Gas cooling of test masses for future gravitational-wave observatories
Christoph Reinhardt, Alexander Franke, Jörn Schaffran, Roman Schnabel, Axel Lindner
GGas cooling of test masses for futuregravitational-wave observatories
Christoph Reinhardt , Alexander Franke , J¨orn Schaffran ,Roman Schnabel and Axel Lindner Deutsches Elektronen Synchrotron (DESY), 22607 Hamburg, Germany Institut f¨ur Laserphysik und Zentrum f¨ur Optische Quantentechnologien derUniversit¨at Hamburg, Hamburg, GermanyE-mail: [email protected]
January 2021
All figures and pictures by the authors under a CC BY 4.0 license
Abstract.
Recent observations made with Advanced LIGO and Advanced Virgohave initiated the era of gravitational-wave astronomy. The number of events detectedby these “2 nd Generation” (2G) ground-based observatories is partially limited by noisearising from temperature-induced position fluctuations of the test mass mirror surfacesused for probing space time dynamics. The design of next-generation gravitational-wave observatories addresses this limitation by using cryogenically cooled test masses;current approaches for continuously removing heat (resulting from absorbed laser light)rely on heat extraction via black-body radiation or conduction through suspensionfibers. As a complementing approach, we investigate cooling via helium gas impingingon the test mass in free molecular flow. We present analytical models for cooling powerand related displacement noise, validated by comparison to numerical simulations.Applying this theoretical framework with regard to the conceptual design of theEinstein Telescope (ET), we find a cooling power of 10 mW at 18 K for a gas pressurethat increases the ET design strain noise goal by at most a factor of ∼ ∼
1. Introduction
The first detection of gravitational waves by LIGO in September 2015 has unlockeda new source of information about the universe [1]. So far, LIGO together withVirgo has observed 15 confirmed events and 35 candidate events of gravitational wavesoriginating from mergers of two black holes, two neutron stars, as well as pairs of oneblack hole and one neutron star [2, 3]. In order to increase the rate and range ofdetections, a “3 rd Generation” (3G) of ground-based observatories is currently beingdeveloped [4, 5, 6, 7]. Research targets increasing the GW signal as well as reducing a r X i v : . [ phy s i c s . i n s - d e t ] J a n as cooling of test masses for future gravitational-wave observatories ∼
300 kg mirrors that are suspended with ratherthin fibres [4, 5, 7] is a major technological challenge. Heat load due to absorbedblack-body radiation from the (room-temperature) kilometre-scale vacuum tubes hasto be suppressed to a minimum. During operation, the test masses are constantlyheated by partial absorption of laser light. Mirror substrate and coating materialsneed to show extremely low optical absorption in the range of a few parts per million(ppm). At the same time the materials need to have high mechanical quality factorsto channel the remaining thermal energy in narrow well-defined mechanical resonances.The problem of operating cryogenically cooled GW observatories is how to continuouslyget thermal energy out of mirrors that are in vacuum and mechanically maximallydecoupled from the environment, while the observatory is taking data. Another keyaspect for cryogenic operation is minimizing durations of cool-down and warm-up, tomaximize the observatory’s duty cycle. In this context, achieving faster cool-downs byusing an exchange gas has been examined [10, 11].Here, we investigate the potential of using helium gas in the free molecular flowregime to extract the heat imparted during observational runs in cryogenic test massesof future gravitational wave observatories. Sec. 2 describes the conceptual setup. Sec. 3establishes the relation between gas-induced cooling power and added strain noise, basedon corresponding analytical models, which are validated by comparison to numericalsimulations. Sec. 4 presents an application of the theoretical framework described inthe previous section with regard to the design of the Einstein Telescope. Sec. 5 givesconcluding remarks. as cooling of test masses for future gravitational-wave observatories
2. Conceptual setup for gas cooling applied to suspended test masses
To establish models for cooling power and corresponding thermal displacement noiserelated to helium gas interacting with a suspended test mass (TM), we consider the setupshown in Fig. 1. Here, a TM suspended by thin fibers (not shown) is heated by partialabsorption of laser light and thermal radiation. The imparted heat is transferred to aclose-by frame, with temperature T frame , by virtue of helium gas and thermal radiation,thereby keeping the TM’s temperature T TM constant. HeliumRadiation
Radiation
Laser Light Frame ( T frame )Test Mass ( T TM ) d Figure 1.
Schematic of a heat transfer model for gas cooling of a suspended test mass(TM). Partial absorption of laser light and thermal radiation from the environmentheat the TM. The TM’s temperature T TM is kept constant by virtue of heat transfer toa frame, at distance d and temperature T frame , via helium gas and thermal radiation.Gas cooling occurs in the free molecular flow regime (no interaction between heliumatoms). Furthermore, we assume that helium atoms reach thermal equilibrium withmirror and frame. Helium atoms are re-emitted from surfaces in random directions,where the probability of emission in a particular direction follows the Knudsen cosinelaw (see main text for details). The following underlying assumptions are made for the interaction between heliumgas and surfaces of TM and frame: The helium gas is in the free molecular flow regime,where interactions between atoms are negligible; we apply the common definition forfree molecular flow based on the Knudsen number: Kn ≡ λ/d >
10, with mean freepath (MFP) λ and distance d between TM and frame. The MFP corresponds tothe average distance traveled by atoms between successive collisions. In Sec. 4 it isshown, that this assumption leads to technically feasible values of d . The transfer ofenergy (i.e., heat) between helium atoms and surfaces is specified by a correspondingaccommodation coefficient α E , which represents the fraction of incident atoms reachingthermal equilibrium with the surface. Similarly, the transfer of momentum betweenhelium atoms and surfaces is specified by an accommodation coefficient α M , whichrepresents the fraction of incident atoms transferring their momentum to the surface.These atoms are said to be diffusely reflected, where the direction of re-emissionfrom the surface is randomly distributed according to the Knudsen cosine law (seeSec. 3.1 for details). The fractions of atoms 1 − α E and 1 − α M , being reflectedwithout exchange of energy or momentum with the surface, are said to be specularlyreflected. We assume the duration between adsorption and desorption of an atom tobe negligible. Throughout this article, we consider T frame = 5 K and T TM = 18 K,with accommodation coefficients for helium α E (5 K) = 1 . α E (18 K) ≡ α E, TM = 0 . as cooling of test masses for future gravitational-wave observatories α M (5 K) = α M (18 K) = 1 .
3. Models for heat transfer and strain noise
The cooling power acting on the TM due to the helium gas is given by P gas = α E, TM (cid:16) ˙ Q out − ˙ Q in (cid:17) , (1)where ˙ Q out (cid:16) ˙ Q in (cid:17) is the heat flux emitted (absorbed) by the TM, in the case of unityaccommodation, and α E, TM represents the fraction of incident atoms contributing to theheat transfer (see Sec. 2). The heat flux per surface element ∆ A is calculated as productof the flux density of emitted (absorbed) atoms φ out ( φ in ) and their mean kinetic energy,˙ Q i ∆ A = φ i (cid:90) m He v i ρ ( v i ) dv i (cid:37) ( θ i ) d Ω i , (2)with i ∈ { out , in } , helium mass m He , speed of emitted atoms v i , speed distribution ρ ( v i ), angle between surface normal and speed vector of atom θ i , angular distribution (cid:37) ( θ i ), and incremental solid angle d Ω i . The assumptions underlying Eq. 2 are describedin Sec. 2. The corresponding speed and angular distributions are given by [15] ρ ( v i ) = v i v T,i exp (cid:32) − v i v T,i (cid:33) (3)and (cid:37) ( θ i ) = cos θ i π , (4)respectively. Here v T,i ≡ (cid:113) k B T i /m He is the characteristic thermal velocity, with T in = T frame and T out = T TM . The contribution from temperature-induced positionfluctuations of the TM to the relative speed between TM and gas particles is insignificantand, therefore, is not taken into account.To validate the model presented above, we compare it with a numerical simulationimplemented in the Molecular Flow Module of COMSOL Multiphysics. We consider thesimple case of two parallel plates, for which Eq. 2 gives ˙ Q i / ∆ A = 2 k B T i φ i . Taking intoaccount equilibrium conditions, with equal incoming and outgoing flux φ out = φ in ≡ φ ,gives ˙ Q i ∆ A = 2 k B T i φ. (5)As a next step, the relation between heat transfer and helium pressure is established:The pressure caused by incoming/outgoing atoms is given by the product of φ i and themean momentum along the surface normal p i = φ i (cid:90) m He v i ⊥ ρ ( v i ) dv i (cid:37) ( θ i ) d Ω i , (6) as cooling of test masses for future gravitational-wave observatories v i ⊥ = v i cos θ i . Evaluating the integral for two parallel plates gives p i = φ (cid:115) πm He k B T i . (7)Gas components coming from the frame and TM have different pressures resulting fromdifferent temperatures. This is a consequence of the assumptions detailed in Sec. 2(i.e., helium atoms do not interact with each other and reach thermal equilibrium withsurfaces of TM and frame). By combining Eqs. 1, 5, and 7 the cooling, acting on theplate at T TM , can be written ‡ P gas = α E, TM (cid:115) k B πm He T frame p in ∆ A ( T TM − T frame ) , ( T TM > T frame ) , (8)where ∆ A corresponds to the surface area of each plate.In the following, the maximum allowed value for the distance d between frame andTM (see Sec. 3.2 for details) is derived. The upper bound follows from the requirementof staying in the free molecular flow regime, where d < λ/
10 (see Sec. 2). The totalMFP corresponds to the average of the MFP of “cold” atoms moving from the frametoward TM, λ in , and the MFP of “hot” (“cold”) particles moving from TM towardframe, λ out ( λ out ): λ = ( n in λ in + n out λ out + n out λ out ) / ( n in + n out + n out ), where n in and n out ( n out ) is the number density of incoming and outgoing “warm” (“cold”)atoms, respectively. Here, the splitting in “warm” and “cold” particles, moving fromTM toward frame, is a consequence of α E, TM <
1. Making use of Eq. 7 and the idealgas law p i = n i k B T i yields λ = λ in + α E, TM (cid:113) T frame /T TM λ out + (1 − α E, TM ) λ out α E, TM (cid:16)(cid:113) T frame /T TM − (cid:17) . (9)The mean free path of a particular “species” i ∈ { in , out , out } of atoms is given by λ i = (cid:104) v i (cid:105) τ i , where (cid:104) v i (cid:105) is the mean speed and τ i is the average time between successivecollisions. Atoms can collide either with atoms of their own species, characterized bycollision time τ i,i , or atoms of the other species, characterized by collision times τ i,j ( i (cid:54) = j ). The total collision time for a particular species i is calculated by summingthe contributing collision rates: τ i − = (cid:80) j τ − i,j . The rate of each collision processes iscalculated by multiplying the volume of interaction per unit time by the number densityof target gas particles: τ − i,j = πδ n j (cid:104) v i,j (cid:105) , where δ is the kinetic diameter of a Heliumatom and (cid:104) v i,j (cid:105) is the mean relative speed between an atom of species i and an atom ofspecies j . Combining the previous considerations yields λ in = k B T frame (cid:104) v in (cid:105) πδ p in (cid:104) (cid:104) v in , in (cid:105) + α E, TM (cid:113) T frame /T TM (cid:104) v in , out (cid:105) + (1 − α E, TM ) (cid:104) v in , out (cid:105) (cid:105) , (10) λ out = k B T TM (cid:104) v out (cid:105) πδ p in (cid:104) α E, TM (cid:113) T frame /T TM (cid:104) v out , out (cid:105) + (cid:104) v out , in (cid:105) + (1 − α E, TM ) (cid:104) v out , out (cid:105) (cid:105) , (11) ‡ There is an additional contribution from internal degrees of freedom, adding ∼ . as cooling of test masses for future gravitational-wave observatories λ out = k B T frame (cid:104) v out (cid:105) πδ p in (cid:104) (1 − α E, TM ) (cid:104) v out , out (cid:105) + (cid:104) v out , in (cid:105) + α E, TM (cid:113) T frame /T TM (cid:104) v out , out (cid:105) (cid:105) . (12)Here, the mean speed is given by (cid:104) v i (cid:105) = (cid:115) πk B T i m He , (13)with T in = T out = T frame and T out = T TM , and the mean relative speed, defined as (cid:104) v i,j (cid:105) = (cid:113) (cid:104) ( (cid:126)v i − (cid:126)v j ) (cid:105) = (cid:113) (cid:104) v i (cid:105) + (cid:104) v i (cid:105) − (cid:104) (cid:126)v i · (cid:126)v j (cid:105) , is given by (cid:104) v in , in (cid:105) = (cid:104) v out , out (cid:105) = (cid:115) k B T frame m He (8 − π ) (14) (cid:104) v out , out (cid:105) = (cid:115) k B T TM m He (8 − π ) (15) (cid:104) v out , out (cid:105) = (cid:104) v out , out (cid:105) = (cid:115) k B m He (cid:20) T frame + T TM ) − π (cid:113) T frame T TM (cid:21) (16) (cid:104) v in , out (cid:105) = (cid:104) v out , in (cid:105) = (cid:115) k B m He (cid:20) T frame + T TM ) + π (cid:113) T frame T TM (cid:21) (17) (cid:104) v in , out (cid:105) = (cid:104) v out , in (cid:105) = (cid:115) k B T frame m He (8 + π ) . (18)The mean values are calculated based on the distribution functions given by Eqs. 3and 4. Maximum Plate Separation (m)10 Incoming Pressure (Pa)10 C oo li n g P o w e r ( W ) AnalyticalCOMSOL
Figure 2.
Heat transfer versus helium gas pressure for two parallel plates of area50 cm ×
50 cm, unity accommodation, and temperatures 5 K and 18 K, respectively. Theblue line shows the analytical model (Eq. 8). Values from the numerical simulation(orange circles) are lower by 0.02 %. The upper abscissa indicates the separationbetween the plates, where shown values correspond to the upper bound compatiblewith free molecular flow (see main text for details). as cooling of test masses for future gravitational-wave observatories p for a pair ofparallel square plates, with edge length 50 cm and unity accommodation at both plates.The blue line corresponds to the analytical expression (Eq. 8) and the orange circlesshow the values predicted by the numerical simulation. The analytical values exceedthe numerical values by 0.02 %. Increasing the pressure lowers the MFP, which requiresa reduced separation between the plates, to be compatible with the free molecular flowregime (Kn > Here, we derive the displacement noise arising from helium atoms impinging on theTM [17]. As dominant noise source we consider a single degree of freedom of the TM,corresponding to motion along the direction of the incident laser light (see Fig. 1) withvelocity V (cid:107) .In addition to impinging gas atoms, diffusive gas flow is a potential source of noiseacting on a TM in a constrained volume. The corresponding noise becomes significantif the channel limiting the flow (i.e. a gap between the TM and a nearby surface) iscomparable to the dimension of the TM [18, 19]. Here, we assume that the gap betweenTM and frame (see. Fig. 1) is not limiting the flow resulting from pressure differencescaused by thermal motion of the TM along the optical axis. Therefore, we consider thenoise contribution from diffusive flow negligible.The time-averaged force per surface element ∆ A of the TM’s side faces is calculatedas mean value of the gas particles’ flux density of momentum parallel to the TM motion F (cid:107) ∆ A = φ in (cid:90) m He v in , (cid:107) ρ V (cid:107) ( v in ) dv in (cid:37) ( θ in ) d Ω in , (19)with speed distribution ρ V (cid:107) ( v in ) = v v T exp − v , ⊥ + (cid:16) v in , (cid:107) + V (cid:107) (cid:17) v T . (20)Here, v in , ⊥ = v in cos θ in (cid:126)e z and v in , (cid:107) = v in sin θ in (cos ϕ in (cid:126)e x + sin ϕ in (cid:126)e y ) are the gas particles’speed components orthogonal and parallel to V (cid:107) , respectively, where ϕ in is the azimuthalangle. Solving the integral gives F (cid:107) = − V (cid:107) p in ∆ A (cid:115) m He πk B T frame ≡ − V (cid:107) β, (21)where the last step defines the damping coefficient β . Note that atoms re-emittedfrom the TM do not cause a net force. This is because emission occurs isotropically.Assuming the TM to represent a damped harmonic oscillator, with frictional dampingforce F (cid:107) , and applying the fluctuation-dissipation theorem yields the displacement noisespectrum [20] x ( ω ) = 4 k B T βm ( ω − ω ) + β ω , (22) as cooling of test masses for future gravitational-wave observatories m TM is the mass of the TM, ω / π is the oscillator’s resonance frequency, and ω/ π is the frequency.To validate our analytical model, we set up a Monte-Carlo simulation. Here, thesuspended TM is modeled as a pendulum (using a small-angle approximation), withlength l and displacement z ( t ). Further assumptions are described in Sec. 2. Foreach impinging atom, we assign a random timestamp and calculate the correspondingmomentum transfer, based on randomly selecting four parameters: in- and outgoingspeed as well as in- and outgoing angle. The underlying probability distributions forspeed and angle are given by Eq. 3 and 4, respectively. The change in the TM’s velocityassociated with the momentum transferred by an adsorbed gas atom is given by δV (cid:107) = m He m TM [ v ( T frame ) · sin θ in + v ( T TM ) · sin θ out ] . (23)The TM’s trajectory is obtained based on energy conservation, giving: z ( t ) ≈ l (cid:18)(cid:113) ξ + 124 (cid:113) ξ (cid:19) sin (cid:18) ω t + z l (cid:19) , (24)with acceleration due to gravity g , velocity added to the TM at the last hit δV (cid:107) , TMdisplacement at the last collision z , and ξ = 12 l (cid:34) z l + 1 g (cid:16) V (cid:107) + δV (cid:107) (cid:17) (cid:35) . (25)The power spectral density of corresponding displacement noise is obtained by Fouriertransforming the time series of TM displacements.Fig. 3 shows the simulated displacement spectral density (orange) together with theanalytical model (blue, given by Eq. 22) for a cubical toy model TM with mass 300 kg.The average deviation between numerical simulation and analytical model is 19 %. Theexample illustrated here just serves the comparison of analytical and numerical model;the resulting displacement noise values are of no relevance whatsoever for gravitationalwave detectors. This is because, with reasonable computational resources we are not ableto simulate collision rates (in excess of 10 s − ) relevant for realistic cooling scenarios(see Sec. 4). Therefore, in the present case, we simulate TM movement with 1 . × s − .Mirror movement is simulated for a time of 5 s. Combining Eqs. 8, 21, and 22 yields the following expression (for ω (cid:29) ω ), directlyrelating the cooling power, provided by the helium atoms, to corresponding thermalnoise: P gas = α E, TM ω x ( ω )2 N TM m m He ( T TM − T frame ) T frame , (26)where N TM is the number of TMs in the detector (in all current and next generationobservatories, N TM = 4). Interestingly, this expression is independent of the heliumpressure p in and the surface area ∆ A of the TM, which is exposed to helium gas atoms.This is a consequence of the fact that for increasing/decreasing either p in or ∆ A , the as cooling of test masses for future gravitational-wave observatories Frequency (Hz)10 − − − − − − T e s t m a ss d i s p l a ce m e n t( m / √ H z ) Analytical modelNumerical model (1 σ ) Figure 3.
Thermal noise from helium gas impinging on four side faces of a cubicaltoy model TM with mass 300 kg, suspension length 2 m, and rate of impinging heliumatoms 1 . × s − . (The displacement noise values shown here are of no relevance forrealistic cooling scenarios, due to a significantly lower collision rate. See main text fordetails). The numerical simulation (orange) is obtained by averaging five individualnoise spectra; the width corresponds to the standard deviation. The analytical model(Eq. 22) is shown in blue. The average deviation between both models is 19 %. effects from increasing/decreasing both heat flux and thermal noise cancel. HeavierTMs mitigate the effect from impinging helium atoms on resulting noise. Increasing thetemperature of the TMs with respect to the frame leads to larger cooling power for agiven amount of added noise. In that regard, increasing the accommodation coefficientis beneficial too. Furthermore, the cooling power for a given amount of noise increasesfor lighter gas atoms and lower frame temperature.
4. Cooling power and added thermal strain noise with regard to the ETdesign
To assess the potential of gas cooling for future gravitational wave observatories,we examine a setup similar to the design of the cryogenic interferometer for thelow-frequency ET [4, 5]. As for the ET design, we assume the interferometer tocomprise four cylindrical TMs made out of silicon; each TM has diameter D TM =45 cm, thickness t TM = 57 cm, and mass m TM = 211 kg. The expected heat loadfrom absorbed laser light and thermal radiation is ∼
100 mW, which, accordingto the baseline design, is fully extracted by conduction through the TM’s siliconsuspension fibers. In the case of gas cooling as complementing cooling strategy, theimparted heat is transferred to a frame surrounding the barrel of a TM (see Sec. 2).Here, we assume TM and frame temperatures of T TM = 18 K and T frame = 5 K,respectively. At this value for T TM , the coefficient of thermal expansion for siliconvanishes, thereby eliminating noise from thermoelastic damping. The cooling power as cooling of test masses for future gravitational-wave observatories A = A barrel = πD TM t TM . The contribution from radiation is calculated based on the Stefan-Boltzmannlaw σ ( ε barrel A barrel + ε face πd /
2) ( T − T ), with Stefan-Boltzmann constant σ ,emissivity of the TM’s barrel ε barrel = 0 .
9, and emissivity of the TM’s front and backside ε TM = 0 . Frequency (Hz)10 S t r a i n ( / H z ) Gas Damping, 10 mW Cooling PowerGas Damping, 30 mW Cooling PowerGas Damping, 100 mW Cooling PowerET-D Design Sensitivity
Figure 4.
Thermal noise from gas cooling for the Einstein Telescope design. The solid,dashed, and dotted blue curves show simulated strain noise spectra caused by heliumatoms impinging on the four test masses, with corresponding total cooling powers pertest mass of 10 mW, 30 mW, and 100 mW, respectively. The green curve shows thedesign sensitivity of the Einstein Telescope.
Figure 4 shows the noise associated with gas cooling together with the sensitivity ofthe ET design (ET-D) [4, 5]. All values are given in terms of strain, which is defined as x ( ω ) /L , where L is the distance between the two TMs in an arm of the interferometer.Here, we assume the value of the ET design: L = 10 km [4, 5]. For 10 mW of totalcooling power, with equal contributions from helium gas and radiation, the noise fromcooling (solid blue line) exceeds the ET-D sensitivity (green) by at most a factor of 2.3in a narrow frequency band (width of 8 Hz) centered at 7 Hz. The corresponding heliumpressure is 2 × − Pa, which imposes the bound d <
66 cm on the distance between TMand frame, for compatibility with free molecular flow. Extracting a heat load of 30 mW(100 mW) requires 25 mW (95 mW) of cooling power provided by the helium gas. Thecorresponding helium pressure is 12 × − Pa (46 × − Pa), which imposes the bound d <
12 cm ( d < as cooling of test masses for future gravitational-wave observatories m TM ∼
500 kg, m TM ∼ m TM ∼ Mirror Temperature (K)10 C oo li n g P o w e r ( W ) Radiative CoolingGas Cooling 1Sum 1Gas Cooling 2Sum 2Gas Cooling 3Sum 3
Figure 5.
Cooling power versus test mass temperature for the Einstein Telescopedesign. The solid, dashed, and dotted blue curves show the cooling power providedby helium gas according to Eq. 26, where the corresponding strain noise is shown inFig. 4 by curves of the same formatting. The orange line shows the contribution fromradiative heat transfer (details provided in the main text). The solid, dashed, anddotted green curves show the sum of contributions from cooling by gas and radiation.The vertical gray line indicates a temperature of the test mass mirrors of 18 K.
Figure 5 shows the total cooling power and its contributors versus TM temperature.The solid, dashed, and dotted blue curves represent the contribution from helium gas,where the corresponding noise is shown by the curves of similar formatting in Fig. 4.For helium pressures corresponding to 10 mW, 30 mW, or 100 mW of cooling power at T TM = 18 K, radiative cooling dominates over gas cooling for T TM >
17 K, T TM >
31 K,or T TM >
50 K, respectively.
5. Conclusion
Based on a conceptual setup of a suspended test mass mirror in future GW observatories,we have established a relation between gas-induced cooling power and correspondingadded observatory strain noise. In this process, we have developed analytical models forcooling power and noise, which we compared to numerical simulations, finding excellentagreement within 1 % (19 %) for the heat transfer (noise) model; our noise model is alsoconsistent with the one presented in Ref. [17]. For the considered setup, where heat istransferred between the mirror’s cylinder barrel and a close-by frame, we have shownthat the gas-induced cooling power, for a fixed amount of added mirror displacement as cooling of test masses for future gravitational-wave observatories ∼ ∼
11 in a frequency band of width 26 Hz centeredat 7 Hz. This indicates that gas cooling might be an interesting addition to conductivecooling via suspension fibers. With regard to the baseline cooling concept for theET project, the discussed cooling contributions of test masses via helium gas in themolecular flow regime does not indicate an additional benefit. The uncertainty in theaccommodation coefficient makes an experimental test of the proposed cooling approachdesirable.
Acknowledgements
We thank Sandy Croatto, Michael Hartman, and Mikhail Korobko for helpful dis-cussions. This work was supported and partly financed (AF) by the DFG underGermany’s Excellence Strategy EXC 2121 ”Quantum Universe” – 390833306.
References [1] Benjamin P Abbott, Richard Abbott, TD Abbott, MR Abernathy, Fausto Acernese, KendallAckley, Carl Adams, Thomas Adams, Paolo Addesso, RX Adhikari, et al. Observation ofgravitational waves from a binary black hole merger.
Physical review letters , 116(6):061102,2016.[2] BP Abbott, R Abbott, TD Abbott, S Abraham, F Acernese, K Ackley, C Adams, RX Adhikari,VB Adya, C Affeldt, et al. Gwtc-1: a gravitational-wave transient catalog of compact binarymergers observed by ligo and virgo during the first and second observing runs.
Physical ReviewX , 9(3):031040, 2019.[3] R Abbott, TD Abbott, S Abraham, F Acernese, K Ackley, A Adams, C Adams, RX Adhikari,VB Adya, C Affeldt, et al. Gwtc-2: Compact binary coalescences observed by ligo and virgoduring the first half of the third observing run. arXiv preprint arXiv:2010.14527 , 2020.[4] Matt Abernathy, F Acernese, P Ajith, B Allen, P Amaro Seoane, N Andersson, S Aoudia, P Astone,B Krishnan, L Barack, et al. Einstein gravitational wave telescope conceptual design study.2011.[5] ET Steering Committee Editorial Team. Einstein telescope design report update 2020. 2020.[6] R X Adhikari, K Arai, A F Brooks, C Wipf, O Aguiar, P Altin, B Barr, L Barsotti, R Bassiri,A Bell, G Billingsley, R Birney, D Blair, E Bonilla, J Briggs, D D Brown, R Byer, H Cao,M Constancio, S Cooper, T Corbitt, D Coyne, A Cumming, E Daw, R deRosa, G Eddolls,J Eichholz, M Evans, M Fejer, E C Ferreira, A Freise, V V Frolov, S Gras, A Green, H Grote,E Gustafson, E D Hall, G Hammond, J Harms, G Harry, K Haughian, D Heinert, M Heintze, as cooling of test masses for future gravitational-wave observatories F Hellman, J Hennig, M Hennig, S Hild, J Hough, W Johnson, B Kamai, D Kapasi, K Komori,D Koptsov, M Korobko, W Z Korth, K Kuns, B Lantz, S Leavey, F Magana-Sandoval, G Mansell,A Markosyan, A Markowitz, I Martin, R Martin, D Martynov, D E McClelland, G McGhee,T McRae, J Mills, V Mitrofanov, M Molina-Ruiz, C Mow-Lowry, J Munch, P Murray, S Ng,M A Okada, D J Ottaway, L Prokhorov, V Quetschke, S Reid, D Reitze, J Richardson, R Robie,I Romero-Shaw, R Route, S Rowan, R Schnabel, M Schneewind, F Seifert, D Shaddock,B Shapiro, D Shoemaker, A S Silva, B Slagmolen, J Smith, N Smith, J Steinlechner, K Strain,D Taira, S Tait, D Tanner, Z Tornasi, C Torrie, M Van Veggel, J Vanheijningen, P Veitch,A Wade, G Wallace, R Ward, R Weiss, P Wessels, B Willke, H Yamamoto, M J Yap, andC Zhao. A cryogenic silicon interferometer for gravitational-wave detection.
Classical andQuantum Gravity , 37(16):165003, jul 2020.[7] David Reitze, Rana X Adhikari, Stefan Ballmer, Barry Barish, Lisa Barsotti, GariLynn Billingsley,Duncan A Brown, Yanbei Chen, Dennis Coyne, Robert Eisenstein, et al. Cosmic explorer: theus contribution to gravitational-wave astronomy beyond ligo. arXiv preprint arXiv:1907.04833 ,2019.[8] A Buikema, C Cahillane, GL Mansell, CD Blair, R Abbott, C Adams, RX Adhikari, A Ananyeva,S Appert, K Arai, et al. Sensitivity and performance of the advanced ligo detectors in the thirdobserving run.
Physical Review D , 102(6):062003, 2020.[9] F Acernese, T Adams, K Agatsuma, L Aiello, A Allocca, A Amato, S Antier, N Arnaud, S Ascenzi,P Astone, et al. Advanced virgo status. In
Journal of Physics: Conference Series , volume 1342,page 012010. IOP Publishing, 2020.[10] Brett Shapiro, Rana X Adhikari, Odylio Aguiar, Edgard Bonilla, Danyang Fan, Litawn Gan, IanGomez, Sanditi Khandelwal, Brian Lantz, Tim MacDonald, et al. Cryogenically cooled ultralow vibration silicon mirrors for gravitational wave observatories.
Cryogenics , 81:83–92, 2017.[11] Edgard Bonilla and Brian Lantz. Improving the cool-down times for third generation gravita-tional wave observatories (lvc). , 2019. LIGO document,LIGO-G1900526-v1.[12] RJ Corruccini. Gaseous heat conduction at low pressures and temperatures.
Vacuum , 7:19–29,1959.[13] Amit Agrawal and SV Prabhu. Survey on measurement of tangential momentum accommodationcoefficient.
Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films ,26(4):634–645, 2008.[14] Bing-Yang Cao, Min Chen, and Zeng-Yuan Guo. Temperature dependence of the tangentialmomentum accommodation coefficient for gases.
Applied Physics Letters , 86(9):091905, 2005.[15] Franck Celestini and Fabrice Mortessagne. Cosine law at the atomic scale: toward realisticsimulations of knudsen diffusion.
Physical Review E , 77(2):021202, 2008.[16] Tamas I Gombosi and Atmo Gombosi.
Gaskinetic theory , chapter 7.2.3. Number 9. CambridgeUniversity Press, 1994.[17] A Cavalleri, G Ciani, R Dolesi, M Hueller, D Nicolodi, D Tombolato, S Vitale, PJ Wass, andWJ Weber. Gas damping force noise on a macroscopic test body in an infinite gas reservoir.
Physics Letters A , 374(34):3365–3369, 2010.[18] Stephan Schlamminger. Comparison of squeeze film damping simulations for the advanced ligogeometry. , 2010.[19] A Cavalleri, G Ciani, R Dolesi, A Heptonstall, M Hueller, D Nicolodi, S Rowan, D Tombolato,S Vitale, PJ Wass, et al. Increased brownian force noise from molecular impacts in a constrainedvolume.
Physical review letters , 103(14):140601, 2009.[20] Peter R Saulson. Thermal noise in mechanical experiments.