Deceleration of High-velocity Interstellar Photon Sails into Bound Orbits at α Centauri

We consider a solar sail of mass M and reflective surface area A approaching a star at an instantaneous distance ${\boldsymbol{r}}$ and with an instantaneous velocity ${\boldsymbol{v}}$ in an x–y plane (Figure 1). Its injection speed is ${v}_{\infty }$. The sail is subject to both the gravitational attractive force between it and the star and subject to the stellar radiation pressure, which, assuming a radially oriented sail and a uniformly bright, finite angular-sized stellar disk without limb darkening (McInnes & Brown 1990), is

$$\begin{eqnarray}&&P(r)=\displaystyle \frac{{L}_{\star }}{3\pi {{cR}}_{\star }^{2}}\left[1-{\left[1-{\left(\displaystyle \frac{{R}_{\star }}{r}\right)}^{2}\right]}^{3/2}\right],\end{eqnarray} \tag{ 1 }$$

where R_⋆ is the stellar radius and L_⋆ is the stellar luminosity. We assume that the magnitude of the photon force on the sail is proportional to the photon pressure times the cosine of the pitch angle α, hence

$$\begin{eqnarray}&&F(\alpha ,r)=\displaystyle \frac{{L}_{\star }A}{3\pi {{cR}}_{\star }^{2}}\left[1-{\left[1-{\left(\displaystyle \frac{{R}_{\star }}{r}\right)}^{2}\right]}^{3/2}\right]\times \cos (\alpha ).\end{eqnarray} \tag{ 2 }$$

We also explored the JPL numerical parametric force model (Wright 1992), where the reflectivity dependence on α is more complex, but these results agreed with those obtained via Equation (2) within 10 to 20% of ${v}_{\infty ,\max }$.

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Integration of Equation (2) with α = 0° over a sail path from r_min (the sail's minimum distance to the star) to $\infty$ yields the maximum possible reduction of kinetic energy from the photon pressure during a frontal approach:

$$\begin{eqnarray}{E}_{\mathrm{kin},{\rm{p}}} & = & {\displaystyle \int }_{{r}_{\min }}^{\infty }{drF}{(r)}_{| \alpha =0}\\ & = & \displaystyle \frac{{L}_{\star }A}{3\pi {{cR}}_{\star }^{2}}\ {\displaystyle \int }_{{r}_{\min }}^{\infty }{dr}\left[1-{\left[1-{\left(\displaystyle \frac{{R}_{\star }}{r}\right)}^{2}\right]}^{3/2}\right].\end{eqnarray} \tag{ 3 }$$

For a full stop, Equation (3) is reduced by the kinetic energy gained by the sail through the conversion of potential into kinetic energy. Hence, the maximum possible reduction of the sail velocity for a full stop is given by

$$\begin{eqnarray}&&{v}_{\mathrm{red}}=\sqrt{\displaystyle \frac{2{E}_{\mathrm{kin},{\rm{p}}}}{M}}-\sqrt{\displaystyle \frac{2{{GM}}_{\star }}{{r}_{\min }}}.\end{eqnarray} \tag{ 4 }$$

As an example, for an M = 1 gram sail with an area of A = 10 m² that passes α Cen A at r_min = 5 R_⋆ (assuming R_⋆ = 1.224 solar radii and L_⋆ = 1.522 L_⊙; L_⊙ being the solar luminosity), we obtain v_red ≈ 1200 km s⁻¹, which corresponds almost precisely to the value of ${v}_{\infty ,\max }$, achievable via a full stop trajectory as we found using our modified photogravitational trajectory code (see Section 2.3). The total maximum deceleration during a fly-by is slightly higher because the sail must climb out of the star's gravitational well after its passage, and therefore it slows down further.

We use a modified N-body code to model the gravitational pull by the stars under the additional effects of the stellar photon pressure onto the sail. As a key feature of our simulations, we solve the problem of continuously adjusting the sail's orientation to maximize the deceleration due to the photon pressure.³ Such an adaptive orientation could be achieved if the sail were able to modify its reflectivity across its surface, e.g., using technologies akin to the nanocrystal-in-glass approach (Llordes et al. 2013; Ma et al. 2016). An asymmetric reflectivity distribution would induce a torque on the sail, which would start to rotate it (Kislov 2004; Hu et al. 2016).

The (x, y) components of the photon force are calculated per

$$\begin{eqnarray}&&{\boldsymbol{F}}(\alpha ,r)=F(\alpha ,r)(\cos (\alpha +\varphi ),\sin (\alpha +\varphi )),\end{eqnarray} \tag{ 5 }$$

where $\varphi =\arctan (y/x)$ (see Figure 1). The maximum deceleration of the sail is achieved if the force component

$$\begin{eqnarray}&&{F}_{\nu }=F(\alpha ,r)\cos (\nu )\end{eqnarray} \tag{ 6 }$$

acting into the opposite direction of the sail's velocity vector is maximized, where

$$\begin{eqnarray}&&\nu =\arccos \left(\displaystyle \frac{{F}_{x}{v}_{x}+{F}_{y}{v}_{y}}{\sqrt{{F}_{x}^{2}+{F}_{y}^{2}}\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}}\right)\end{eqnarray} \tag{ 7 }$$

is the angle enclosed by ${\boldsymbol{F}}$ and ${\boldsymbol{v}}$. We enter Equation (7) into Equation (6) and numerically determine the value of α that yields maximum deceleration. The resulting trajectory is here referred to as a photogravitational assist, along which the photon pressure is used to reduce speed while both the photon pressure and the gravitational tug from the star determine the sail's deflection angle.

The orbits of the α Cen stellar triple system are non-coplanar, so we simulate each fly-by in a new x–y plane. For a broad range of ${v}_{\infty }$, we numerically integrate hundreds of sail trajectories, each of which has a slightly different horizontal offset (in steps of 0.1 R_⋆) in the x–y plane.

Figure 2(a) illustrates an example of such a trajectory iteration for an M = 1 gram, A = 10 m² sail approaching α Cen A with ${v}_{\infty }=1270\,\mathrm{km}\,{{\rm{s}}}^{-1}$. All trajectories with an initial x-offset ≲ 3 R_⋆ lead to a physical encounter of the sail with the star, i.e., the sail is lost (type I trajectory). For values of ${v}_{\infty }\lesssim 1200\,\mathrm{km}\,{{\rm{s}}}^{-1}$, the stellar photon pressure can stop the sail beyond 5 R_⋆, a distance that we use as a fiducial value to prevent the sail from destruction (see Section 4). We refer to such a trajectory as a full stop (type II). A full stop route allows the sail to reorient itself at r_min to then leave the stellar system against gravity into arbitrary directions by means of the photon pressure. A type III trajectory leads the sail into a bound elliptical orbit around the star, from where it could then depart in an arbitrary direction. Type IV trajectories are what we refer to as photogravitational fly-bys. Figure 2(b) shows a detailed example of the optimum photogravitational assist around α Cen A toward α Cen B with ${v}_{\infty }=1270\,\mathrm{km}\,{{\rm{s}}}^{-1}$, with the instantaneous speed of the sail shown along its trajectory and the instantaneous acceleration shown in the color bar at the right.

As an important feature of this trajectory, note that the photon pressure acts to initially accelerate the sail toward positive x values, that is, in the direction opposite to the intended deflection. The additional effect due to the gravitational force, however, can ultimately deflect the sail into very different directions. Somewhat larger deflection angles via photogravitational assists could be achieved if the trajectory were not determined according to the maximization of the instantaneous deceleration as per Equation (6). Although this would come at the price of less efficient deceleration, future simulations could show if such a trade-off could effectively increase the maximum possible injection speed.

Close stellar encounters necessarily invoke the risk of impacts of high-energy particles and of thermal overheating. On the one hand, impacts of high-energy particles could damage the physical structure of the sail, its science instruments, its communication systems, or its navigational capacities. On the other hand, if those impacts could be effectively absorbed by the sail, they could even help to decelerate it. As shown in Section (3), heating from the stellar thermal radiation will not have a major effect on a highly reflective sail. However, the electron temperature of the solar corona is >100,000 K at a distance of five solar radii. The Solar Probe Plus (planned to launch in mid-2018) is expected to withstand these conditions for tens of hours (Fox et al. 2015), although the shielding technology for an interstellar sail would need to be entirely different (Hoang et al. 2016), possibly integrated into the highly reflective surface covering.

The effects of special relativity can be neglected in our calculations, with the Lorentz factor being $1\leqslant \gamma \ ={(1-{v}^{2}/{c}^{2})}^{-1/2}\lt 1.005$ for v < 10% c. We also assumed that the stars are perfectly spherically symmetric. In reality, however, the acceleration of the probe will depend on the non-uniform mass distribution within the rotating star and possibly on effects from General Relativity (Kezerashvili & Vázquez-Poritz 2010). Stellar oblateness and limb darkening affect the photon pressure by <1% at distances >5 R_⋆ (McInnes & Brown 1990), so these effects do not change the results of this study significantly. Yet, they must be considered for the planning of real missions.

Regarding the nautical issues of an A-B-C trajectory, communication among sails within a fleet could support their navigation during stellar approach, as it will be challenging for an individual sail to perform parallel observations of both the approaching star and its subsequent target star or of other background stars. Course corrections will need to be calculated live on board. In particular, the location of r_min will need to be determined on-the-fly because it will depend on the actual velocity and approach trajectory, and hence on the local stellar radiation pressure and magnetic fields (Reiners & Basri 2008) along this trajectory.

Although we focused our simulations on the injection of light photon sails into bound orbits around Proxima, we also discovered that trajectory types II and III open up the opportunity for sample return missions to Earth (see Figure 2(a)). What is more, trajectory types II, III, and IV allow multi-fly-by missions at α Cen A, B, and C. Among those, type IV will generally deliver ${v}_{\infty ,\max }$.

Photogravitational assists can, of course, also be performed in the solar system. Once the technological implementation of a sail capable of photogravitational assists has been achieved, it seems natural to accelerate it to interstellar velocities using solar photons rather than using additional expensive technologies such as ground-based laser launch systems. Using Equation (4) and numerical integrations of Equation (3) from 5 solar radii to $\infty$, we estimate the maximum photonic ejection speed from the solar system to be about ${v}_{\odot ,\max }=\mathrm{11,500}\;\mathrm{km}{{\rm{s}}}^{-1}$ for an 8.6 × 10⁻⁴ gram m⁻² sail, implying travel times to α Cen of about 115 years. Beyond that, other nearby stars offer more favorable conditions than the α Cen triple for the deceleration of incoming photon sails. Sirius A, as an example, at just about twice the distance from the Sun as α Cen, offers a power of about 25 L_⊙ for deceleration. Consequently, the maximum possible injection speed of an 8.6 × 10⁻⁴ gram m⁻² sail that can be absorbed (44,600 km s⁻¹ or 14.9% c) exceeds v_⊙,max by far. Thus, a laser launch system or alternative technologies would be required to accelerate photon sails from the solar system to these maximum possible speeds, maybe in combination with acceleration by the solar photon pressure.

In multi-stellar systems, successive fly-bys at the system members can leverage the additive nature of photogravitational assists. For multiple assists to work, however, the stars need to be aligned within a few tens of degrees along the incoming sail trajectory of the sail. Such a successive braking is particularly interesting for multi-stellar systems, where bright stars can be used as photon bumpers to decelerate the sail into an orbit around a low-luminosity star, such as Proxima (0.0017L_⊙) in the α Cen system or the white dwarf Sirius B (0.056 L_⊙) around Sirius A.