Damping separation of finite open systems in gravity-related experiments in the free molecular flow regime

The residual gas damping of the test mass (TM) in the free molecular flow regime is studied in the finite open systems for high-precision gravity-related experiments. Through strict derivation, we separate the damping coefficients for two finite open systems, i.e., the biplate system and the sensor core system, into base damping and diffusion damping. This elucidates the relationship between the free damping in the infinite gas volume and the proximity damping in the constrained volume, unifies them into one microscopic picture, and allows us to point out three pathways of energy dissipation in the biplate gap. We also provide the conditions that need to be met to achieve this separation. In applications, for space gravitational wave detection, our results for the residual gas damping coefficient for the 4TM torsion balance experiment are the closest to the experimental and simulation data compared to previous models. For the LISA mission, our estimation for residual gas acceleration noise at the sensitive axis is consistent with the simulation result, within about 5% difference. In addition, in the test of the gravitational inverse-square law, our results suggest that the constraint on the distance between TM and the conducting membrane can be reduced by about 28%.

Figure 1

Schematic side view of the biplate finite open system. The TM moves downward parallel to the $x$ axis, with a gap spacing denoted as $h$. Gas molecules can exchange with the environment through the gap boundary. The pure light gray gas molecules are from the fixed plate surface or the environment, while the gas molecules represented by light gray and black desorb from the TM surface. The color black represents more or less momentum or energy carried compared to the case when the TM is stationary. The effective molecular number density on the windward and leeward sides is denoted as $n_{\mathrm{W}}^{\prime }$ and $n_{\mathrm{L}}^{\prime }$, respectively.

Figure 2

Schematic side view of the sensor core in GRS. The sensor core consists of a TM in the middle and an electrode housing surrounding it, with a gap between them. The gap can be divided into six biplate gaps, i.e., bottom gaps $G_{x1}$, $G_{x2}$, and lateral gaps $G_{z1}$, $G_{z2}$, $G_{y1}$, $G_{y2}$, and the last two are not shown.

Figure 3

Schematic drawing of the 4TMs torsion balance experiment setup (not to scale). (a) A cross-shaped structure is suspended by a wire, with four hollow cubic TMs attached at each end. The side length of each TM is $s=46\mathrm{mm}$, and the pendulum arm length is $r=0.1065\mathrm{m}$. (b) The gap spacings between the GRS’s TM and the electrode housing (EH) in the $x$, $y$, and $z$ axes are 4.0 mm, 2.9 mm, and 3.5 mm, respectively. (c) The gap spacings between the S2’s TM and the EH in the $x$, $y$, and $z$ axes are 8.0 mm, 6.0 mm, and 8.0 mm, respectively [31].

Figure 4

Comparison of different rotational damping coefficients in the 4TM torsion balance experiment. The black dashed line ($+$) and black dashed line ($\times$) represent the experimental measured values and simulation values from the UTN, respectively [31]. The green solid line ($+$) represents the theoretical values from Eq. (28), i.e., our finite open model in this paper, and the short dashed gray line ($\times$) represents the theoretical values from the constrained volume model [39]. The short dashed gray line represents the theoretical result of damping coefficients in an infinite volume [36].

Figure 5

Schematic drawing of the test of the ISL experiment setup (not to scale). An $I$-shaped pendulum is suspended from the bottom of the torsion balance through tungsten wire, and the pendulum faces the position of the attractor. The middle part of the pendulum is a glass block ($M$), and two glass bases ($G_1$, $G_2$) are symmetrically installed at both ends. Two glass substrates ($G{t}_1$, $G{t}_2$), two tungsten TMs ($W{t}_1$, $W{t}_2$), and two gravitational compensation pieces ($Wt{c}_1$, $Wt{c}_2$) are glued to the glass bases opposite the attractor. A special glass clamp is installed on the top of the middle glass block, and it can fix the suspended tungsten wire to the center of the pendulum. On the left side of the conducting membrane, there is an attractor not shown here. The attractor consists of eight tungsten source masses and eight tungsten compensation masses, arranged alternately on a rotatable glass disk [19].

Figure 6

Comparison of gap spacing constraints under different pressures between the finite open model and the nonisothermal model. The shaded area below the green solid line represents the constraints given by Eq. (31), i.e., our biplate finite open model in this paper, and the shaded area below the gray dashed line represents the constraints given by the nonisothermal model [38].

Figure 7

Schematic diagram of the torque exerted on the TM. (a) The coordinate system where the TM is located; (b) Top view of the TM, the torque along the $z$ axis.