THE AXION-ELECTRON COUPLING AND SENSITIVITY TO ga 10-26 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 In the main text we have10 shown10 the limit10 on g10a gaee10that follows10 from10 X-ray10 observations10 10 of MWD10 stars. Here we would | | like to translate this into a limit on simply ga. For any value of ga there is a “reasonable range” of values for gaee: it is bounded from above by direct observation, and it is boundedma from[eV below] because an axion-electron coupling can be induced 18 Time-varying ﬂux [Burleigh et. al. (1999)]

Time-resolved spectroscopy gives a model for the B-field structure è Best described by an offset dipole è Leads to time-varying flux at O(50%)

rotaon axis dipole axis 3

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 B~ 4

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 a a g p

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

0 0 1 2 3 4 5 6 ϕ dipolar strength = 363 MG rotaon-to-dipole = 19 deg rotaon-to-viewing = 51 deg 19 dipole oﬀset = -0.19 RWD Volume averaging

In the main text we assume … è dipole profile (B ~ r-3) è axion trajectories emerge radially (BT does not change direction)

We can generalize these assumptions …

10-21 RE J0317-853

3 10-22 CAST (95% CL) ] 1 - SN1987a g γγ + CAST g γγ + WD g

4 a a aee

10 - GeV 10 23

× 2 |[

→γ cooling hints a aee g p 10-24 × γγ a axion g

1 | -25 Suzaku - fiducial 10 axion trajectories originate at center of MWD DFSZ Suzaku - alt B-field

starting location averaged over interior KSVZ 0 10-26 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

θ ma [eV]