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Dust as a solar shield [1]

['Benjamin C. Bromley', 'Department Of Physics', 'Astronomy', 'University Of Utah', 'Salt Lake City', 'Ut', 'United States Of America', 'Sameer H. Khan', 'Scott J. Kenyon', 'Smithsonian Astrophysical Observatory']

Date: 2023-02

We revisit dust placed near the Earth–Sun L 1 Lagrange point as a possible climate-change mitigation measure. Our calculations include variations in grain properties and orbit solutions with lunar and planetary perturbations. To achieve sunlight attenuation of 1.8%, equivalent to about 6 days per year of an obscured Sun, the mass of dust in the scenarios we consider must exceed 10 10 kg. The more promising approaches include using high-porosity, fluffy grains to increase the extinction efficiency per unit mass, and launching this material in directed jets from a platform orbiting at L 1 . A simpler approach is to ballistically eject dust grains from the Moon’s surface on a free trajectory toward L 1 , providing sun shade for several days or more. Advantages compared to an Earth launch include a ready reservoir of dust on the lunar surface and less kinetic energy required to achieve a sun-shielding orbit.

Data Availability: The only actual data specific to our study is now available in S1 Table . We generate figures from the data in-core with Python scripts available now as a tar file in a new Github repository: https://github.com/benjbromley/Dust-as-a-solar-shield .

This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

Here, we revisit the reduction of sunlight received by Earth that results from the placement of dust at or near the inner Lagrange point, L 1 , lying directly between Earth and the Sun, including gravitational perturbations from the Moon and other planets. While unstable, these corotating orbits allow for the possibility of temporarily shading Earth. We start by assessing the shadows produced by various types of dust; then we numerically determine orbits that persist near L 1 , including the impact of radiation pressure and solar wind. Our main results are a connection between the quantity and quality of dust and the attenuation of sunlight at Earth on achievable orbits near L 1 . To compare with previous work, we target a reduction in solar irradiance of 1.8%, or 6 attenuation-days per year.

Variations on the original proposals to shade Earth with artificial sun shields include the use of dust. Clouds of micron-size gains at the Earth–Sun L 1 point [ 15 – 17 ], at Lagrange points of the Moon-Earth system [ 11 , 18 ], and in orbit around Earth [ 19 – 21 ] have all shown some promise, albeit with limitations. The potential sources of dust include terrestrial and lunar mines and near-Earth asteroids. In all cases, masses ≳ 10 10 kg are necessary to have climate impact.

Space-baced approaches for solar radiation management provide an alternative. Objects in space—a large screen [ 10 – 12 ] or a swarm of small artificial satellites [ 13 , 14 ]—that are well-positioned at the L 1 Lagrange point between Earth and the Sun can efficiently shade our planet. Challenges includes maintaining orbits in the face of radiation pressure from sunlight. The optical properties of the orbiters are thus chosen to mitigate this problem. A high degree of forward scattering allows light to be deflected without transferring much photon momentum, a feat accomplished with refractive, non-absorbing material. A second challenge is that the amount of material required to provide climate-impacting shade exceeds 10 9 kg, which is roughly a hundred time more mass than humans have sent into space to date. However, strategies have been identified that are feasible [ 13 ].

Climate change on Earth is an existential threat. Increased entrapment of solar energy, the result of changes in composition of the Earth’s atmosphere, is acknowledged to be a severe problem [ 1 – 3 ]. One class of strategies for combating climate change is to reduce the solar irradiance by intercepting sunlight before it reaches Earth [ 4 ]. The target attenuation of sunlight, based on modeling, is approximately 1–2% [ 5 , 6 ]. For historical context, aerosols in the Earth’s atmosphere can potentially serve as light reflectors or absorbers that redistribute solar radiation [ 7 ]. However, their overall impact may be difficult to predict owing to uncertainties in how they may be circulated and their interaction with clouds. Regional weather changes, the nature of deployment programs, and long-term environmental effects will inevitably cause uneven society hardships and benefits, arguing that other approaches should be prioritized [ 8 , 9 ].

Shading by dust near L 1

A particle orbiting the Sun can shield Earth by absorbing or scattering radiation. A shield’s overall effectiveness depends on its ability to sustain an orbit that casts a shadow on Earth. In principle, a small shield placed at L 1 can remain in close alignment between Earth and the Sun, offering hope that it can provide steady sun shade. The distance between L 1 and Earth, d 1 , is approximately the Hill radius, (1) where a semi is the semimajor axis of the Earth’s orbit around the Sun, the masses M ⊕ and M ⊙ correspond to Earth and the Sun, respectively, and 1 au ≈ 1.496 × 1011 m. Knowing this distance allows us to estimate the effectiveness of a shield, whether it is a solid thin film or, as we consider here, a cloud of dust. The first step in this determination is how an individual dust grain interacts with sunlight.

Particle scattering The amount of sunlight removed by a particle is quantified by a cross section, Q ext πr p 2, where r p is the particle’s radius, and (2) is the extinction efficiency factor, broken down in terms of absorption and scattering efficiencies (Q abs and Q sca , respectively). Light scattered by reflection and refraction is characterized by a phase function, Φ(θ), giving the amount of incident light that is scattered at angle θ relative to the direction of travel of the incident beam. The anisotropy parameter, (3) where the integration is over all solid angles, indicates whether light is, on average, back-scattered (−1 ≤ g < 0), forward-scattered (0 < g ≤ 1) or neither (g = 0, as for an isotropic scattering). While the effectiveness of a particle as a sun shield is governed by Q ext , the scattering anisotropy may also be important. A small particle that is impacted by radiation pressure feels a stronger force if it back-scatters photons, reversing their angular momentum, than if the particle forward-scatters them, with only modest changes to the flow of photon momentum [22]. Scattering outcomes depend strongly on particle size. For particles larger than a micron, bigger than the typical wavelength of sunlight, λ ∼ 0.5 μm, the scattering efficiency is Q ext ≈ 2; particles of this size block incident sunlight as a result of their geometric cross section A p , equal to for spheres, augmented by interference effects from their internal response to the incident radiation [23]. At distance d 1 , the Earth-L 1 separation, even objects as large as m will be in the Frauenhofer diffraction regime [24]. Particles with sub-micron radii are smaller than the typical wavelength of sunlight. The interaction of these small grains with light can be complicated, though the theories of Rayleigh, Lorentz, and Mie are powerful starting points. From Rayliegh theory, the efficiency factors for nanoparticles are (4) (5) where K = (n2 − 1)/(n2 + 2) and n is the index of refraction. In this limit, scattered light is isotropic. For smaller particles in this regime, extinction efficiencies are low; for example, 0.01 μm silver nanoparties have Q ext ∼ 0.01 [25]. Small molecules, with physical radii of a few tenths of a nanometer and absorption cross sections typically smaller than 10−4 nm2 when averaged across the solar spectrum, have extinction efficiencies below roughly 10−3. In the intermediate regime, where grains have radii of 0.1–1 μm, the extinction efficiencies are higher and admit the possibility of forward scattering. We use Mie theory to estimate scattering properties of these grains, incorporating the miepython package (github.com/scottprahl/miepython) to obtain average Q ext , Q sca , and g for particles of coal dust, sea salt, glass, and gold in sunlight, modelled with a blackbody spectrum running from 300 nm to 1500 nm. Table 1 provides a list of parameters. PPT PowerPoint slide

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TIFF original image Download: Table 1. Optical properties. https://doi.org/10.1371/journal.pclm.0000133.t001 Other dust or grain configurations include particles with elongated shapes (e.g., ice crystals) or in fluffy agglomerates. When the geometric limit applies, the average scattering cross section of a randomly oriented, long cylinder (length L and diameter D) is (6) When the wavelength of light is comparable to D, the cross-section is much larger [27]. Fluffy aggregates with high porosity may also increase the scattering efficiencies compared to objects with the same bulk composition and mass. The Bruggeman mixing rule allows for an estimate of n eff the effective index of refraction of a non-absorbing composite material through the solution of (7) where f is the filling factor, defined to be a volume fraction of the aggregate material within the fluffy particle. By varying composition and structure (shape, porosity), the optical properties of a material may be adjusted to optimize effectiveness as a solar shield.

Shadowing Earth If a light-scattering particle lies directly between Earth and the Sun, it will diminish the solar irradiance at Earth by an amount that depends on its angular size, which depends on the cross-section and the distance from Earth [13]. In the limit where the particle is very close to Earth, just above the atmosphere for example, every photon scattered or absorbed by the particle would have hit Earth. The attenuation in sunlight, defined here as the fractional reduction of the total radiative power of the Sun received by Earth, is then the ratio of the cross-section of the particle to the area of the illuminated face of our planet. When the particle is farther from Earth, well beyond the L 1 point, its shadow, a penumbra, may extend beyond the Sun-facing disk of Earth. Many of the photons that the particle deflects would not have reached Earth. The attenuation in this case is comparatively small. In general, the attenuation of solar irradiance at Earth, stemming from a particle with geometric cross section σ, is (8) where f • is the fraction of solar photons deflected by the particle that would have otherwise reached Earth. This “shadowing efficiency” quantifies the effects of geometry, specifically the fraction of the particle’s penumbral shadow that is occupied by the Sun-facing disk of Earth. Toward estimating f • , we define R • , the radius of the particle’s penumbra when projected onto the surface of a sphere about the Sun that contains the center of Earth: (9) where a semi is the Earth–Sun distance, and d p is the distance from the Earth’s center to the particle. Close to Earth, R • is small, and grows as the particle is moved toward the Sun. At some optimal distance from Earth, (10) the penumbra exactly covers the Sun-facing disk of Earth. As in the rightmost expression in the equation, d • is curiously close to the distance between Earth and L 1 , d 1 ≈ R Hill [13]. When Earth, the Sun, and a light-scattering particle are not aligned, the center of the penumbral shadow of the particle will fall some distance from the center of the Earth’s Sun-facing disk given by (11) where is the particle position relative to Earth, and is the Earth’s displacement from the Sun. With these definitions, the shadowing efficiency is (12) When a light-scattering particle is interior to d • ≈ 0.008 au and aligned with Earth and the Sun. the shadowing efficiency is unity. Further out, at L 1 (b ≤ R • −R ⊕ , f • ≈ 0.82). To estimate the impact of dust on the solar radiance, we consider the attenuation from a spherical particle with a radius of 1 μm, located at L 1 and aligned with Earth and the Sun: (13) While this value is small for an individual absorbing particle, a substantial ensemble of dust particles may have a significant impact on the solar radiation received on Earth. If the particle were moved toward the Sun to a new distance d p > d 1 , then the attenuation would fall off roughly as the square of d p . This decrease with distance may be an important consideration when establishing a swarm of particles at some new L 1 point in response to non-gravitational forces like radiation pressure. A scatterer near L 1 in line with Earth and the Sun attenuates solar radiation by deflecting light out of the narrow cone aimed at the illuminated face of Earth. Light scattered into this cone, with a solid angle ΔΩ, amplifies the sunlight received by Earth by a factor (14) where angular braces denote the phase function averaged over solid angle ΔΩ in the Earth’s direction. For a particle near L 1 , aligned between Earth and the Sun, Earth appears as a disk with an angular radius of ΔΘ = R ⊕ /d 1 ≈ 0.24°, filling a solid angle of ΔΩ ≈ 5.7 × 10−5 steradians; the phase function in this case is evaluated around a scattering angle of 0°. In practice we estimate the average value of the phase function in ΔΘ using the miepython routine i_unpolarized and a 7-point Simpson’s 3/8s rule integrator, since the phase function can be sharply peaked at θ = 0°. For micron size grains at L 1 , the amplification is generally below 0.1% of the level of the attenuation, even for strongly forward-scattering particles (g ≳ 0.8). For larger particles with r p ≳ 10 μm, the phase function is so sharply peaked at θ = 0° that the amplification can exceed over 10% of the magnitude of attenuation. Particles that forward-scatter light amplify the sunlight received on Earth even when they provide no shade at all. While the magnitude of the effect is small compared with attenuation, an accumulation of dust particles just beyond the disk of the Sun from the Earth’s perspective could overwhelm the impact of attenuation by those orbiting in front of the Sun. It is important to avoid a situation where a significant mass in dust lingers just outside the solar disk, brightening the terrestrial sky. Two factors mitigate this risk. First, as particles move away from alignment with Earth and the Sun, the scattering angle toward Earth increases, typically leading to a drop in the amplification. Second, particles displaced from L 1 tend to move away from that point quickly as compared with the metastable orbits at the Lagrange point. It seems unlikely to accumulate much dust in the broad space between Earth and the Sun.

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[1] Url: https://journals.plos.org/climate/article?id=10.1371/journal.pclm.0000133

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