Propulsion of Spacecraft to Relativistic Speeds Using Natural Astrophysical Sources

Although we will deal with weakly relativistic light sails for the majority of our analysis, it is instructive to tackle the relativistic case first; this scenario was first modeled by Marx (1966). For a light sail powered by an isotropic astrophysical source of constant luminosity (L_⋆),⁷ and supposing that the sail reflectance is close to unity (R ≈ 1), the corresponding equation of motion is derivable from Equation (2) of Macchi et al. (2009) and Equation (9) of Kulkarni et al. (2018):

$$\begin{eqnarray}&&{\gamma }^{3}\displaystyle \frac{d\beta }{{dt}}\approx \displaystyle \frac{{L}_{\star }}{2\pi {r}^{2}{{\rm{\Sigma }}}_{s}{c}^{2}}\left(\displaystyle \frac{1-\beta }{1+\beta }\right),\end{eqnarray} \tag{ 1 }$$

where β = v/c, $\gamma =1/\sqrt{1-{\beta }^{2}}$, and Σ_s is the mass per unit area of the sail; we adopt the fiducial value of Σ₀ ≈ 2 × 10⁻⁴ kg m⁻² as it could be feasible in the near future (Parkin 2018). Note that v denotes the instantaneous velocity of the sail, and r represents the time-varying distance between the sail and the astrophysical source.

In formulating this equation, we have not accounted for the inward gravitational acceleration, but this term is negligible provided that L_⋆ ≳ 0.01 L_⊙ (Lingam & Loeb 2020). Likewise, the drag force has been neglected, as it does not alter the results significantly in the limits of $\beta \to 0$ and $\beta \to 1$ (Hoang 2017). We have also presumed that the light sail preserves a constant orientation relative to the source at all points during its acceleration. This requires the selection of suitable sail architecture (Manchester & Loeb 2017) as well as the deployment of nanophotonic structures for self-stabilization (Ilic & Atwater 2019). Lastly, we implicitly work with the scenario in which the payload mass (M_pl) is distinctly smaller than, or comparable to at most, the sail mass (M_s).

Next, after employing the relation dt = dr/(βc) and integrating (1), we arrive at

$$\begin{eqnarray}&&\displaystyle \frac{1}{3}\left[1+\displaystyle \frac{\sqrt{1+{\beta }_{T}}\left(-1+2{\beta }_{T}\right)}{{\left(1-{\beta }_{T}\right)}^{3/2}}\right]\approx \displaystyle \frac{{L}_{\star }}{2\pi {{\rm{\Sigma }}}_{s}{c}^{3}{d}_{0}},\end{eqnarray} \tag{ 2 }$$

where d₀ represents the initial distance from the source (i.e., when the light sail is launched) and β_T is the normalized terminal velocity achieved by the light sail. Instead of calculating β_T, it is more instructive to express our results in terms of the spatial component of the 4-velocity, namely, U_T = β_Tγ_T because ${U}_{T}\to {\beta }_{T}$ for β_T ≪ 1 and ${U}_{T}\to {\gamma }_{T}$ for β_T → 1.

The next aspect to consider is the initial launch distance. While this appears to be a free parameter, it will be constrained by thermal properties in reality (McInnes 2004, Chapter 2.6). We introduce the notation ε for the absorptance (note that ε ≪ 1 under optimal circumstances) and denote the sail temperature at the initial location by T_s. If we suppose that the sail behaves as a blackbody, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{\varepsilon {L}_{\star }}{4\pi {d}_{0}^{2}}\approx \sigma {T}_{s}^{4},\end{eqnarray} \tag{ 3 }$$

provided that the total sail emissivity associated with its front and back sides is close to unity.⁸ As we shall hold T_s fixed henceforth at d₀, it is reasonable to presume that the initial emissivity—which is dependent on T_s (Ancona & Kezerashvili 2017, Equation (9))—is crudely invariant for all astrophysical sources. This equation can be duly inverted to solve for d₀, thus yielding

$$\begin{eqnarray}&&{d}_{0}\approx 0.17\,\mathrm{au}{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{1/2}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-2}.\end{eqnarray} \tag{ 4 }$$

We have introduced the fiducial values of ε ≈ 0.01 and T_s ≈ 300 K. The normalization factor for ε is optimistic because this value embodies the aggregate across all wavelengths, but it might be realizable through the use of multilayer stacking techniques (Atwater et al. 2018, Figure 3). The temperature of 300 K was chosen on the premise that it represents a comfortable value for organic lifeforms as well as electronic instrumentation. After combining (3) and (2), the latter is expressible as

$$\begin{eqnarray}&&\displaystyle \frac{1}{3}\left[1+\displaystyle \frac{\sqrt{1+{\beta }_{T}}\left(-1+2{\beta }_{T}\right)}{{\left(1-{\beta }_{T}\right)}^{3/2}}\right]\approx \displaystyle \frac{{T}_{s}^{2}}{{{\rm{\Sigma }}}_{s}{c}^{3}}\sqrt{\displaystyle \frac{{L}_{\star }\sigma }{\pi \varepsilon }},\end{eqnarray} \tag{ 5 }$$

and upon substituting the previously specified parameters, the above equation simplifies to

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{L}_{\star }}{{L}_{\odot }} & \approx & 5.8\times {10}^{11}\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}\\ & & \times \ {\left[1+\displaystyle \frac{\sqrt{1+{\beta }_{T}}\left(-1+2{\beta }_{T}\right)}{{\left(1-{\beta }_{T}\right)}^{3/2}}\right]}^{2}.\end{array}\end{eqnarray} \tag{ 6 }$$

If we know the terminal speed that we wish to achieve using a suitable astrophysical source, we can employ this equation to estimate the requisite luminosity of the object. Before proceeding further, it is useful to consider two limiting cases. First, in the nonrelativistic regime corresponding to β_T ≪ 1, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\approx 1.3\times {10}^{12}\,{\beta }_{T}^{4}\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}.\end{eqnarray} \tag{ 7 }$$

Next, if we consider the ultrarelativistic regime wherein γ_T ≫ 1, we find that (6) reduces to

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\approx 9.3\times {10}^{12}\,{\gamma }_{T}^{6}\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}.\end{eqnarray} \tag{ 8 }$$

Hence, anticipating later results, it is evident that attaining the ultrarelativistic regime is very difficult because it necessitates L_⋆ ≫ 10¹³ L_⊙.

In Figure 1, we have plotted the luminosity of the astrophysical source as a function of U_T. We have restricted the lower bound to 0.01 L_⊙ because gravitational acceleration becomes important below this luminosity as noted previously, and the upper bound has been chosen based on the most luminous quasars. In the case of U_T ≪ 1, we observe that the luminosity requirements are relatively modest. For example, we find that L_⋆ ≈ L_⊙ leads to β_T ≈ 10⁻³. However, once we approach the regime of U_T ∼ 1, the associated luminosity becomes very large, eventually exceeding that of virtually all known astrophysical objects. By inspecting the figure, it is observed that the plot behaves as a power law with an exponent of +4 up to U_T ≳ 0.1, as expected from (7).

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At this point, it is useful to address some long-lived astrophysical sources in more detail. First, we consider the hottest and most massive stars in the universe, whose luminosity can be roughly approximated by the Eddington luminosity (Kippenhahn et al. 2012, Equation (22.10)). When expressed in terms of the stellar mass M_⋆, the luminosity is given by

$$\begin{eqnarray}&&{L}_{\star }\approx 3.8\times {10}^{4}\,{L}_{\odot }\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right).\end{eqnarray} \tag{ 9 }$$

Hence, upon specifying M_⋆ ≈ 200 M_⊙, given that it seems characteristic of certain massive Wolf–Rayet stars in the Large Magellanic Cloud, the above scaling yields L_⋆ ≈ 7.6 × 10⁶ L_⊙ and thereby evidences reasonable agreement with observations (Hainich et al. 2014; Crowther et al. 2016). From Figure 1, we find that this luminosity yields a terminal speed of β_T ≈ 0.05.

Along similar lines, final speeds of ∼0.01c are attainable by light sails near low-mass X-ray binaries because these objects have bolometric luminosities of ≲10⁶L_⊙; these objects have the additional benefit of being long-lived, as their lifespans can reach ∼0.1 Gyr (Gilfanov 2004). Another class of objects that give rise to similar speeds are a particular category of X-ray binaries, known colloquially as microquasars (Becker 2008). As these sources comprise black holes with masses of ∼1–10 M_⊙ (Mirabel 2001; Cherepashchuk et al. 2005), the use of (9) suggests that their typical luminosities are on the order of 10⁵–10⁶L_⊙, thereby giving rise to β_T ∼ 0.01.

The next class of objects to consider are active galactic nuclei (AGNs), whose luminosities are estimated via (9); the only difference is that M_⋆ should be replaced with the mass (M_BH) of the supermassive black hole (SMBH; Krolik 1999). As per theory and observations, M_BH ∼ 10¹¹ M_⊙ constitutes an upper bound for SMBHs in the current universe (McConnell et al. 2011; Inayoshi & Haiman 2016; Dullo et al. 2017; Pacucci et al. 2017; Inayoshi et al. 2019). When this limit is substituted into (9) after invoking the fact that the Eddington factor is typically around unity during the quasar phase (Marconi et al. 2004),⁹ we find L_⋆ ∼ 3.8 × 10¹⁵ L_⊙. By plugging this value into (6), we end up with γ_T ≈ 2.9. In other words, the most luminous AGNs are capable of driving light sails into the relativistic regime, but not to ultrarelativistic speeds.

Next, we turn our attention to supernovae (SNe). There are many categories of supernovae, each powered by different physical mechanisms, owing to which the identification of a characteristic luminosity is rendered difficult. A general rule of thumb is to assume a peak luminosity of 10⁹L_⊙ (Branch & Wheeler 2017, Chapter 1), which yields β_T ≈ 0.15 after making use of (6); in other words, typical SNe may accelerate light sails to mildly relativistic speeds⁶. It is, however, important to recognize that a special class of supernovae, known as superluminous supernovae (SLSNe), have peak luminosities that are ≳100 times larger than normal events (Gal-Yam 2019; Inserra 2019). Calculations based on numerical simulations and empirical data suggest that the upper bound on the peak luminosity of SLSNe is approximately 5.2 × 10¹² L_⊙ (Sukhbold & Woosley 2016). By applying (6), we obtain β_T ≈ 0.66, thereby suggesting that extreme SLSNe could accelerate light sails to significantly relativistic speeds.

The previous consideration of SNe brings up a crucial caveat that merits further scrutiny. Hitherto, we have implicitly operated under two implicit assumptions concerning the astrophysical object: (1) it has a constant luminosity (L_⋆), and (2) it remains functional for a sufficiently long time to effectively enable the light sail to attain speeds that are close to the terminal value calculated in (6). It is apparent that these two assumptions will be violated for objects that are highly luminous but remain so only for a transient period of time; examples of such objects are SNe and gamma-ray bursts (GRBs). In contrast, massive stars and AGNs are functional over long timescales (≳10⁶ yr).

Hence, it becomes necessary to address another major question: what is the time required for a light sail to achieve a desired final velocity (v_F)? We will adopt v_F ∼ 0.1c because this is close to the terminal speeds associated with several high-energy astrophysical phenomena, as well as comparable to the speed of laser-powered light sails such as Breakthrough Starshot. Moreover, as this speed is weakly relativistic, it is ostensibly reasonable to utilize the nonrelativistic counterpart of (1) without the loss of much accuracy (McInnes 2004, Chapter 7.3). Upon integrating the nonrelativistic version of (1), by taking the limit β ≪ 1, we get

$$\begin{eqnarray}&&{v}^{2}\approx \displaystyle \frac{{L}_{\star }}{\pi c{{\rm{\Sigma }}}_{s}{d}_{0}}\left(1-\displaystyle \frac{{d}_{0}}{r}\right).\end{eqnarray} \tag{ 10 }$$

Hence, the distance covered by the light sail before it attains the desired speed of v_F is defined as Δr = r_F−d₀, where r_F is the location at which v = v_F is attained. Hence, upon further simplification, we end up with

$$\begin{eqnarray}&&{\rm{\Delta }}r\approx {d}_{0}{\left[\displaystyle \frac{{L}_{\star }}{\pi {d}_{0}{c}^{3}{\beta }_{F}^{2}{{\rm{\Sigma }}}_{s}}-1\right]}^{-1},\end{eqnarray} \tag{ 11 }$$

where we have introduced the notation β_F = v_F/c. By making use of (3), the above equation reduces to

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}r & \approx & 0.17\,\mathrm{au}{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{1/2}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-2}\\ & & \times \ \left[8.8\times {10}^{-5}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{2}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{-1}{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{-1/2}\right.\\ & & \times \ {\left.{\left(\displaystyle \frac{{\beta }_{F}}{0.1}\right)}^{-2}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{1/2}-1\right]}^{-1}.\end{array}\end{eqnarray} \tag{ 12 }$$

It is apparent from inspecting the above equation that Δr > 0 necessitates very high luminosities. This requirement is expected, because Figure 1 illustrates that reaching a terminal speed on the order of 0.1c is feasible only for highly luminous sources. We have plotted Δr as a function of L_⋆ in Figure 2. To begin with, we notice that Δr > 0 only for sufficiently high luminosities, as explained previously. Second, at large luminosities, it is found that Δr becomes independent of L_⋆. This trend is discernible from (12) after assuming that the first term inside the square brackets is much larger than unity.

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It is convenient to define the following variable for the ensuing analysis:

$$\begin{eqnarray}\begin{array}{rcl}{v}_{\infty } & \equiv & \sqrt{\displaystyle \frac{{L}_{\star }}{\pi c{{\rm{\Sigma }}}_{s}{d}_{0}}}\approx 9.4\times {10}^{-4}\,c{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{1/4}{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{-1/4}\\ & & \times \ \left(\displaystyle \frac{{T}_{s}}{300\,{\rm{K}}}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{-1/2}.\end{array}\end{eqnarray} \tag{ 13 }$$

By integrating (10) and invoking the definition of ${v}_{\infty }$, we end up with

$$\begin{eqnarray}&&r\sqrt{1-\displaystyle \frac{{d}_{0}}{r}}+{d}_{0}\,{\tanh }^{-1}\left(\sqrt{1-\displaystyle \frac{{d}_{0}}{r}}\right)\approx {v}_{\infty }t.\end{eqnarray} \tag{ 14 }$$

In particular, we are interested in calculating Δt, which is defined as the time at which r = r_F and v = v_F. This timescale is determined by substituting r = r_F in (14), but the final expression proves to be tedious (albeit straightforward to calculate), owing to which the explicit formula is not provided herein.

Figure 3 shows Δt as a function of L_⋆ for different choices of v_F. We observe that Δt is initially large but rapidly reaches an asymptotic value, which is independent of L_⋆. By considering the formal mathematical limit of ${L}_{\star }\to \infty$ and employing standard asymptotic techniques (Olver 1974), one arrives at ${\rm{\Delta }}t\sim 2{d}_{0}{v}_{F}/{v}_{\infty }^{2}$. After using (4) and (13) in this asymptotic expression for Δt, we find that the dependence on L_⋆ cancels out, thereby providing the explanation as to why Δt attains a value independent of L_⋆ in Figure 3.

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From an inspection of Figure 3, it is evident that AGNs comfortably satisfy the requirements for Δt because they are typically active over timescales comparable to the Salpeter time, which has characteristic values of ∼10–100 Myr (Shen 2013). In the case of SNe, we see that Δt ∼ 0.6 yr is necessary to achieve a speed of ∼0.1c, but this number can be lowered further by tuning the other parameters; for example, increasing the temperature by 30% yields Δt ∼ 77 days. This estimate is comparable to the typical peak luminosity timescale for most classes of SNe, which is potentially a few months (Sukhbold & Woosley 2016). Hence, it is conceivable that the timescale over which SNe are operational suffices to power light sails to weakly relativistic speeds.

The situation is rendered very different, however, when we consider GRBs. In theory, the peak luminosities of GRBs are sufficiently high to enable U_T ≫ 1 to be achieved in accordance with (6) and Figure 1. This is because most GRBs that have been detected are characterized by peak values of L_⋆ > 10¹⁶L_⊙, although low-luminosity GRBs have also been identified (Zhang et al. 2018). However, the real bottleneck is the timescale over which these phenomena are active: even the ultralong GRBs have timescales of ∼10⁴ s (Gendre et al. 2013; Kumar & Zhang 2015). Hence, upon comparison with Figure 3, we see that this timescale is insufficient to accelerate the light sails to ∼0.1c. In addition, the close proximity of light sails to GRBs may cause damage to instruments and putative biota due to the high fluxes of ionizing radiation (Melott & Thomas 2011). Finally, as explained in the subsequent sections, higher values of Σ_s and ε are potentially necessary at these wavelengths, thereby suppressing β_T by orders of magnitude.

Broadly speaking, our model is characterized by the existence of three control parameters. Of the trio, a conservative choice was adopted for T_s, the sail temperature at the launch location. In fact, choosing T_s ≈ 400 K would enable the attainment of higher terminal speeds and not cause much damage to silicon-based electronics in the process. The damage to organic lifeforms could be more pronounced, but several authors have suggested that technological entities capable of interstellar travel may be postbiological in nature (Freitas 1980; Dick 2003, 2008; Smart 2012), in which case the significance of this limitation would be diminished.

The other two parameters are the area density (Σ_s) and absorptance (ε). In our calculations, we have normalized Σ_s by Σ₀, which effectively amounts to a sail thickness of ≲0.1 μm. The astrophysical sources we have considered herein, however, span a wide range of luminosities and exhibit correspondingly different spectral energy distributions (SEDs). Thus, the challenge is to design sails that have low absorptance (i.e., high reflectance) while also maintaining a sail thickness compatible with Σ₀. As the terminal speed decreases monotonically with Σ_s and ε, we emphasize that the results derived earlier represent optimistic upper bounds that are probably not realizable in practice. Furthermore, the SEDs of most astrophysical objects are broadband, unlike artificial (i.e., laser or maser) sources, thus making sail specification and design a challenging endeavor. Hence, unless advanced technology can make light sails effective across the range where a substantial fraction of photon output occurs, they will be unable to reach the high speeds obtained in the previous sections.

Studies dealing with reflectance in the γ-ray regime are relatively few in number. Hence, it is instructive to focus on X-rays as they represent the adjacent range. In the case of Si and SiO₂, two materials considered for Breakthrough Starshot, the reflectance (R) has ranged between ∼0.1 and 1 at wavelengths of ≲100 Å (Tripathi et al. 2002); the issue, however, is that the corresponding grazing angles were low (0°–10°), implying that the sail would have to maintain an unusual orientation continuously with respect to the incoming radiation. At low grazing angles and wavelengths of ∼1 nm, peak reflectances of R ≈ 0.2–0.3 have been obtained for various multilayers (Cao et al. 1994; Stoev & Sakurai 1999; Voronov et al. 2015; Burcklen et al. 2016).

Although most of the prior studies were carried out at low grazing angles, several experimental studies indicate that X-ray mirrors with R ≳ 0.2 are realizable at near-normal incidence, albeit at select wavelengths of typically ∼1–10 nm (Bilderback & Hubbard 1982; Trebes et al. 1987; Stearns et al. 1991; Montcalm et al. 1996; Mertins et al. 1998; Eriksson et al. 2003; Lumb et al. 2007; MacDonald et al. 2009; Luo et al. 2018); multilayers comprised the likes of W/C, Ni/Ti, W/B₄C, Cr/Sc, Si/Mo, and Mo/Y. In most of the cases referenced hitherto, the total thickness of the multilayer was comparable to the thickness associated with Σ₀.

Looking beyond soft X-rays, we note that high reflectances have been achieved even for hard X-rays and soft γ-rays (whose energies are ≳10 keV) under very specific circumstances. Shvyd'Ko et al. (2011) demonstrated via experiments and theory that ∼1 mm thick diamond crystals were capable of achieving >99% reflectivity for photons of energies 13.7 and 23.9 keV via Bragg diffraction and backscattering at normal incidence; in general, the typical thickness of the crystal ought to be ≳50 μm to permit near-complete reflection (Shvyd'ko & Lindberg 2012). Another option to achieve $R\to 100 \%$ is to ensure that the diamond crystal is perpetually positioned close to grazing incidence (Shvyd'Ko et al. 2010).

Hitherto, we have discussed only the reflectance, but the results for the absorptance are complementary; if the transmittance is minimal, we have ε = 1 − R. In other words, when R is much smaller than unity, we note that $\varepsilon \to 1$. Although this translates to an increase in ε by two orders of magnitude compared to the fiducial choice, this does not pose a major thermal issue. The reason stems from (3) and (4), which imply that the launch distance is chosen such that the initial sail temperature T_s is fixed at ∼300 K. Hence, if ε is elevated for certain astrophysical SEDs and sail materials, we find that d₀ is correspondingly increased, and vice versa. In all cases, however, the sail temperature ought to remain within the specified thermal limit (in theory), but this benefit comes at the cost of reduced sail accelerations and terminal speeds for higher values of ε.

There are two vital points to bear in mind concerning the above studies. The peak reflectance was governed by not only the grazing angle but also the chosen wavelengths. Given that real-world sails would be confronted with sail stabilization and control as well as the broadband SEDs of astrophysical sources, it appears unlikely that the fiducial values of Σ_s and ε will be attained in practice. It is instructive to gauge how β_T will change if more conservative choices of the parameters are adopted. Upon substituting Σ ≈ 10⁶ Σ₀ (McAlister 2018),¹⁰  ε ≈ 1, and T_s ≈ 400 K in (7), we find that the new values of β_T are reduced by three orders of magnitude compared to the prior estimates of this quantity (for a given L_⋆).

In turn, we note that the acceleration distance and time derived in (2.3) would also be duly modified. Let us work with the ansatz ${v}_{F}=\delta {v}_{\infty }$, implying that δ < 1 constitutes the fraction of the terminal speed achieved. In this event, we find that the asymptotic values are Δr ∼ δ²d₀ and ${\rm{\Delta }}t\sim 2\delta {d}_{0}/{v}_{\infty }$. Thus, adopting the parametric choices outlined in the previous paragraph, after using (4) and (13), we find that Δr increases by a factor of ∼5, whereas Δt is elevated by four orders of magnitude, provided that δ is held fixed.

Hence, it seems necessary to view the estimates in Table 1 as highly optimistic, suggesting that at least some of them (e.g., GRBs and low-mass X-ray binaries) have to be downgraded by a few orders of magnitude. However, for certain high-energy sources—such as massive stars, AGNs (Vasudevan & Fabian 2009; Balbi & Tombesi 2017), supernovae (Branch & Wheeler 2017), and gamma-ray afterglows (Piran 2005; Kumar & Zhang 2015)—a reasonably high fraction of photons are emitted at near-ultraviolet, optical, and infrared wavelengths. For this class of systems, it is conceivable that the estimates in Table 1 are not vastly inaccurate. Moreover, we caution that our prior conclusions regarding β_T were based on current human technology. In principle, therefore, if a technologically advanced species is capable of absolute sail control and utilizes sophisticated nanomaterials, perhaps it might have the capacity to attain speeds that are not very far removed from those listed in Table 1 for some objects.

During the phase where the light sail is accelerated to its final velocity of v_F in the vicinity of the astrophysical source, several key constraints are imposed by the ambient gas and dust present in the environment.

For starters, the following condition must hold true in order to prevent significant slowdown via the cumulative accrual of gas (Bialy & Loeb 2018):

$$\begin{eqnarray}&&1.4{m}_{p}{\int }_{{d}_{0}}^{{r}_{F}}n(r){dr}\lt {{\rm{\Sigma }}}_{s},\end{eqnarray} \tag{ 15 }$$

where m_p is the proton mass, n(r) represents the number density, and Δr is the acceleration distance estimated in (12); the factor of 1.4 accounts for the contribution of helium to the mass density of the gas. In this section, we will assume that the gas density obeys n(r) ≈ n₀ (d₀/r)², which constitutes a reasonable assumption for certain astrophysical sources such as massive stars (Beasor & Davies 2018), thereby simplifying (15) to

$$\begin{eqnarray}&&1.4{m}_{p}{n}_{0}{d}_{0}{\left(\displaystyle \frac{{v}_{F}}{{v}_{\infty }}\right)}^{2}\lesssim {{\rm{\Sigma }}}_{s},\end{eqnarray} \tag{ 16 }$$

after making use of (10). Upon further simplification, the above equation reduces to

$$\begin{eqnarray}&&{n}_{0}\lesssim 2.9\times {10}^{8}\,{{\rm{m}}}^{-3}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{4}{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{-1}{\left(\displaystyle \frac{{\beta }_{F}}{0.1}\right)}^{-2}.\end{eqnarray} \tag{ 17 }$$

In comparison, note that the characteristic value of the number density in the local interstellar medium (ISM) is around 10⁶ m⁻³. Two striking features emerge from (17): it does not depend on the luminosity of the source, nor does it depend on the area density of the light sail. However, this statement is valid only if β_F is held fixed. Instead, if we presume that β_F = δβ_T, we can utilize (7) to accordingly obtain

$$\begin{eqnarray}\begin{array}{rcl}{n}_{0} & \lesssim & 3.3\times {10}^{12}\,{{\rm{m}}}^{-3}\,{\delta }^{-2}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{-1/2}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)\\ & & \times \ {\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{2}{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{-1/2}.\end{array}\end{eqnarray} \tag{ 18 }$$

Thus, it is evident that n₀ increases monotonically with Σ_s, whereas it declines when L_⋆ is increased, both of which seem consistent with expectations.

Another major process responsible for the damage of light sails is ablation caused by impacts with dust grains. The limit on mass ablation is constructed from Bialy & Loeb (2018, Equation (13)), thereby yielding

$$\begin{eqnarray}&&\displaystyle \frac{1.4{m}_{p}\chi {\varphi }_{{dg}}\bar{m}}{{ \mathcal U }}{\int }_{{d}_{0}}^{{r}_{F}}n(r){v}^{2}(r){dr}\lt {{\rm{\Sigma }}}_{s},\end{eqnarray} \tag{ 19 }$$

wherein χ = 0.2 is the fraction of kinetic energy of the dust grain used to vaporize the sail material, φ_dg is the dust-to-grain mass ratio, $\bar{m}$ is the mean atomic weight of the ablated material, and ${ \mathcal U }$ is the vaporization energy. In formulating this expression, it was assumed that the dust grains are moving at much lower speeds than the light sail. After simplifying the integral, we end up with

$$\begin{eqnarray}&&\displaystyle \frac{0.7{m}_{p}\chi {\varphi }_{{dg}}\bar{m}{n}_{0}{d}_{0}{v}_{\infty }^{2}}{{ \mathcal U }}{\left(\displaystyle \frac{{v}_{F}}{{v}_{\infty }}\right)}^{4}\lesssim {{\rm{\Sigma }}}_{s},\end{eqnarray} \tag{ 20 }$$

and we will tackle the case where β_F = δβ_T. Using this scaling, the above equation is expressible as

$$\begin{eqnarray}\begin{array}{rcl}{n}_{0} & \lesssim & 1.4\times {10}^{12}\,{{\rm{m}}}^{-3}\,{\delta }^{-4}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}{\left(\displaystyle \frac{\chi }{0.2}\right)}^{-1}\\ & & \times \ {\left(\displaystyle \frac{{\varphi }_{{dg}}}{0.01}\right)}^{-1}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-1}\left(\displaystyle \frac{{ \mathcal U }}{4\,\mathrm{eV}}\right).\end{array}\end{eqnarray} \tag{ 21 }$$

A more comprehensive analysis of the drag, as well as the ablation caused by dust grains and gas on weakly relativistic light sails, has been undertaken in the context of the ISM by Hoang et al. (2017).

The constraints on n₀ set by the astrophysical source environment are jointly embodied by (18) and (21). If all the other parameters are held fixed, we note that (21) constitutes the more stringent constraint for L_⋆ > L_c, whereas (18) becomes the more crucial constraint in the regime where L_⋆ < L_c. The critical luminosity L_c that demarcates these two regimes is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{c} & \approx & 0.17{L}_{\odot }\,{\delta }^{-4}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{\chi }{0.2}\right)}^{-2}\\ & & \times \ {\left(\displaystyle \frac{{\varphi }_{{dg}}}{0.01}\right)}^{-2}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-2}{\left(\displaystyle \frac{{ \mathcal U }}{4\mathrm{eV}}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 22 }$$

Hence, if all the parameters are held fixed at their fiducial values, we find that L_⋆ > L_c is valid for most astrophysical objects of interest provided that δ is not much smaller than unity. In other words, the primary constraint on n₀ is apparently set by (21). We will, therefore, use this result in our subsequent analysis.

The constraint on the number density translates to a limit on the mass-loss rate ($\dot{M}_{\star }$) of the source via

$$\begin{eqnarray}&&\dot{M}_{\star }\approx {\rm{\Omega }}{r}^{2}{\rho }_{w}(r){u}_{w}(r),\end{eqnarray} \tag{ 23 }$$

under the assumption of spherical symmetry. Note that Ω denotes the solid angle over which the mass-loss rate occurs, whereas ρ_w(r) and u_w(r) are the mass density and the velocity of the wind. At distances >d₀, we will suppose that u_w(r) remains approximately constant, which appears to be reasonably valid for stars (Vink et al. 2000; Gombosi et al. 2018). We specify r = d₀ and utilize ρ(d₀) = 1.4m_pn₀ in parallel with (18), thus arriving at

$$\begin{eqnarray}\begin{array}{rcl}\dot{M}_{\star } & \lesssim & 2\times {10}^{-10}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}{\delta }^{-4}\left(\displaystyle \frac{{\rm{\Omega }}}{4\pi }\right)\left(\displaystyle \frac{{u}_{w}}{{u}_{\odot }}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}\\ & & \times \ {\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{\chi }{0.2}\right)}^{-1}{\left(\displaystyle \frac{{\varphi }_{{dg}}}{0.01}\right)}^{-1}\\ & & \times \ {\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-1}\left(\displaystyle \frac{{ \mathcal U }}{4\,\mathrm{eV}}\right).\end{array}\end{eqnarray} \tag{ 24 }$$

In comparison, the current solar mass-loss rate is given by $\dot{M}_{\odot }\approx 2\times {10}^{-14}{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Linsky 2019). Here, we have opted to normalize u_w by u_⊙ = 500 km s⁻¹, as it corresponds to the solar wind speed near the Earth (Marsch 2006). The most striking aspect of (24) is the fact that L_⋆ is absent therein, which implies that the upper bound on M_⋆ is independent of the source luminosity.

Next, we shall direct our attention to massive stars. Observations indicate that the terminal value of u_w (denoted by ${u}_{\infty }$) is close to the escape speed (v_esc) from the star (Vink et al. 2001; Cranmer & Saar 2011). Thus, it is possible to determine ${u}_{\infty }$ by utilizing the relationship ${u}_{\infty }\approx 1.3{v}_{\mathrm{esc}}$ (Vink et al. 2000), as follows:

$$\begin{eqnarray}&&\displaystyle \frac{{u}_{\infty }}{{u}_{\odot }}\approx {\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{0.22},\end{eqnarray} \tag{ 25 }$$

where we have employed the mass–radius relationship for massive stars (Demircan & Kahraman 1991, p. 320). By combining this relationship with (24), we have obtained a heuristic upper bound on the stellar mass-loss rate that permits the efficient functioning of light sail acceleration. The empirical mass-loss rates for massive stars exhibit significant scatter and depend on a number of parameters, such as the pulsation period, gas-to-dust mass ratio, and the luminosity (Goldman et al. 2017). We shall, however, adopt the simple prescription provided in Beasor & Davies (2018, Equation (3)) for massive stars at their end stages, which is expressible as

$$\begin{eqnarray}&&\dot{M}_{\star }\approx 2.8\times {10}^{-25}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{3.92}.\end{eqnarray} \tag{ 26 }$$

In the case of intermediate-mass stars, we adopt the mass–luminosity scaling from Eker et al. (2015, Table 3) and combine it with (24) and (25) to arrive at

$$\begin{eqnarray}&&{M}_{\max }\approx 9\,{M}_{\odot }\,{\delta }^{-0.38},\end{eqnarray} \tag{ 27 }$$

with the rest of the parameters in (24) held fixed at their characteristic values. The relevance of M_max stems from the fact that stars with M_⋆ ≳ M_max are potentially incapable of accelerating light sails to their terminal speeds without causing excessive damage in the process. The above expression suggests that smaller choices of δ can increase this threshold to some degree; for instance, if we choose δ ∼ 0.1, we end up with M_max ≈ 21.6 M_⊙.

There is another method by which we can deduce M_max. By inspecting (25), we see that ${u}_{\infty }/{u}_{\odot }\approx 2$ for a star with mass ∼10 M_⊙. By substituting this relation in (24), we arrive at $\dot{M}_{\star }\,\lesssim 4\times {10}^{-10}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}{\delta }^{-4}$. Hence, if we specify the range δ ≈ 0.1–0.5, we end up with $\dot{M}_{\star }\lesssim 6.4\times {10}^{-9}\mbox{--}4\times {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Upon comparing these maximal mass-loss rates with those observed for O- and B-type stars (Kobulnicky et al. 2018, Tables 1 and 2), we determine that M_max ∼ 10 M_⊙. This estimate for the maximum stellar mass is consistent with the one obtained in the prior paragraph.

It is, however, necessary to appreciate that the ambient gas density and the mass-loss rate associated with massive stars (or AGNs) are not spherically symmetric because they exhibit a strong angular dependence relative to the rotation axis of the central object (Puls et al. 2008; Smith 2014). Hence, through the selection of launch sites where the density of gas and dust is comparatively lower, the above limit on M_max could be enhanced to a significant degree. We will not present an explicit estimate of this boost herein because of the inherent complexity of mass loss from massive stars.

The next astrophysical objects of interest that we delve into are SNe. The ejecta produced during the explosion move at typical speeds of ∼0.1c (Kelly et al. 2014; Branch & Wheeler 2017). By substituting this estimate for u_w in (24), we end up with $\dot{M}_{\star }\lesssim 1.2\times {10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\delta }^{-4}$ when all the other parameters are held fixed. In comparison, the mass-loss rate of the progenitor just prior to the explosion is ∼0.01–0.1 M_⊙ yr⁻¹ (Kiewe et al. 2012), and it increases by a few orders of magnitude during the explosion. Hence, unless δ is sufficiently small, it is likely that SNe will cause significant damage to light sails situated in their vicinity.

Lastly, we turn our attention to AGNs.¹¹ Two contrasting phenomena are at work, namely, the inflow of gas that powers SMBHs and feedback-driven outflows (Veilleux et al. 2005; Fabian 2012; Zhang 2018). These two processes are not mutually exclusive and are simultaneously at play in regions such as the AGN torus, thereby rendering modeling very difficult (Hickox & Alexander 2018). Hence, for the sake of simplicity, we will suppose that the accretion occurs almost entirely within the Bondi radius (r_B), whose magnitude is given by (Di Matteo et al. 2003, Equation (1))

$$\begin{eqnarray}&&{r}_{B}\approx 4.6\times {10}^{-2}\,\mathrm{pc}\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{6}\,{M}_{\odot }}\right){\left(\displaystyle \frac{{T}_{\mathrm{gas}}}{{10}^{7}{\rm{K}}}\right)}^{-1},\end{eqnarray} \tag{ 28 }$$

where T_gas represents the temperature of the gas. This approach is consistent with the fact that AGN-driven outflows may play important roles at distances as small as ∼0.1 pc (Arav et al. 1994; Hopkins et al. 2016). By comparing this result with (4), after using (9) and assuming an Eddington factor of roughly unity (Marconi et al. 2004), we find d₀ > r_B for SMBHs. Hence, we will restrict ourselves to the consideration of outflows.

The accretion of gas in AGNs is accompanied by wide-angle (i.e., noncollimated) outflows whose velocities vary widely. Although many quasars exhibit outflows with speeds of ∼0.1c (Krolik 1999; Moe et al. 2009; Tombesi et al. 2015), observations of other AGNs have identified winds and outflows at ≲0.01c (Fabian 2012, Section 2.3). Upon substituting the optimistic case given by u_w ∼ 0.1c into (24), we end up with

$$\begin{eqnarray}&&\dot{M}_{\star }\lesssim 1.2\times {10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\delta }^{-4}.\end{eqnarray} \tag{ 29 }$$

In order to model the outflow mass-loss rate, we will employ a simple prescription, namely, that the outflow rate is proportional to the inflow (i.e., accretion) rate; the latter, in turn, is modeled using the Eddington accretion rate (Shen 2013). The proportionality constant ζ exhibits significant scatter: it ranges in ≲0.1–1000 (Crenshaw & Kraemer 2012), although values of ζ ∼ 100 are seemingly more common (Kurosawa & Proga 2009; DeBuhr et al. 2012; Hopkins et al. 2016). As per the preceding assumptions, the mass-loss rate arising from AGN outflows is expressible as

$$\begin{eqnarray}\begin{array}{rcl}\dot{M} & \approx & 2.2\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{{\rm{\Gamma }}}_{e}\left(1-{\epsilon }_{\mathrm{BH}}\right)\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{6}\,{M}_{\odot }}\right)\\ & & \times \ \left(\displaystyle \frac{\zeta }{100}\right){\left(\displaystyle \frac{{\epsilon }_{\mathrm{BH}}}{0.1}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 30 }$$

where Γ_e is the Eddington ratio and _BH represents the radiative efficiency of the SMBH. Hence, by comparing this expression with (29), we see that AGN outflows could cause significant damage to light sails in the event that δ is not much smaller than unity.

In view of the preceding discussion, it would appear as though there are noteworthy hindrances to deploying light sails in the vicinity of many high-energy astrophysical objects. However, there exist at least two avenues by which the aforementioned issues are surmountable in principle. First, by carefully selecting the timing at which the light sail is "unfurled," one might be able to operate in an environment where most of the ambient gas and dust have been cleared out (e.g., by shock waves), thus leaving behind radiation pressure to drive the spacecraft. Second, as we have seen, most of the hindrances arise from high ambient gas and dust densities. Hence, if the spacecraft is equipped with a suitable system to deflect these particles (provided that they possess a finite electrical charge or dipole moment) by means of electric or magnetic forces, one may utilize this device to prevent impacts and the ensuing ablation.

This physical principle is essentially identical to the one underlying magnetic (Zubrin & Andrews 1991) and electric (Janhunen 2004) sails, which are reliant upon the deflection of charged particles and the consequent transfer of momentum to the spacecraft. Thus, not only could one potentially bypass the dangers delineated thus far but also achieve a higher final speed in the process, albeit under ideal circumstances. We will not delve into this topic further as we briefly address electric sails in Section 3. Likewise, it might also be feasible in principle to utilize an interstellar ramjet (Bussard 1960; Crawford 1990; Blatter & Greber 2017) for the dual purposes of scooping up particles and gainfully employing them to attain higher speeds in the process.

We have not considered the slowdown arising from the hydrodynamic drag herein. This is because, as we shall demonstrate in Section 2.6, the drag force is potentially less effective in comparison to slowdown arising from the direct accumulation of gas; in particular, the reader is referred to (36) and (39) for the details. In a similar vein, we have not tackled the damage from sputtering as it contributes to the same degree as slowdown from gas accumulation (Bialy & Loeb 2018); see also (36) and (40) in the following section.

Finally, we turn our attention to the cascades caused by the impact of high-energy photons and particles with energies ≳1 GeV. Laboratory experiments have determined that each cascade displaces ≲10³ atoms when the colliding particle has energies of order of gigaelectronvolts (Was 2017, pp. 77–130). We denote the average number flux of particles (taken here to be photons) in the source environment with ≳1 GeV energies by ${\bar{F}}_{\mathrm{GeV}}$. If a fraction μ_GeV of particles triggers cascade formation, the ensuing constraint on the particle flux is expressible as

$$\begin{eqnarray}&&{\bar{F}}_{\mathrm{GeV}}\cdot {\mu }_{\mathrm{GeV}}\cdot {10}^{3}\cdot \bar{m}\cdot {\rm{\Delta }}t\lesssim {{\rm{\Sigma }}}_{s},\end{eqnarray} \tag{ 31 }$$

where ${\rm{\Delta }}t\sim 2\delta {d}_{0}/{v}_{\infty }$ is the asymptotic acceleration time to reach the requisite final speed, after which the sail could be folded or discarded (see Section 2.6). After substituting the appropriate quantities, we arrive at

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{\bar{F}}_{{\rm{G}}{\rm{e}}{\rm{V}}} & \lesssim & 5.5\times {10}^{13}\,{{\rm{m}}}^{-2}\,{{\rm{s}}}^{-1}\,{\delta }^{-1}{\mu }_{{\rm{G}}{\rm{e}}{\rm{V}}}^{-1}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{-1/4}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{1/2}\\ & & \times \,{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{-3/4}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{3}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-1}.\end{array}\end{array}\end{eqnarray} \tag{ 32 }$$

Note that the photon flux obeys $F(r)={F}_{0}{\left({d}_{0}/r\right)}^{-2}$, where F₀ signifies the flux at r = d₀. Therefore, the average flux during the acceleration phase is given by

$$\begin{eqnarray}&&\bar{F}\approx \displaystyle \frac{1}{{\rm{\Delta }}r}{\int }_{{d}_{0}}^{{r}_{F}}F(r){dr}\approx {F}_{0}\left(1-{\delta }^{2}\right),\end{eqnarray} \tag{ 33 }$$

where the last equality follows after employing the definitions of Δr and r_F. Even for a reasonably high value of δ = 0.3, we see that $\bar{F}\approx {F}_{0}$, owing to which this relationship is adopted for all wavelength ranges. We denote the fraction of the total luminosity comprising photons with energies ≳1 GeV by κ_GeV. Hence, ${\tilde{L}}_{\star }=4\pi {d}_{0}^{2}{\bar{F}}_{\mathrm{GeV}}(1\,\mathrm{GeV})/{\kappa }_{\mathrm{GeV}}$ represents a heuristic upper bound on the total luminosity of the source. Hence, by employing (32), we end up with

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{\tilde{L}}_{\star } & \lesssim & 0.19{L}_{\odot }\,{\delta }^{-1}{\mu }_{{\rm{G}}{\rm{e}}{\rm{V}}}^{-1}{\kappa }_{{\rm{G}}{\rm{e}}{\rm{V}}}^{-1}{\left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right)}^{3/4}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{1/2}\\ & & \times \,{\left(\displaystyle \frac{\varepsilon }{0.01}\right)}^{1/4}{\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-1}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-1}.\end{array}\end{array}\end{eqnarray} \tag{ 34 }$$

By dropping the tilde (i.e., setting ${\tilde{L}}_{\star }={L}_{\star }$), we can invert the above relationship to estimate an upper bound on the luminosity of the source as follows:

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{L}_{\star } & \lesssim & 1.3\times {10}^{-3}{L}_{\odot }\,{\delta }^{4}{\left({\mu }_{{\rm{G}}{\rm{e}}{\rm{V}}}{\kappa }_{{\rm{G}}{\rm{e}}{\rm{V}}}\right)}^{-4}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{2}\\ & & \times \,\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-4}.\end{array}\end{array}\end{eqnarray} \tag{ 35 }$$

Hence, for fiducial choices of the free parameters, we see that most high-energy astrophysical objects do not outwardly appear to meet this criterion albeit in the highly extreme (and unrealistic) limit of ${\mu }_{\mathrm{GeV}}\to 1$ and ${\kappa }_{\mathrm{GeV}}\to 1$. If we increase Σ_s or ε in accordance with Section 2.4, we see that the upper bound on L_⋆ increases further. At first glimpse, this formula appears counterintuitive because it appears to rule out most stars, although the feasibility of stellar sailing is well documented (McInnes 2004; Vulpetti 2012). The answer lies in the fact that κ_GeV is many orders smaller than unity for all stars (Böhm-Vitense 1989), thereby ensuring that (35) preserves consistency with expectations.

We assume henceforth that the light sail enters the ISM at the velocity v_F; depending on the interval over which the source remains active, v_F may be close to the terminal velocity, as explained earlier. Once the light sail enters the ISM, it will be subject to impacts by gas, dust, and cosmic rays. This subject has been extensively studied by Hoang et al. (2017) and Hoang & Loeb (2017), but we will adopt the heuristic analysis by Bialy & Loeb (2018) instead.

The first effect that merits consideration is the slowdown engendered by the accumulation of gas by the light sail. The mean number density of protons in the ISM along the spacecraft's trajectory is denoted by $\langle n\rangle$ and normalized in terms of 10⁶ m⁻³ as noted previously. The maximum distance that is traversed by the spacecraft prior to experiencing significant slowdown (D_a) is

$$\begin{eqnarray}&&{D}_{a}\approx 2.8\,\mathrm{pc}{\left(\displaystyle \frac{\langle n\rangle }{{10}^{6}{{\rm{m}}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right).\end{eqnarray} \tag{ 36 }$$

The next effect that we address is collisions with dust grains, as they cause mass ablation upon impact. The corresponding maximal distance (D_d) is expressible as

$$\begin{eqnarray}\begin{array}{rcl}{D}_{d} & \approx & 5\times {10}^{-5}\,\mathrm{pc}{\left(\displaystyle \frac{\langle n\rangle }{{10}^{6}{{\rm{m}}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right){\left(\displaystyle \frac{{v}_{F}}{0.1c}\right)}^{-2}\\ & & \times \ \left(\displaystyle \frac{{ \mathcal U }}{4\,\mathrm{eV}}\right){\left(\displaystyle \frac{\chi }{0.2}\right)}^{-1}{\left(\displaystyle \frac{{\varphi }_{{dg}}}{0.01}\right)}^{-1}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 37 }$$

An alternative expression for D_d at weakly relativistic speeds (i.e., for v_F > 0.1c) is derivable from Hoang (2017, Equation (29)) as follows:

$$\begin{eqnarray}&&{D}_{d}\approx 54.8\,\mathrm{pc}{\left(\displaystyle \frac{\langle n\rangle }{{10}^{6}{{\rm{m}}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{{{ \mathcal R }}_{\min }}{1\mathrm{nm}}\right)}^{1/2},\end{eqnarray} \tag{ 38 }$$

wherein ${{ \mathcal R }}_{\min }$ is the minimum size of interstellar dust grains. It must be noted, however, that the above equation was derived specifically for very thin light sails.

As the light sail moves through the ISM, it will experience hydrodynamic drag due to the ambient gas. At low speeds, the drag force is linearly proportional to the kinetic energy of the sail (Draine 2011), but this scaling breaks down at higher speeds. The maximum distance that can be covered by a weakly relativistic light sail before major slowdown due to hydrodynamic drag (D_g) is estimated from Hoang (2017, Equation (28)):

$$\begin{eqnarray}\begin{array}{rcl}{D}_{g} & \approx & 4.3\times {10}^{4}\,\mathrm{pc}{\left(\displaystyle \frac{\langle n\rangle }{{10}^{6}{{\rm{m}}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)\\ & & \times \ {\left(\displaystyle \frac{{v}_{F}}{0.1c}\right)}^{2.6}{\left(\displaystyle \frac{{\rm{\Delta }}{\ell }}{0.1\mu {\rm{m}}}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 39 }$$

where Δℓ represents the thickness of the light sail. The last effect that we shall tackle entails sputtering due to gas, as it causes the ejection of particles from the light sail and thereby depletes its mass. The maximum travel distance before sputtering becomes a major hindrance (D_s) is expressible as follows (Bialy & Loeb 2018, Equation (17)):

$$\begin{eqnarray}&&{D}_{s}\approx 3\,\mathrm{pc}{\left(\displaystyle \frac{\langle n\rangle }{{10}^{6}{{\rm{m}}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right){\left(\displaystyle \frac{{ \mathcal Y }}{0.1}\right)}^{-1}{\left(\displaystyle \frac{\bar{m}}{12{m}_{p}}\right)}^{-1},\end{eqnarray} \tag{ 40 }$$

where ${ \mathcal Y }$ represents the total sputtering yield, with the associated normalization factor chosen in accordance with Tielens et al. (1994, Figure 10). Aside from sputtering, mechanical torques arising from collisions with ambient gas can result in spin-up and subsequent rotational disruption. At high speeds, however, this mechanism is apparently less efficient than sputtering in terms of causing damage unless the thickness of the light sail is <0.01 μm (Hoang & Lee 2019, Figure 5).

An inspection of (36)–(40) reveals that the upper bound on the distance is potentially ≲1 pc for the parameter space described in the previous sections. Hence, at first glimpse, it would appear as though light sails moving at high speeds are not capable of traveling over interstellar distances. There is, however, a crucial factor that has been hitherto ignored. If the sail is "folded" in some fashion (e.g., retracted or deflated) or dispensed with altogether, the area density will be elevated by orders of magnitude. To see why this claim is valid, we shall consider the limiting case wherein the payload mass is roughly equal to the sail mass.¹² The size of the sail is denoted by ${{ \mathcal L }}_{s}$, whereas the density and size of the payload are ρ_pl and ${{ \mathcal L }}_{\mathrm{pl}}$, respectively. As the case delineated above amounts to choosing ${{\rm{\Sigma }}}_{s}{{ \mathcal L }}_{s}^{2}\approx {\rho }_{\mathrm{pl}}{{ \mathcal L }}_{\mathrm{pl}}^{3}$, reformulating this equation appropriately yields

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{ \mathcal L }}_{s}}{{{ \mathcal L }}_{\mathrm{pl}}}\right)}^{2}\approx 1.8\times {10}^{6}{\left(\displaystyle \frac{{{ \mathcal L }}_{s}}{1\mathrm{km}}\right)}^{2/3}{\left(\displaystyle \frac{{\rho }_{\mathrm{pl}}}{{\rho }_{0}}\right)}^{2/3}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{s}}{{{\rm{\Sigma }}}_{0}}\right)}^{-2/3},\end{eqnarray} \tag{ 41 }$$

where ρ_pl has been normalized in units of ρ₀ ≈ 4.5 ×10² kg m⁻³, namely, the mean density of the International Space Station.¹³ The significance of (41) is a consequence of the fact that this represents the amplification of the effective area density (stemming from the decrease in cross-sectional area) provided that the sail is completely folded. In other words, one would need to replace Σ_s with ${({{ \mathcal L }}_{s}/{{ \mathcal L }}_{\mathrm{pl}})}^{2}\,{{\rm{\Sigma }}}_{s}$ in (36)–(40). Hence, by closing the light sail, it ought to be feasible in principle for the spacecraft to travel distances on the order of kiloparsecs without being subject to major damage from the impediments arising from the ISM.

Lastly, even if the sail is folded, the collision of high-energy particles with the spacecraft will trigger the onset of cascades and thereby pose radiation hazards to electronics (and perhaps organics) on board the spacecraft (Semyonov 2009). It was estimated by Hippke et al. (2018) that spacecraft traveling at ∼0.1c would exhibit atomic depletion rates of ∼10¹⁷ m⁻³ yr⁻¹ due to cascades arising from cosmic-ray impacts. However, it should be noted that the atomic densities of most solid materials are ∼10²⁹ m⁻³. Hence, this factor is unlikely to be important during the passage through the ISM.

However, if the time-averaged flux of gigaelectronvolt particles during the journey is ∼10⁷ times higher than the cosmic-ray flux near the Sun, this issue may prevent interstellar travel across kiloparsec distances at speeds of ∼0.1c. Given that most of the high-energy particle flux ought to exist toward the beginning of the voyage, that is, in the vicinity of the astrophysical source, it is not clear as to whether such a high average flux would be prevalent; although this subject does necessitate further study, a detailed analysis is beyond the scope of the paper.

Hitherto, we have focused on analyzing the constraints on a single light sail. It should be noted, however, that a sufficiently advanced technological species could opt to place many light sails in the vicinity of the source and accelerate them to high speeds. By doing so, they can take advantage of the economies of scale (Stigler 1958), as the extra cost per additional light sail ought to be relatively low. We will delve into this possibility briefly and highlight a few of the accompanying caveats.

Earlier, we have commented that the spacecraft are launched from r = d₀. Hence, we consider a sphere of this radius and introduce the variable ϕ_s to denote the fraction of the sphere's surface that is covered by the light sails. Hence, the total number of spacecraft per source (${{ \mathcal N }}_{s}$) is roughly estimated as

$$\begin{eqnarray}&&{{ \mathcal N }}_{s}\approx {\phi }_{s}\left(\displaystyle \frac{4\pi {d}_{0}^{2}}{{{ \mathcal L }}_{s}^{2}}\right).\end{eqnarray} \tag{ 42 }$$

By making use of (4), the above expression simplifies to

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal N }}_{s} & \approx & 2.5\times {10}^{3}\left(\displaystyle \frac{{\phi }_{s}}{{\phi }_{0}}\right)\left(\displaystyle \frac{\varepsilon }{0.01}\right){\left(\displaystyle \frac{{T}_{s}}{300{\rm{K}}}\right)}^{-4}\\ & & \times \ \left(\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\right){\left(\displaystyle \frac{{{ \mathcal L }}_{s}}{1\mathrm{km}}\right)}^{-2},\end{array}\end{eqnarray} \tag{ 43 }$$

where we have chosen to normalize ϕ_s in units of ϕ₀ ≈ 3 × 10⁻¹³; the latter parameter embodies the fraction of light blocked by geostationary and geosynchronous satellites orbiting Earth (Socas-Navarro 2018, Section 3.1). In principle, we could adopt much higher values of ϕ_s, such as ϕ_s ≈ 10⁻⁴ (Socas-Navarro 2018), which would elevate ${{ \mathcal N }}_{s}$ by many orders of magnitude. For a supernova with L_⋆ ≈ 10⁹ L_⊙, if we hold all of the other parameters fixed at their fiducial values, we find that ${{ \mathcal N }}_{s}\approx 2.5\times {10}^{12}$ for kilometer-sized light sails. Hence, at least in principle, it is possible to ensure that the number of spacecraft launched per high-energy source considerably exceeds the number of stars in the Milky Way by tuning the free parameters in (43) accordingly.

There are, however, a couple of limitations to bear in mind. First, if ${{ \mathcal N }}_{s}$ becomes exceedingly high, transporting such a large number of light sails to the astrophysical source from the parent system would pose difficulties; alternatively, one may attempt to construct them in situ, but this is contingent on the availability of raw materials. Second, we have deliberately adopted a conservative value of ϕ_s in (43), but even this miniscule fraction has a certain risk of the spacecraft colliding with one another (Lucken & Giolito 2019) and potentially triggering a collisional cascade known as the "Kessler syndrome" (Kessler & Cour-Palais 1978). While this risk is probably not a major concern for ϕ_s ≈ ϕ₀ (Bradley & Wein 2009; Drmola & Hubik 2018), it will become increasingly prominent at higher choices of ϕ_s (Sallmen et al. 2019), unless it can be overcome by an exceptional degree of spacecraft coordination and control.

In order to determine the acceleration produced by electric sails, one must calculate the force per unit length (dF_s/dz) and the mass per unit length (dM_s/dz). The former is difficult to estimate because it entails a complex implicit equation (Janhunen et al. 2010). However, to carry out a simplified analysis, it suffices to make use of Janhunen & Sandroos (2007, Equation (8)) and Toivanen & Janhunen (2009, Equation (3)). The force per unit length for the electric sail is expressible as

$$\begin{eqnarray}&&\displaystyle \frac{{{dF}}_{s}}{{dz}}\approx 2{ \mathcal K }{m}_{p}n{\left(v-{u}_{w}\right)}^{2}{r}_{D},\end{eqnarray} \tag{ 44 }$$

where ${ \mathcal K }$ is a dimensionless constant of order unity and r_D is the Debye length, which is defined as

$$\begin{eqnarray}&&{r}_{D}=\sqrt{\displaystyle \frac{{\epsilon }_{0}{k}_{B}{T}_{e}}{{{ne}}^{2}}},\end{eqnarray} \tag{ 45 }$$

wherein ₀ is the permittivity of free space and T_e signifies the electron temperature. In reality, (44) has been simplified because we neglected a term that is not far removed from unity, as it would otherwise make the analysis much more complicated (see Lingam & Loeb 2020 for additional details); the resulting expression for the acceleration is functionally identical to that of Janhunen (2004). In addition, the factor of v − u_w occurs in lieu of v, because prior studies were solely concerned with the regime where v ≪ u_w was valid. The mass per unit length for the sail material is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{s}}{{dz}}=\pi {{ \mathcal R }}_{s}^{2}{\rho }_{s},\end{eqnarray} \tag{ 46 }$$

where ${{ \mathcal R }}_{s}$ and ρ_s denote the radius and density of the wire that comprises the electric sail. In order to maintain the sail at a constant potential, an electron gun is required, but we will suppose that its mass is smaller than (or comparable to) the sail mass; this assumption is reasonably valid at large distances from the source (Lingam & Loeb 2020). The acceleration can be calculated by dividing (44) with (46).

There are, however, some major issues that arise even when it comes to analyzing the spherically symmetric case. First, the density profile does not always obey the canonical n ∝ r⁻² scaling; instead, it varies across jets, winds, or outflows associated with different astrophysical sources. For example, the classic Blandford–Payne model for winds from magnetized accretion disks obeys n ∝ r^−3/2 (Blandford & Payne 1982), whereas the outflows from Seyfert galaxies are characterized by n ∝ r^−α with α ≈ 1–1.5 (Bennert et al. 2006; Behar 2009). Second, the scaling of the temperature is also not invariant: the Blandford–Payne model yields a power-law exponent of −1, while the solar wind exhibits an exponent of roughly −0.5 near the Earth (Le Chat et al. 2011). Lastly, the velocity u_w is not independent of r in the regime of interest (namely, r ≳ d₀), although it eventually reaches an asymptotic value (denoted by ${u}_{\infty }\ne 0$) as per both observations and models (Parker 1958; Vlahakis & Tsinganos 1998; Beskin 2010).

Thus, this complexity stands in contrast to light sails, where the radiation flux falls off with distance as per the inverse square law. Hence, at first glimpse, it would appear very difficult to derive generic properties of electric sails. We will, however, show that a couple of useful identities can nonetheless be derived. First, we consider the limiting case of ${u}_{w}\approx {u}_{\infty }$ as it constitutes a reasonable assumption at large enough values of r. We will also introduce the scalings n ∝ r^−α and T_e ∝ r^−ξ and leave the exponents unfixed. To simplify our analysis, we employ the normalized variables $\tilde{v}\equiv v/{u}_{\infty }$ and $\tilde{r}\equiv r/{d}_{0}$. Using these relations along with (44)–(46), we arrive at

$$\begin{eqnarray}&&\tilde{a}\equiv \tilde{v}\displaystyle \frac{d\tilde{v}}{d\tilde{r}}\approx {{ \mathcal C }}_{E}{\left(\tilde{v}-1\right)}^{2}{\tilde{r}}^{-(\alpha +\xi )/2},\end{eqnarray} \tag{ 47 }$$

where ${{ \mathcal C }}_{E}$ is a dimensionless constant that encapsulates the material properties of the electric sail as well as certain astrophysical parameters (e.g., source luminosity). In formulating the above expression, we have neglected the gravitational acceleration and hydrodynamic drag for reasons elucidated in Section 2.1. After integrating this equation, we end up with

$$\begin{eqnarray}&&\mathrm{ln}(1-\tilde{v})+\displaystyle \frac{\tilde{v}}{1-\tilde{v}}\approx \displaystyle \frac{2{{ \mathcal C }}_{E}}{\alpha +\xi -2}\left(1-{\tilde{r}}^{-(\alpha +\xi -2)/2}\right)\end{eqnarray} \tag{ 48 }$$

after specifying $\tilde{v}=0$ at $\tilde{r}=1$.

Due to the uncertainty surrounding ${{ \mathcal C }}_{E}$, α, and ξ, we have plotted the normalized acceleration distance (given by $\tilde{r}-1$) as a function of the final speed for various choices of these parameters in Figure 4. The right-hand panel, which satisfies the criterion α + ξ < 2, yields results that are consistent with intuition. As the sail speed approaches ${u}_{\infty }$, the acceleration distance diverges. On the other hand, the left-hand panel exhibits slightly unusual behavior that is dependent on ${{ \mathcal C }}_{E}$. At large values of ${{ \mathcal C }}_{E}$, we observe that the acceleration distance diverges in the limit of $\tilde{v}\to 1$ as before. However, when we have ${{ \mathcal C }}_{E}\lesssim 1$, we noticed that the acceleration distance becomes singular at sail speeds that are conspicuously smaller than ${u}_{\infty }$. In other words, this implies that one cannot reach speeds close to ${u}_{\infty }$, irrespective of the distance traveled by the spacecraft. We will not estimate the acceleration time because reducing (48) to quadrature is not straightforward to accomplish.

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Next, we shall formalize the above results by carrying out a mathematical analysis of (48) for two distinct cases. The first scenario corresponds to α + ξ ≤ 2, and applying this limit to (48) yields $\tilde{v}\to 1$. In other words, we end up with ${v}_{\infty }\approx {u}_{\infty }$ in this regime, which was also proposed in Janhunen (2004). However, for a number of astrophysical systems (e.g., stellar winds) as well as classic theoretical models such as Blandford & Payne (1982), we must address the case with α + ξ > 2. By taking the limit of $\tilde{r}\to \infty$, the solution of (48) is

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\infty }}{{u}_{\infty }}\approx 1+{\left[W\left(-\displaystyle \frac{1}{e}\exp (-{\rm{\Upsilon }}\right)\right]}^{-1},\end{eqnarray} \tag{ 49 }$$

where W(x) is the Lambert W function (Corless et al. 1996; Valluri et al. 2000), and we have introduced the auxiliary variable ${\rm{\Upsilon }}=2{{ \mathcal C }}_{E}/(\alpha +\xi -2)$. Before analyzing (49) in detail, it is important to recognize a subtle point. By inspecting (48), we see that $0\leqslant \tilde{v}\leqslant 1$ because $\tilde{v}\gt 1$ would lead to the logarithmic function giving rise to nonreal (i.e., complex) values. In other words, to ensure the existence of physically consistent solutions, we require ${v}_{\infty }/{u}_{\infty }\leqslant 1$ to be valid; in turn, this inequality states that the upper bound on ${v}_{\infty }$ is the terminal wind speed.

Depending on the magnitude of ${{ \mathcal C }}_{E}$ (and therefore ϒ), there are two different regimes that require explication. First, let us consider the physically relevant scenario where ${{ \mathcal C }}_{E}\gg 1$ holds true, which is potentially applicable to astrophysical sources with high luminosities. As this choice is essentially equivalent to taking the limit ϒ ≫ 1, employing the latter yields

$$\begin{eqnarray}&&{v}_{\infty }\approx \left[\displaystyle \frac{{\rm{\Upsilon }}+\mathrm{ln}\left({\rm{\Upsilon }}+1\right)}{{\rm{\Upsilon }}+1+\mathrm{ln}\left({\rm{\Upsilon }}+1\right)}\right]{u}_{\infty },\end{eqnarray} \tag{ 50 }$$

which reduces further to ${v}_{\infty }\approx {u}_{\infty }$ when ${\rm{\Upsilon }}\to \infty$. Next, suppose that we consider the opposite case wherein ${{ \mathcal C }}_{E}\ll 1$. As this limit is tantamount to working with ϒ ≪ 1, applying standard asymptotic techniques for the Lambert W function near the branch point (de Bruijn 1958; Corless et al. 1996) leads to

$$\begin{eqnarray}&&{v}_{\infty }\approx \left[\sqrt{2{\rm{\Upsilon }}}-\displaystyle \frac{4{\rm{\Upsilon }}}{3}\right]{u}_{\infty },\end{eqnarray} \tag{ 51 }$$

and substituting ${\rm{\Upsilon }}\to 0$ implies that ${v}_{\infty }\to 0$.

In summary, we found that choosing α + ξ ≤ 2 gave rise to ${v}_{\infty }={u}_{\infty }$. On the other hand, for the physically pertinent case of α + ξ > 2 and ${{ \mathcal C }}_{E}\gg 1$, we approximately arrived at the same result; this is evident upon inspecting (50). Hence, without much loss of generality, it is safe to assume that the terminal speed of an electric sail for a given astrophysical object is set by the asymptotic value of the wind speed. In principle, one could also analyze the acceleration time and distance along the lines of Section 2.3 and assess the constraints set by the source environment and the ISM.¹⁴ However, we refrain from undertaking this study for two reasons: (1) many of the parameters as well as the scalings are nonuniversal and poorly determined, and (2) the equation of motion is much more complicated, as seen from (48), which makes subsequent analysis difficult.

For the aforementioned reasons, we shall confine ourselves to listing the observed values of ${u}_{\infty }$ for various astrophysical systems. It is natural to commence our discussion with stellar winds. By inspecting (25), it is apparent that ${u}_{\infty }$ only varies by a factor of ∼3 even when M_⋆ is increased by two orders of magnitude. Hence, insofar as stellar winds are concerned, the terminal wind speeds are on the order of 10⁻³c in most cases; note that this statement also holds true for low-mass stars such as M-dwarfs (Dong et al. 2017a, 2017b, 2018; Lingam & Loeb 2019a). Next, we consider SNe because the ejecta expelled during the explosion move at speeds of ∼0.1c, as noted in Section 2.5. Hence, this could serve as a rough measure of the final speeds attainable by electric sails in such environments.

In the case of AGNs, there are two phenomena that need to be handled separately. The first are diffuse outflows that are characterized by ${u}_{\infty }\lesssim 0.1c$ (Merritt 2013, Equation (2.44)). These outflows have been identified in most quasars through the detection of broad absorption lines at ultraviolet wavelengths (Murray et al. 1995; Gibson et al. 2009; Tombesi et al. 2013; King & Pounds 2015). In contrast, relativistic jets from AGNs (i.e., blazars) typically exhibit Lorentz factors of ${ \mathcal O }(10)$ (Padovani & Urry 1992; Worrall 2009; Blandford et al. 2019); it is suspected that the observed jet emission is powered by magnetic reconnection (Sironi et al. 2015). Hence, at least in principle, it is possible for electric sails to attain such speeds provided that the relationship ${v}_{\infty }\approx {u}_{\infty }$ is still preserved.¹⁵ The Lorentz factors for jets arising from microquasars are usually of order unity (Mirabel & Rodríguez 1999; Romero et al. 2017), suggesting that they also constitute promising sources for accelerating electric sails to relativistic speeds.

Pulsar wind nebulae (PWNe) will be the last example that we shall study here. PWNe comprise highly energetic winds that are powered by a rapidly rotating and highly magnetized neutron star (Gaensler & Slane 2006; Kargaltsev et al. 2015). The energy loss is caused by the magnetized wind emanating from the neutron star and is expressible as (Slane 2017, Equation (2))

$$\begin{eqnarray}&&\dot{E}=-\displaystyle \frac{{B}_{p}{R}_{p}^{6}{\omega }_{p}^{4}}{6{c}^{3}}{\sin }^{2}{\rm{\Theta }},\end{eqnarray} \tag{ 52 }$$

where B_p is the dipole magnetic field strength at the poles, R_p and ω_p are the radius and rotation rate of the pulsar, and Θ is the angle between the pulsar magnetic field and rotation axis. The minimum particle current ($\dot{N}$) that is necessary for the sustenance of a charge-filled magnetosphere is estimated using Gaensler & Slane (2006, Equation (10)), which equals

$$\begin{eqnarray}&&\dot{N}=\displaystyle \frac{{B}_{p}{R}_{p}^{3}{\omega }_{p}^{2}}{{ \mathcal Z }{ec}},\end{eqnarray} \tag{ 53 }$$

where ${ \mathcal Z }e$ represents the ion charge; this relationship was first determined by Goldreich & Julian (1969). The maximum Lorentz factor (γ_max) that is achievable in pulsar winds occurs near the termination shock, the location at which the ram pressure of the wind and the ambient pressure in the PWN balance each other, and has been estimated to be (Slane 2017, p. 2164)

$$\begin{eqnarray}&&{\gamma }_{\max }\approx 8.3\times {10}^{6}{\left(\displaystyle \frac{\dot{E}}{{10}^{31}{\rm{J}}}\right)}^{3/4}{\left(\displaystyle \frac{\dot{N}}{{10}^{40}{{\rm{s}}}^{-1}}\right)}^{-1/2}.\end{eqnarray} \tag{ 54 }$$

It is important to note, however, that γ is typically on the order of 100 just outside the light cylinder, which is defined as c/ω_p (Gaensler & Slane 2006, Section 4.4). The analysis of data from young PWNe in conjunction with spectral evolution models yielded bulk Lorentz factors of γ ∼ 10⁴–10⁵ for the pulsar winds (Tanaka & Takahara 2011, Table 2). It is worth noting that the characteristic synchrotron emission lifetime of particles in PWNe is ∼10³ yr (Slane 2017, Equation (10)). Most PWNe that have been detected are young (with ages of ∼10³ yr), but some PWNe discovered by the Suzaku X-ray satellite have ages of ∼10⁵ yr and are apparently still active (Bamba et al. 2010). Hence, the lifetime over which PWNe are functional may suffice to accelerate putative electric sails close to the bulk speeds of pulsar winds.

Lastly, another chief advantage associated with electric sails merits highlighting. Hitherto, we have seen that a variety of sources are capable of accelerating light sails or electric sails to relativistic speeds on the order of 0.1c. However, after the spacecraft has been launched toward the target planetary system, it will need to eventually slow down and attain speeds of order of tens of km s⁻¹ to take part in interplanetary maneuvers. Electric sails are a natural candidate for enforcing comparatively rapid slowdown through the process of momentum transfer with charged particles in the ISM. More specifically, Perakis & Hein (2016) concluded that spacecraft with total masses of ∼10⁴ kg could be slowed down from 0.05c to interplanetary speeds over decadal timescales by utilizing an electric sail.¹⁶