Asteroids for $\mu \mathrm{Hz}$ gravitational-wave detection

A major challenge for gravitational-wave (GW) detection in the $\mu \mathrm{Hz}$ band is engineering a test mass (TM) with sufficiently low acceleration noise. We propose a GW detection concept using asteroids located in the inner Solar System as TMs. Our main purpose is to evaluate the acceleration noise of asteroids in the $\mu \mathrm{Hz}$ band. We show that a wide variety of environmental perturbations are small enough to enable an appropriate class of $\sim 10\mathrm{km}$-diameter asteroids to be employed as TMs. This would allow a sensitive GW detector in the band $(\mathrm{few})\times {10}^{-7}\mathrm{Hz}\lesssim f_{\mathrm{GW}}\lesssim (\mathrm{few})\times {10}^{-5}\mathrm{Hz}$, reaching strain $h_c\sim {10}^{-19}$ around $f_{\mathrm{GW}}\sim 10\mu \mathrm{Hz}$, sufficient to detect a wide variety of sources. To exploit these asteroid TMs, human-engineered base stations could be deployed on multiple asteroids, each equipped with an electromagnetic transmitter/receiver to permit measurement of variations in the distance between them. We discuss a potential conceptual design with two base stations, each with a space-qualified optical atomic clock measuring the round-trip electromagnetic pulse travel time via laser ranging. Trade space exists to optimize multiple aspects of this mission: for example, using a radio-ranging or interferometric link system instead of laser ranging. This motivates future dedicated technical design study. This mission concept holds exceptional promise for accessing this GW frequency band.

Figure 1

Sketch of the mission concept outlined in Sec. 2 (not to scale). Two 10 km-class asteroids act as inertial test masses. Base stations deployed on the asteroids exchange electromagnetic pulses via a transmitter/receiver link system. The round-trip travel time for the pulses is recorded by referencing a local space-qualified optical atomic clock. This creates a ranging-type gravitational-wave detector. A more sophisticated setup, with the base stations held in orbit around the asteroids but continually referenced to reflectors deployed on the asteroid surfaces, is also discussed in Sec. 2. The projected strain sensitivity for this concept is shown in Fig. 9. Image credits [52]: NASA/JPL (433 Eros) [53], NASA/Goddard/University of Arizona (101955 Bennu) [54].

Figure 2

Individual noise contributions to overall characteristic strain sensitivity around the $\mu \mathrm{Hz}$ band. In every panel (a)–(h), the solid thick gray lines denote two smoothed estimates of our combined sensitivity reach [Sec. 6]; see Fig. 9. Each panel shows one or more of the noise estimates detailed in Secs. 3 and 4: (a) solar intensity c.m. noise [Sec. 3a1; see also Sec. 3b1]; (b) solar wind c.m. noise (green band, with smoothed result as a solid green line) [Sec. 3a2; see also Sec. 3b1]; (c) thermal expansion [Sec. 3d2]; (d) asteroid GGN [65] simulation (solid dark blue) and close-pass estimate (shaded light blue), in both cases plotted only where $h_c\ge {10}^{-20}$ [Sec. 3a3]; (e) clock noise (long-dashed salmon) (not included in enveloped noise curve; see text) [Sec. 4c]; (f) link noise for two different laser powers in a laser pulsing setup (solid and dashed gold being 1 W and 3 W of laser power transmitted, respectively) [Sec. 4a], and in a radio pulsing setup (dotted teal) [Sec. 4b]; (g) collisional c.m. noise using in-quadrature (realistic; solid turquoise) or linear (worst-case; dashed turquoise) summation [Sec. 3a4; see also Sec. 3b3], and a variety of magnetic torquing estimates using both high (dotted lines) and low (433 Eros-like; solid lines) values for the asteroid specific magnetic moment, for torques both parallel (purple lines) and perpendicular (orange bands with smoothed orange lines) to the angular momentum axis of the asteroid [Sec. 3b2]; and (h) expected excluded bands (taken to be 5% wide) around 4 hr (light gray) and 5 hr (darker blue-gray) asteroid rotational periods, and their first 10 harmonics [Sec. 3c2]. In panels (a) and (c), the different colored curves denote the results of using different measured solar intensity fluctuation power spectral densities; the different lines are detailed in the relevant sections of Sec. 3a. These results assume asteroid TMs with 8 km radii and densities of $2.5\mathrm{g}/{\mathrm{cm}}^3$ (mass of $5.4\times {10}^{15}\mathrm{kg}$), located 1.5 AU from the Sun, with a fixed 1 AU baseline; see discussion in Sec. 3a1. A combined plot is given in Fig. 8; an enveloped noise curve is given in Fig. 9.

Figure 3

Upper panel: integral number-flux density $I(m)$, defined as at Eq. (15), of dust in the interplanetary medium in the vicinity of the Earth’s orbit. The various colored lines are the dust model adopted in Ref. [157] (various shades of red, as annotated), and both the lunar (green) and interplanetary (dashed blue) flux models from Ref. [154] (identified in the legend as “Grün, et al.”). We adopt the conservative estimate shown in thick, dashed black; the exact construction of this curve is discussed in the text. Lower panel: The mass-weighted difference number-flux density (black). We take $\overline{m}=\sqrt{m_1{m}_2}$ and $\mathrm{\Delta }I(m)\equiv I(m_1)-I(m_2)$, where $m_2>m_1$ are the bin edges (we present these results using two bins per decade of mass as measured in grams). Note also that $I(m)$ is a decreasing function of $m$: $I(m_1)>I(m_2)$. Each vertical dotted blue line shows the upper edge of the largest mass bin for which one collision with 314159 Alice can be expected in the amount of time annotated on the line; fewer than one collision in the indicated time is expected to occur for objects with masses in each mass bin in the blue shaded regions to the right of each of these lines. We draw lines at ${10}^4\mathrm{s}$ (approximately the highest frequency of interest for our proposal), ${10}^6\mathrm{s}$, 1 year, 10 years (a typical mission duration), 100 yrs, 1000 yrs, the surface age of 433 Eros (approximately 400 Myr) [153] (Eros itself has however likely spent a significant fraction of this period in the Main Belt, where its surface cratering/collisional rates would be much higher than in its current near-Earth environment [116]), the age of the Solar System (4.6 Gyr) [158], and the age of the Universe (13.8 Gyr) [159]. Excluding rare events that would not be expected to occur within a mission duration, the dominant mass-weighted differential number flux is in the region $m\sim {10}^{-6}-{10}^{-5}\mathrm{g}$.

Figure 4

Contributions to the strain noise $(h_c{)}_i$ given in Eq. (17) from each mass-bin $i$, presented using two mass bins per decade of mass as measured in grams. The red, green, and blue lines are, respectively, results for GW frequencies $f_{\mathrm{GW}}={10}^{-6}$, ${10}^{-5}$, and ${10}^{-4}\mathrm{Hz}$. The lines are drawn solid up to and including the bin containing the maximum mass ($m_{\max }$) object as defined at Eq. (19) in the text; the maximum mass for each frequency is denoted by the thin vertical dashed line of like color. Above the bin containing the maximum mass, the per-bin strain results $(h_c{)}_i$ are shown by dotted lines; this estimate is actually not correct in that mass range, and we do not use it. The overall strain estimate at Eq. (18) only includes contributions up to and including the maximum-mass bin (i.e., we use only the solid parts of the various colored lines). Because $(h_c{)}_i$ is an increasing function of $m$, the rarest and largest collisions dominate this noise estimate.

Figure 5

As for Fig. 4, but using instead the conservative noise estimate given at Eq. (20) and discussed in the text. For this estimate, collisions with objects with $m\sim {10}^{-5}\mathrm{g}$ always dominate the estimate for the GW frequencies in our band.

Figure 6

One-sided PSD of the one-hour-averaged heliospheric magnetic field (HMF) as measured by the ACE satellite [137, 138] at its orbital location around the Earth-Sun L1 Lagrange point over the time period 1997–2021 (light-red line merging into the light-red band). The sliding average of the PSD taken over a Gaussian kernel in log-frequency space with a width of $0.1{\log }_{10}[\mathrm{Hz}]$ is shown by the thick red line. The solid blue line shows $S[⟨B_{\mathrm{HMF}}⟩]=60{\mathrm{nT}}^2/\mathrm{Hz}\times (f/\mathrm{mHz}{)}^{-5/3}$, while the dotted blue line is $5\times$ that same analytical expression; see discussion in text. The horizontal green line shows the value $S[⟨B_{\mathrm{HMF}}⟩]=1.5\times {10}^6{\mathrm{nT}}^2/\mathrm{Hz}$, with the green band covering a range from half to twice that value (i.e., a factor-of-2 variation).

Figure 7

For each of four pairs $i$, $j$ of asteroids selected from Table 1, and annotated in each quadrant of this figure, we present two sets of results. Upper panel for each asteroid pair: we show $\widehat{h}[x]$ as defined in the text, for the quantities $x$ taken to be the baseline separation distance $L_{i,j}(t)$ between the asteroids (red lines), and the Solar-System-barycentric radius of each asteroid orbit $r_k(t)$ for $k=i$, $j$ (blue and green lines, respectively). These results are presented for two separate, contiguous simulated 10-year mission durations: one set of results as solid lines and the other as dashed lines. Note that these results are obtained after processing the time-series data through a window function $w(t)={\sin }^8[\pi (t-t_0)/T]$ to reduce spectral leakage and increase the dynamic range of the PSD [65, 123], but are corrected upward by a factor $\zeta \equiv 128/35$ to partially account for the amplitude-suppression effects of the window (see discussion in text) and are thus directly comparable to results for $h_c$. Also shown in gray is the smoothed and enveloped $h_c$ noise curve arising from all other relevant noise sources, as shown in Fig. 9. We note that a number of the results curves are truncated at values of $\widehat{h}[x]$ either at, or below, the noise curve; this is to avoid showing unphysical results that are impacted by residual spectral leakage effects that our windowing procedure reduced but did not completely eliminate (see discussion in the text). No conclusions are modified by this truncation. Lower panel for each asteroid pair: we show the time-series data for $L_{i,j}(t)$ (red lines) and $r_k(t)$ for $k=i$, $j$ (blue and green lines, respectively), with the two contiguous 10-year mission durations shown again as solid and dashed lines, respectively, and matching the same line styles as used in the upper panels. Also shown in purple (with the same solid/dashed definition) is the envelope of the window function $w(t)$. These results are discussed at length in the text.

Figure 8

Combined plot of all contributing noise sources for our mission concept; see also Fig. 2 for a clearer version of the individual noise contributions shown here and a more detailed explanation of the various lines and bands. The two smoothed envelope lines (solid black thick lines), computed using a log-frequency-space Gaussian smoothing kernel, take into account all noise sources, except for the following estimates (the reasons for their omission are discussed in the text): clock noise, upper bound on collisions, and high specific magnetic moment torquing. At high frequency, the enveloped lines track the conservative and aggressive laser ranging noise curves, respectively.

Figure 9

Smoothed and enveloped projected characteristic-strain sensitivity reach $h_c$ for our mission concept as a function of the gravitational-wave period $f_{\mathrm{GW}}$, assuming either conservative (thick green line, labeled 1) or aggressive (thick red line, labeled 2) laser link noise at high frequency. Also shown for comparison are limits or projections for other existing and proposed facilities: existing NANOGrav limits [15] (teal; converted to $h_c$ per the discussion in Sec. 6), existing EPTA limits [12] (dark green; converted to $h_c$ per the discussion in Sec. 6), the LISA L3 design sensitivity (solid black) [20], the $\mu \mathrm{Ares}$ strawman mission concept projected sensitivity (darker, short-dashed purple, plotted above $\sim 2\mu \mathrm{Hz}$; see text) [6], and projected future astrometric GW detection reach using Gaia or Roman Space Telescope survey data [64] (gold and orange, respectively; converted to $h_c$ per the discussion in Sec. 6). Also shown is the astrophysical/cosmological confusion noise estimate (lighter, long-dashed purple, plotted above $\sim \mu \mathrm{Hz}$; see text) from Ref. [6]. Vertical shaded bands show the rotational frequencies of asteroids with rotational periods of 5 hrs (darker blue-gray) or 4 hrs (light gray), as well as the first ten harmonics of these rotational frequencies (the bands are $\pm 5\%$ wide), which limit the high-frequency reach of our concept.