Relativistic Light Sails

TL;DR

This paper develops new relativistic formulas for light sail acceleration, accounting for ensemble photon effects, and estimates the impact on spacecraft velocity, temperature, and laser efficiency for the Breakthrough Starshot project.

Contribution

It introduces a novel principle of ensemble equivalence for photon impacts on sails, improving velocity predictions at relativistic speeds.

Findings

Previous models underestimate terminal velocity by ~50 m/s.

Sail temperature and diffraction losses are critical for laser propulsion efficiency.

Designs must ensure sail absorption is below 1 in 260,000 photons for effective operation.

Abstract

One proposed method for spacecraft to reach nearby stars is by accelerating sails using either solar radiation pressure or directed energy. This idea constitutes the thesis behind the Breakthrough Starshot project, which aims to accelerate a gram-mass spacecraft up to one-fifth the speed of light towards Proxima Centauri. For such a case, the combination of the sail's low mass and relativistic velocity render previous treatments formally incorrect, including that of Einstein himself in his seminal 1905 paper introducing special relativity. To address this, we present formulae for a sail's acceleration, first in response to a single photon and then extended to an ensemble. We show how the sail's motion in response to an ensemble of incident photons is equivalent to that of a single photon of energy equal to that of the ensemble. We use this 'principle of ensemble equivalence' for both perfect and imperfect mirrors, enabling a simple analytic prediction of the sail's velocity curve. Using our results and adopting putative parameters for Starshot, we estimate that previous relativistic treatments underestimate the spacecraft's terminal velocity by ~50m/s for the same incident energy, sufficient to miss a target by several Earth radii. Additionally, we use a simple model to predict the sail's temperature and diffraction beam losses during the laser firing period, allowing us to estimate that for firing times of a few minutes and operating temperatures below 300C (573K), Starshot will require a sail of which absorbs less than 1 in 260,000 photons.

Equations

$$h{\nu }_i+\frac{m{c}^2}{\sqrt{1-{\beta }_i^2}}$$

$$\frac{h{\nu }_i}{c}+\frac{m{\beta }_{i}c}{\sqrt{1-{\beta }_i^2}}$$

$${\beta }_f$$

$${\nu }_f$$

$$r$$

$$1+z$$

$${\beta }_{i,z=0}$$

$$\underset{{\beta }_i\to 0}{\lim }{\beta }_f$$

$$r_{\mathrm{rel}}$$

$$\Delta E_K$$

$${ϵ}_K\equiv \frac{\Delta E_K}{h{\nu }_i}$$

$${ϵ}_K$$

$$\underset{r\to 0}{\lim }{ϵ}_K$$

$${\beta }_f-{\beta }_i$$

$${\beta }_n$$

$$n$$

$$E_{\mathrm{light}}$$

$${\nu }_f$$

$${\lambda }_f$$

$$\displaystyle=\lambda_{i}\Bigg{(}\frac{1-\beta_{i}}{1+\beta_{i}}\Bigg{)},$$

$$\tilde{t}$$

$${\beta }_f=$$

$$-\frac{(1+\sqrt{3}\mathrm{i})/2}{[{\kappa }^2(-5+2\kappa +6\tilde{t}(\kappa -4-6\tilde{t})){]}^{1/3}},$$

$$t_n$$

$$\beta [\mathrm{this}\mathrm{work}]-\beta [\mathrm{Lubin}2016]$$

$$h{\nu }_i+\frac{m{c}^2}{\sqrt{1-{\beta }_i^2}}$$

$$\frac{h{\nu }_i}{c}+\frac{m{\beta }_{i}c}{\sqrt{1-{\beta }_i^2}}$$

$$\mathcal{M}$$

$$-2r{\beta }_i\sqrt{1-{\beta }_i^2}-r^2(3-{\beta }_i)(1-{\beta }_i)(1+{\beta }_i)$$

Full text

relativistic processes — space vehicles

1 Introduction

One remarkable consequence of electromagnetism is that light carries finite momentum (Maxwell, 1865; Compton, 1923). Consequently, when light reflects off a surface, it imparts a small momentum kick to the surface leading to radiation pressure, an effect hypothesized about since at least the $17^{\mathrm{th}}$ century (Kepler, 1619). Since the early $20^{\mathrm{th}}$ century, it has been recognized that this effect could be utilized to propel spacecraft with large mirror-like sails harvesting the momentum of incident photons111Tsiolkovsky & Zander first discuss this possibility in 1925 as detailed in Zander (1964).. Whilst such sails were originally conceived with solar radiation in mind, the invention of lasers in the 1960s enables efficient laser sailing propulsion systems too (Marx, 1966; Redding, 1967; Forward, 1984).

Recently, the Breakthrough Starshot project (simply Starshot in what follows) announced plans to develop the technology needed for a laser sail nano-satellite capable of flying to the closest stars within a generation. A proposed configuration is to fire an Earth-based array of gigawatt (or greater) lasers onto a gram-mass, microchip-sized satellite which would be accelerated up to approximately one-fifth the speed of light, reaching Proxima Centauri in just over two decades (see Heller & Hippke 2017 for deceleration schemes).

Whilst a great deal of literature, experiments and even space flight demonstrations of solar sailing exist (Kawaguchi et al., 2008; Mori et al., 2009), Starshot is unique for two main reasons. First, the target speeds are relativistic, and thus classical expressions suitable in the context of solar sails become invalid. Second, Starshot is designed to be ultra-light, which means the mass of the sail cannot be assumed to be infinite, as is typically assumed in relativistic calculations of photon exchanges with a mirror (e.g. see Gjurchinovski 2013).

In this work, we first present a simple derivation of the relativistic velocity curve of a light sail in Section 2. We then extend our analysis to consider the effect of imperfect mirrors and subsequent thermal heating of the sail and spacecraft payload in Section 3. We finish with some key conclusions in Section 4 and highlight parts of the calculation requiring further work.

2 Sailing with a Perfect Mirror

2.1 A single photon

We begin by considering the simple case of a single photon of frequency $\nu_{i}$ fired at a normal incident angle towards a perfect mirror, or equivalently a light sail, of mass $m$ moving along the same vector as the photon at speed $c\beta_{i}$, as depicted in Figure 1. The reflection of the photon is assumed to be perfectly elastic, although we later relax this assumption in Section 3. The motion of the mirror and the frequency of the photon can be calculated by requiring the conservation of relativistic energy and momentum.

First, the system’s total energy (the sum of the photon’s energy and the mirror’s energy) must be conserved before and after the reflection. Using the relativistic expressions, one may write that

[TABLE]

Similarly, requiring that the system conserves momentum, we find

[TABLE]

Solving the Equations (1) & (2) simultaneously and simplifying, we find

[TABLE]

and

[TABLE]

where we have defined $r$ as the photon’s “relative energy” using

[TABLE]

Note, that our calculation has ignored the effect of the mirror’s gravity, which in principal imparts a small gravitational frequency shift which is negligible for gram-mass sails.

2.2 Redshift of the reflected photon

Equation (4) may also be expressed in terms of the redshift, $z$, of the reflected photon:

[TABLE]

which we plot in Figure 2 for several choices of $r$. Equation (6) reveals that the reflected photon will have a redshift of zero when

[TABLE]

where the negative sign indicates that the mirror is now coming towards the photon.

We also point out that Equation (6) and Figure 2 reveal that the photon becomes redshifted to infinity (i.e. redshifted out of existence) as $\beta_{i}\to 1$. This result implies that as the mirror moves closer to $c$, the transfer of the photon’s energy into the kinetic energy of the mirror becomes increasingly efficient. This result is verified later in Section 2.4.

2.3 Accelerating a mirror with a single photon

In the limit of $\beta_{i}\to 0$, in other words an initially stationary mirror, one may write that the mirror will be accelerated up to a speed of

[TABLE]

Solving the above for $\beta_{f}=\tfrac{1}{2}$ yields a characteristic relative photon energy, $r$, necessary to impart relativistic motion as

[TABLE]

To first order in $r$, Equation (8) is simply $2r$, which shows that $r\gtrsim\mathcal{O}[10^{-2}]$ to get to even a few percent the speed of light.

2.4 Efficiency

In the case of photon sailing, the primary goal of hitting the sail with photons is to propel a sail in the desired direction. Two useful figures of merit to consider in this context are the kinetic energy and speed of the sail in response to a photon reflection.

Consider first: what is the gain in kinetic energy of the mirror as a function of its initial velocity, $\beta_{i}$? The change in kinetic energy of the mirror is most easily expressed by equating it to the total energy lost by the photon:

[TABLE]

where on the second line we have re-expressed the kinetic energy gain in units of the incident photon’s energy, which can be considered to be the efficiency by which energy is transferred from the photon to the sail. Using Equation (4), we may now write that

[TABLE]

In the limit of $r\to 0$, where the photon’s energy is much less than the rest mass energy of the sail, we find that

[TABLE]

which is a monotonically increasing function from $\beta_{i}=0\to 1$. This result therefore demands that the fraction of the photon’s energy transferred to the sail as kinetic energy increases as $\beta_{i}$ increases. In this sense, the sail becomes more efficient once it has gained some initial momentum, verifying the argument based earlier in Section 2.2.

Consider now the velocity change of the sail as a function of $\beta_{i}$. In the classical framework, the speed ever-increases linearly into the super-luminal regime. At low velocities, one may easily show that our expression in Equation (3) reproduces the classical behavior; for example, in the limit of $r\ll 1$, the related expression Equation (8) simply gives $2r$, as expected. Therefore, the relative velocity change predicted by our formula should decrease at high speeds, in order to reproduce an asymptote towards $c$. One can verify this mathematically by writing

[TABLE]

which reproduces the correct behavior of a velocity change of $2r$ at low $\beta_{i}$ and zero velocity change as $\beta_{i}$ approaches unity.

2.5 Accelerating a mirror with an ensemble of photons

We now consider firing multiple photons at the sail/mirror in order to induce acceleration. In what follows, we ignore the effect of drag forces, such as interstellar dust and even photonic gas drag (Balasanyan & Mkrtchian, 2009). We treat each photon as striking the mirror consecutively, leading to a series of small impulses, each of which increases the velocity of the sail slightly.

We set the initial velocity to $\beta_{0}=0$ and then define $\beta_{n}$, the velocity after the $n^{\mathrm{th}}$ reflection, as $\beta_{f}$ from Equation (3), replacing $\beta_{i}\to\beta_{n-1}$. Writing out the first few terms and simplifying, one may show that $\beta_{n}$ may be written as

[TABLE]

Comparing this expression to $\beta_{f}$ in Equation (3), one can see that

[TABLE]

In other words, the final speed of the light sail after $n$ reflections of identical photons of relative energy $r$ is equal to that expected due to the reflection of single photon of energy $nr$. We refer to this as the principle of ensemble equivalance in the remainder of this work.

In Figure 3, we compare the velocity curve predicted by Equation (15) versus that computed numerically for $10^{6}$ reflections for large choices of $r$. These experiments find the formulae are correct to within machine precision, thereby providing a simple formula to predict the velocity curves of relativistic sails.

By re-arranging Equation (14) to make $n$ the subject, we are able to write down a simple formula for the number of photons needed to accelerate a sail up to a target relativistic speed, $\beta_{\mathrm{targ}}$:

[TABLE]

or, equivalently, that the total light energy needed to strike the mirror is

[TABLE]

As a practical example, we plot the velocity curve of a Starshot-like sail ($m=1$ g) accelerating up to $0.2c$ using our relativistic formula in comparison to the non-relativistic case in Figure 4.

2.6 Previous literature & why Einstein’s formalism is erroneous for Starshot

It is instructive to compare our results to those of the pre-exisiting literature. Our solution calculates two distinct quantities: the redshift of a single photon after reflection (Equation 4) and the resulting velocity change of the sail (Equation 3), which forms the basis to scale up to an ensemble of photons (Equation 14).

A photon’s redshift off a relativistic mirror is a classic problem which has been studied by many previous authors, including Einstein himself in his historic paper introducing special relativity (Einstein, 1905). The corresponding velocity change of the mirror is less commonly derived, although our derivation finds that the solutions must come as a pair. Of course for beamed laser sailing, it is this velocity change which is of greatest interest. Before comparing our velocity predictions to the literature, we first consider the redshift result, due to the rich literature of comparisons at our disposal.

We first compare to Gjurchinovski (2013) who provide a pedagogical derivation of a photon incident upon a relativistic mirror at an angle $\alpha$ but under the explicit assumption of an infinitely heavy mirror ($m\to\infty$). By conserving energy and momentum, Gjurchinovski (2013) obtain

[TABLE]

As expected, the above is equivalent to our Equation (4) in the case of a normal incident photon ($\alpha=0$), as was assumed in our work, and the limit of $r\to 0$ (which is equivalent to Gjurchinovski’s assumption of $m\to\infty$).

Another insightful example to compare to (where it cannot be assumed that $m\to 0$) is for Compton scattering, which is essentially the same problem but where the mirror is replaced with an electron. For an electron initially at rest ($\beta_{i}=0$), the photon’s frequency is shifted to (Equation 7.2 of Rybicki & Lightman 1979):

[TABLE]

where $\theta$ is the scattering angle equal to $\pi$ for an exact reflection back along the original path - as adopted in our work. As expected, Equation (19) is indeed equivalent to our result in Equation (4) for $\beta_{i}=0$ and $\theta=\pi$.

Having established the validity of our redshift formula, we now compare it to that being used in the literature of light sails. Of most relevance is the result curated in the “Roadmap to Interstellar Flight”, a comprehensive review by Lubin (2016), which ultimately inspired the Breakthrough Starshot project (Popkin, 2017). Lubin (2016) report that their relativistic solutions come from Kulkarni et al. (2016), who in Equation (1) have

[TABLE]

which is equivalent to Gjurchinovski (2013) for $\alpha=0$ and also to our Equation (4) in the limit of $r\to 0$. Therefore, although it is not explicitly stated in Kulkarni et al. (2016), the authors appear to have tacitly adopted the infinite mass sail approximation222 In a subsequent paper by Kulkarni et al. (2017), it is explicitly verified that this is indeed an assumption made in their derivation (see Section 2 of that work). .

This can be verified by following the description of their derivation, which unlike this work and Gjurchinovski (2013) uses Lorentz frame transfers rather than balancing conserved quantities. Specifically, the authors first shift the photon to the sail’s frame, then assume it “is emitted with the same wavelength as it is incident with”, before finally transferring back to the original frame. Crucially, this is also the same tacit assumption made by Einstein himself in Einstein (1905), who in Section 8 of that work adopt same derivation procedure of frame transfers, and use the same intermediate step in the sail’s rest frame of $\nu^{\prime\prime}=\nu^{\prime}$ in Einstein’s original notation (which indicates that the reflected light’s frequency equals the incident light’s frequency when viewed in the sail’s frame).

It is with some trepidation that we argue here that Einstein, and indeed all subsequent authors adopting this assumption (e.g. Gjurchinovski 2013; Lightman et al. 1975; Galli & Amiri 2012), must be formally wrong. For a sail (or mirror) at rest, the reflected photon cannot have the same frequency as the incident photon without violating the conservation of energy. The photon has reversed momentum and so the mirror must increase its absolute momentum (from initially zero, since it is defined to be at rest) to conserve total momentum. Since the mirror is now moving, its kinetic energy must have also increased. Therefore, to conserve the total energy of the system, the photon has to lose energy which it can only do so by decreasing in frequency. Ergo, Einstein’s assumption that $\nu^{\prime\prime}=\nu^{\prime}$ violates the conservation of energy (note that this can also be seen by comparison to Compton scattering where this general statement is false; Rybicki & Lightman 1979).

Another way to think about the above is to assume Einstein is correct and that $\nu^{\prime\prime}=\nu^{\prime}$ and then look at the consequences. The equality $\nu^{\prime\prime}=\nu^{\prime}$ means that the photon has lost no energy when reflecting off a mirror at rest (since $E=h\nu$). If this is true, then by conservation of energy, the mirror cannot have gained any kinetic energy. In other words, the mirror does not move. This simple point demonstrates the falsehood of $\nu^{\prime\prime}=\nu^{\prime}$, since it requies that no matter how many photons are incident upon a mirror initially at rest, it will never move. In other words, $\nu^{\prime\prime}=\nu^{\prime}$ would make the entire concept of light sailing impossible, since objects could never be accelerated away from being initially at rest.

Although formally wrong, one might argue that practically speaking this infinite mass mirror assumption is always extremely well justified. In other words, one might reasonably posit that whether this assumption is imposed or not, the resulting predictions will be nearly identical. Remarkably, this appears to be false. Consider the other half of the solution now, the corresponding velocity change of the mirror in response to an ensemble of photons (which we state in Equation 14). This solution does not appear in Einstein (1905) but is derived in Kulkarni et al. (2016), who, recall used the same derivation framework for $\nu$ as Einstein.

Kulkarni et al. (2016) relate the relativistic velocity of a perfect sail in response to a constant beam of power $P$ fully on the sail for a time $t$ as

[TABLE]

where we use $\tilde{t}\equiv(Pt)/(mc^{2})$. Although it was not stated in Lubin (2016) or Kulkarni et al. (2016), we may re-arrange Equation (21) to solve for $\beta$, which leads to a cubic equation with one real root of

[TABLE]

where $\kappa\equiv\sqrt{5+12\tilde{t}(2+3\tilde{t})}$. Our work does not strictly assume a constant laser illumination, which we would argue is an advantage of our prescription, but it can be modified to such a form as follows. In the original version of this manuscript, we accomplished this by taking Equation (14) and replacing $nr=E_{\mathrm{inc}}/(mc^{2})=(Pt)/(mc^{2})=\tilde{t}$. In a reply that version, Kulkarni et al. (2017) correctly point out that this does not account for the time delay for light to reach the sail, leading to an unfair comparison of the two formulae and we correct for this here. One may show that the time of the $n^{\mathrm{th}}$ photon reflection on the sail, accounting for time delays, is given by

[TABLE]

where $t_{0}$ is a reference time and $\delta t$ represents the time between each photon emission (assuming a uniform rate i.e. constant power). Assuming that the sail begins from rest at time $t_{0}$, we may use Equation (14) to write that

[TABLE]

We may now re-arrange the above to make $n$ the subject and replace the real root of the resulting cubic into Equation (14) to give us a formulae for $\beta$ as a function of time under constant laser power. Our formulae, written as a function of time, is compared directly to that of Lubin (2016) and Kulkarni et al. (2016) in the right panel of Figure 4. Although the two equations show close agreement for the fiducial choice of parameters in Figure 4, they are not equivalent - as evident from the residual plot in that figure. Specifically, our formula predicts a slightly faster acceleration, due to the additional recoil accounted for by the photon reflections ignored in the Lubin (2016) formalism.

Although both formulae are fairly unwieldy when expressed in terms of time, we can take the difference between them (the residuals) and perform a series expansion in $r$. This leads to the following expression for the difference between the two

[TABLE]

For a target speed of $0.2c$, this corresponds to a difference of 47 m/s, which would change the arrival time at Proxima Centauri b by 8.7 minutes. Given planet b’s orbital velocity, this would cause the planet to be in a different location by 25,000 km, or around four planetary radii. Although the difference is certainly small, we highlight several key advantages of this work’s formalism of that of Lubin (2016) and Kulkarni et al. (2016):

• •

The formalism of Lubin (2016) and Kulkarni et al. (2016) is predicated on the assumption of no photon recoil on the sail, which technically makes it impossible to ever accelerate the sail from rest.

• •

It is not necessary to assume constant power on the sail with our formalism, any temporal profile (for example one accounting for diffraction) can be employed.

• •

If the arrival position of a relativistic sail to a nearby star needs to be predicted to a precision of several Earth radii or better (for example if attempting a fly-by manoeuvre), then our formula would be favored.

3 Imperfect Sails in Thermal Equilibrium

3.1 Overview

Throughout Section 2, we have explicitly assumed a perfect mirror, one with a reflection coefficient of unity. In such a case, the sail is maximally efficient and thermally stable, absorbing no photons as thermal energy. Accordingly, the time frame over which one fires the photons at the sail is inconsequential, and in principle, the sail can receive the full jolt of energy in a single laser pulse. In practice, even slight imperfections in the reflectivity will both degrade the rate of acceleration and lead to the sail absorbing thermal photons, potentially leading to a catastrophic failure of the sail and/or electronic payload. We here provide a simple derivation of the magnitude of these effects, starting again from the case of a single photon.

3.2 Inelastic photon collisions

We begin by considering a single photon which makes an inelastic collision with the mirror. The picture is therefore the similar to that depicted in Figure 1, except the final photon is not reflected but absorbed into the mirror, slightly increasing the rest mass energy of the mirror. As before, we proceed by balancing the energy

[TABLE]

and momentum

[TABLE]

in the system, which may be solved for $\beta_{f}$ and $\mathcal{M}$, where $\mathcal{M}$ is the relative increase in the rest mass energy of the sail, giving

[TABLE]

and

[TABLE]

Note that we now distinguish between the mirror’s velocity from an absorbed versus reflected photon using the superscripts “abs” and “ref”, respectively. Accordingly, comparing Equations (3) & (29), we can verify the classical result that

[TABLE]

which states that a reflected photon imparts twice the momentum as an absorbed photon (which can be seen to not hold in the relativistic regime).

Consider a sail that is accelerated to relativistic speeds exclusively by absorbed photons, but maintained a constant temperature via thermal equilibrium. This means that although the mirror’s rest mass temporarily increases after the absorption, it immediately re-radiates this excess energy isotropically, thereby returning to a rest mass $m$. Since isotropic re-radiation of the sail does not affect its velocity (else anything moving and at non-zero temperature would feel a constant drag/acceleration force), we may use the principle of ensemble equivalnce used earlier in Section 2.5 to show that the velocity curve is

[TABLE]

3.3 Accounting for reflectivity

We now need to combine the two cases, reflection and absorption, into a single model described by a reflection coefficient, $R$. In what follows, we define $R$ as being the fraction of incident photon power which is reflected elastically by the mirror, with the remaining fraction $\bar{R}=(1-R)$ being absorbed inelastically. For simplicity, we will also assume that the reflection coefficient is achromatic.

We first point out that trying to derive this formula in the case of a single photon appears intractable, on the basis that we have two conserved quantities (energy and momentum) but three unknowns (final mirror velocity, final frequency of the photon, and final rest mass of the mirror).

In order to make progress, we adopt the following approximate model. We assume that a single photon can be split into two components, one of energy $Rh\nu_{i}$ which reflects off the sail, and the other of energy $\bar{R}h\nu_{i}$ which is absorbed. Let’s assume the elastic collision occurs first, followed by the inelastic collision; in each independent collision, we can analytically solve the final state of the system. In the time between this “pair” of photons and the next, we assume that the sail re-radiates the excess absorbed energy, i.e. it is in thermal equilibrium. Using this model, we can combine the results found earlier in Section 2.1 & 3.2 to write that, for an initial velocity of $\beta_{0}=0$, the speed after the $n=1^{\mathrm{st}}$ incident photon is

[TABLE]

where the superscript “mix” on the left-hand side denotes that this velocity change is a mixture model of both elastic and inelastic components. Using our principle of ensemble equivalence (i.e. that a series of photon impacts is equivalent to one cumulative energetic photon collision), we may write that

[TABLE]

As expected, Equation (33) can be easily demonstrated to reproduce Equation (14) in the limit of $R\to 1$ and Equation (31) in the limit of $R\to 0$.

3.4 Numerical verification

Equation (33) is derived by assuming that each photon can treated as a pair of dummy photons. We test here the validity of this assumption through numerical simulations.

In each simulation, we consider firing $10^{5}$ incident photons at a mirror where the photon has a probability $R$ of being an elastic photon and $\bar{R}$ of being inelastic333Practically speaking, we simply generate a pseudo-random number between 0 and 1 and compare it to these probabilities at each iteration. Starting from rest, we numerically compute the velocity curve of the mirror using Equation (3) for elastic collisions and Equation (29) for those which are inelastic. After each inelastic collision, we assume the mirror re-radiates the absorbed energy isotropically before the next photon arrives (i.e. thermal equilibrium), such that the rest mass of the mirror does not evolve.

Since the simulations are intrinsically stochastic via the reflection probabilities, we repeat each simulation 1000 times and take the mean. Because we have assumed a small number of incident photons (just $10^{5}$), we use several large choices of $r=10^{-6}$, $10^{-5}$ & $10^{-4}$ in order to accelerate the mirror to relativistic speeds. We set the reflectivity to $R=0.9$, representing a fairly poorly optimized sail. Comparing to the predictions of Equation (33), we estimate the expression is accurate to within 0.04% for all reflectivities $R>0.9$.

3.5 Velocity losses due to non-unity reflectivities

We may now compare the velocity curve predicted by Equation (33) to that of a perfect mirror in Equation (14), in order to quantify the losses due to non-unity reflectivities:

[TABLE]

which reduces to the classical result of

[TABLE]

Since $nr$ dictates the final velocity of the mirror, we may replace $nr$ with the target velocity, $\beta_{\mathrm{targ}}$, and Taylor expand to first order in $\bar{R}$ to yield

[TABLE]

Using the cumulative energy needed to accelerate to a perfect sail to $0.2c$, the final speed of the sail is reduced by 4.4% for a 90% reflectivity and 0.044% for a 99.9% reflectivity. We therefore conclude that velocity losses due to imperfect reflectivities are fairly modest and unlikely to be a limiting design constraint on the sail.

3.6 Energy and thermal requirements

We also consider here the energy which is absorbed by the sail thermally. One may re-arrange Equation (33) to write that the cumulative number of photons needed to accelerate a sail up to a target velocity $\beta_{\mathrm{targ}}$ is given by

[TABLE]

where $r_{\mathrm{tot}}=nr$. The above can also be expressed as an energy given by

[TABLE]

Note that this is the energy incident upon the sail and does not account for beam losses due to diffraction or scattering between the laser source and the sail. In total, we assume that the sail has absorbed a fraction $\bar{R}$ of this energy as thermal photons over a time $t$. Time $t$ corresponds to the time that the photon is actually received, not when it is emitted, due to light travel time. Time $t$ will always exceed the emission duration but the ratio is extremely close to unity at the start of the acceleration (e.g. see Equation 24), leading to the greatest thermal stress on the sail. Since this regime sets the design constraints on the sail, the time lag is unimportant for this purpose. We also highlight that for sails with finite transmittance, the prescription given here can be easily modified by attenuating the incident energy accordingly.

In the sail’s reference frame, the incident energy is received over a dilated of time $t^{\prime}$. Assuming a constant acceleration (or force) applied to the sail initially at rest, the time dilation factor is (Iorio, 2005)

[TABLE]

For $\beta_{\mathrm{targ}}=0.2$, this time dilation factor is less than a percent and thus practically speaking one may simply assume $t^{\prime}\simeq t$.

As was done earlier, we assume that the sail immediately re-radiates the absorbed energy isotropically. For the sake of simplicity, we assume that the sail emits this thermal energy as a blackbody over the laser firing time of $t$, such that the total energy emitted by the sail is $2\mathcal{A}t\sigma_{B}T^{4}$, where $\mathcal{A}$ is the area of the sail. Note that the sail’s area is not length contracted since it is normal to the direction of motion. Equating the received and emitted powers and then solving for the sail’s temperature, $T$, we have

[TABLE]

where $\Sigma$ is the effective444We use the term “effective” because the rest mass includes the payload surface density of the sail, given by $\Sigma\equiv m/\mathcal{A}$. Note that Equation (40) refers to the temperature of the sail in the Earth’s frame of reference, not in the sail’s frame of reference which we ultimately require. Einstein (1907) and Planck (1908) argue that temperature is covariant, given by $T^{\prime}=T/\gamma$ (where $\gamma=1/\sqrt{1-\beta^{2}}$), but Ott (1963) later challenged this, obtaining the result $T^{\prime}=T\gamma$. Later, Landsberg (1966, 1967) argue that thermodynamic quantities like entropy and temperature should not vary between two reference frames and we adopt this result in our work here too555These disagreements provide an interesting opportunity for experiment onboard Starshot, i.e. $T^{\prime}=T$.

In order to proceed, we assign some parameters appropriate for the Starshot proposal. We choose optimistic but plausible values for the spacecraft mass of $m=1$ g and a sail of area $\mathcal{A}=16$ m2 and assume $\beta_{\mathrm{targ}}=0.2$. The firing time is varied between several options. We plot the resulting temperature of the spacecraft, given by Equation (40), as a function of aborptivity in Figure 5. As noted earlier, these temperatures should be treated as the temperature which the sail rises to during the initial phases of acceleration, but finite light travel time will lead to a cooling effect at later times.

As an example, for $\bar{R}=10^{-5}$ aborptivity, which is plausible with optically coated materials (Rempe et al., 1992), temperatures below $300^{\circ}$C (typical of a high-temperature microsystem; Lien et al. 2011; Chiamori et al. 2014) can be maintained over an 8.6 minute firing period.

Such a case would require just over 10 TJ of incident energy on the sail, or a constant power of 19.6 GW, giving an average flux on the sail of 1.2 GW m*-2* for the adopted 16 m2 area.

3.7 Diffraction Losses

The rapid acceleration of the sail causes it to quickly traverse great distances which poses at least two challenges for the laser system. First, at great distances it may be difficult to maintain accurate pointing on the sail, particularly if atmospheric turbulence introduces small refractive deviations to the optical path. Second, even if perfect pointing is maintained, diffraction of the laser light can introduce significant losses of the beam energy by the time it reaches the target. We tackle this second issue in what follows and assume a stable sail riding the beam throughout (see Manchester & Loeb 2017 for details on this point).

Consider a transmitter of diameter $D_{T}$ producing a laser of wavelength $\lambda$, which strikes its target at a distance of $L$ away from the source. For a diffraction-limited beam, the beam width at distance $L$ will be (Kipping & Teachey, 2016)

[TABLE]

where we have assumed that the final beam width has diffracted to be much greater than the initial width.

For simplicity, we consider a sail which is circular in projection and a Gaussian beam profile. At a distance $L$ then, the integrated fraction of the laser power striking the sail will be

[TABLE]

The above can now be evaluated by replacing $L$ with the corresponding distance expected at some target velocity, $\beta_{\mathrm{targ}}$. To make analytic progress, we will assume that the sail undergoes strictly uniform but relativistic acceleration. Accordingly, the distance the sail has traversed

[TABLE]

where $a_{\mathrm{const.}}$ is the constant acceleration of the sail as observed in the laser’s reference frame, given by $c\beta_{\mathrm{targ}}/t$. Using the above, the fractional power striking the sail is now

[TABLE]

One may now replace $t$ in the above with Equation (40) to relate the power loss at a given target velocity as function of the basic sail properties. Before doing so, we present a quick an order-of-magnitude calculation by using $L\simeq\tfrac{1}{2}\beta_{\mathrm{targ}}ct$ (non-relativistic) to give $L\simeq 2.9$ AU. For a $D_{T}=10$ m transmitter at 650 nm wavelength, the beam width will be $W_{L}\simeq 40$ km at a distance $L$ and thus we should expect $(16/40000)^{2}\sim 10^{-7}$ fractional power striking the sail. Using the full equation, we obtain similar results, as depicted in Figure 6 for four possible choices of $D_{T}$.

Our results imply that any firing time of order hours or greater will lead to very large beam losses of at least a million, increasing the energy demands to ten exa-joules or more. To keep energy losses within a factor of 10, a kilometer sized transmitter could fire for 199 seconds at a sail with an absorptivity satisfying $\bar{R}<3.9\times 10^{-6}$. These calculations argue that key technical requirements for Starshot are a $\bar{R}=10^{-6}$ sail, kilometer-sized lasers achieving 500 GW power over a firing time of a few minutes.

4 Discussion

We have derived an equation for the velocity change of a relativistic moving mirror (or equivalently a light sail) in response to an ensemble of normal incident photons, as well as the corresponding redshift of the reflected photons. Whilst our formulae for the cases of a perfectly reflective and perfectly absorptive mirror are exact, our formula for a mirror with a reflectivity in the range $0<R<1$ should be treated as an excellent approximation rather than being formally true, and we suggest that the solution may in fact be intractable without approximation.

Crucially, our expression for the velocity curve differs from that stated in Lubin (2016), which motivates the Starshot project (Popkin, 2017). The Lubin (2016) result is derived in Kulkarni et al. (2016), who use Lorentz frame transfers and assume that in the sail’s rest frame the frequency of reflection equals that of incidence. We have shown that this assumption, also made by Einstein in his seminal 1905 paper introducing relativity, violates the conservation of energy since the sail must increase it’s momentum (and thus kinetic energy) in response to the reflection and thus the photons must lose energy by becoming redshifted. Since this treatement overestimates the photon’s final energy, it also underestimates the sail’s velocity. For Starshot-like parameters, the difference is small, corresponding to a difference of $\sim 50$m/s in the predicted speed of the sail. If the arrival time of Starshot needs to be predicted to a precision of minutes, or equivalently the arrival position to within a few Earth radii, then our formula should be favored over the formalism of Lubin (2016) and Kulkarni et al. (2016).

Our equations provide an analytic framework to predict the acceleration of a light sail under solar or laser irradiation up to relativistic speeds, as appropriate for the Starshot project for example.

A useful result from our work is that the relativistic velocity curve from a large number of incident photons can be described analytically as that of a single photon with the equivalent energy of the ensemble, for either elastic or inelastic collisions. This insight, which we have used several times and referred to as the principle for ensemble equivalance for convenience, is demonstrated in Section 2.5 and provides a simple analytic approach for modeling sail response functions.

Additionally, we have discussed how the high levels of incident radiation on the sail, necessary to achieve relativistic speeds, will put thermal stress on the sail and payload. Practically speaking, the ideal sail material should be ultra-light, rigid against the radiation pressure inhomogeneities, thermally stable up to hundreds of Kelvin and ultra-reflective. To avoid losing more than a factor of ten of the laser power through diffraction losses, we find that a kilometer-sized transmitter needs to fire for 3.3 minutes or less, excerbating the thermal stress on the sail. For such a case, we estimate that the absorptivity needs to less than $4\times 10^{-6}$ and be able to operate at $300^{\circ}$ C (573 K).

There are numerous effects we have ignored which will further influence the design requirements for Starshot. For example, additional beam losses due to scattering through the Earth’s atmosphere will certainly lead to a higher laser power output requirement than that estimated here. In terms of the sail itself, at least three effects we have ignored will influence the sail’s velocity. First, drag forces from interstellar dust and even photonic gas (Balasanyan & Mkrtchian, 2009) will act to slowly decelerate the sail. Second, we have assumed that the reflection coefficient is achromatic, but in reality, man-made highly reflective surfaces, such as dielectric mirrors, are often extremely sensitive to wavelength (Rempe et al., 1992). Third, we have assumed that the sail and spacecraft chassis are in thermal equilibrium from the first incident photon to the last, whereas in reality some of this energy will not be re-radiated but used to warm up the chassis, potentially leading to material deformations, for example. We highlight these problems to the community for future work.

D.M.K. thanks members of the Cool Worlds Lab for stimulating conversations on this topic. Thanks to Zoltan Haiman, Jules Halpern & Emily Sandford for their helpful comments on early drafts. Special thanks to the anonymous reviewer for encouraging detailed literature comparisons.