Relativistic Generalization of the Incentive Trap of Interstellar Travel with Application to Breakthrough Starshot

As new concepts of sending interstellar spacecraft to the nearest stars are now being investigated by various research teams, crucial questions about the timing of such a vast financial and labor investment arise. If humanity could build high-speed interstellar lightsails and reach the alpha Centauri system 20 yr after launch, would it be better to wait a few years, then take advantage of further technology improvements to increase the speed, and arrive earlier despite waiting? The risk of being overtaken by a future, faster probe has been described earlier as the incentive trap. Based on 211 yr of historical data, we find that the speed growth of human-made vehicles, from steam-driven locomotives to Voyager 1, is much faster than previously believed, about 4.72 % annually or a doubling every 15 yr. We derive the mathematical framework to calculate the minimum of the wait time (t) plus travel time (tau(t)) and extend two exponential growth law models into the relativistic regime. We show that the minimum of t+tau(t) disappears for nearby targets. There is no use of waiting for speed improvements once we can reach an object within about 20 yr of travel, irrespective of the actual speed. In terms of speed, the t+tau(t) minimum for a travel to alpha Centauri will occur once 19.6 % the speed of light (c) become available, in agreement with the 20 % c proposed by the Breakthrough Starshot Initiative. If interstellar travel at 20 % c can be obtained within 45 yr from today and if the kinetic energy could be increased at a rate consistent with the historical record, then humans can reach the ten most nearby stars within 100 yr from today.Comment: 10 pages, 5 col. figures, 1 tabl

Analytic solutions to the maximum and average exoplanet transit depth for common stellar limb darkening laws

The depth of an exoplanetary transit in the light curve of a distant star is commonly approximated as the squared planet-to-star radius ratio, (R_p/R_s)^2. Stellar limb darkening, however, results in significantly deeper transits. Here we derive analytical solutions to the overshoot of the mid-transit depth caused by stellar limb darkening compared to the (R_p/R_s)^2 estimate for arbitrary transit impact parameters. In turn, this allows us to compute the true planet-to-star radius ratio from the transit depth for a given parameterization of a limb darkening law and for a known transit impact parameter. We calculate the maximum emerging specific stellar intensity covered by the planet in transit and derive analytic solutions for the transit depth overshoot. Solutions are presented for the linear, quadratic, square-root, logarithmic, and non-linear stellar limb darkening with arbitrary transit impact parameters. We also derive formulae to calculate the average intensity along the transit chord, which allows us to estimate the actual transit depth (and therefore R_p/R_s) from the mean in-transit flux. The transit depth overshoot of exoplanets compared to the (R_p/R_s)^2 estimate increases from about 15% for A main-sequence stars to roughly 20% for sun-like stars and some 30% for K and M stars. The error in our analytical solutions for R_p/R_s from the small planet approximation is orders of magnitude smaller than the uncertainties arising from typical noise in real light curves and from the uncertain limb darkening. Our equations can be used to predict with high accuracy the expected transit depth of extrasolar planets. The actual planet radius can be calculated from the measured transit depth or from the mean in-transit flux if the stellar limb darkening can be properly parameterized and if the transit impact parameter is known. Light curve fitting is not required.Comment: 7 pages, 3 figures (2 col, 1 b/w), published in A&

Transits of extrasolar moons around luminous giant planets

Beyond Earth-like planets, moons can be habitable, too. No exomoons have been securely detected, but they could be extremely abundant. Young Jovian planets can be as hot as late M stars, with effective temperatures of up to 2000 K. Transits of their moons might be detectable in their infrared photometric light curves if the planets are sufficiently separated ($\gtrsim10$ AU) from the stars to be directly imaged. The moons will be heated by radiation from their young planets and potentially by tidal friction. Although stellar illumination will be weak beyond 5 AU, these alternative energy sources could liquify surface water on exomoons for hundreds of Myr. A Mars-mass H$_2$O-rich moon around $\beta$ Pic b would have a transit depth of $1.5\times10^{-3}$, in reach of near-future technology.Comment: 2 colored figures, 4 pages, in press at A&A (http://dx.doi.org/10.1051/0004-6361/201527496

Hot Moons and Cool Stars

The exquisite photometric precision of the Kepler space telescope now puts the detection of extrasolar moons at the horizon. Here, we firstly review observational and analytical techniques that have recently been proposed to find exomoons. Secondly, we discuss the prospects of characterizing potentially habitable extrasolar satellites. With moons being much more numerous than planets in the solar system and with most exoplanets found in the stellar habitable zone being gas giants, habitable moons could be as abundant as habitable planets. However, satellites orbiting planets in the habitable zones of cool stars will encounter strong tidal heating and likely appear as hot moons.Comment: submitted as Proceedings to the ROPACS meeting "Hot Planets and Cool Stars" (Nov. 2012, Garching), 4 pages, 2 colored figure

How to determine an exomoon's sense of orbital motion

We present two methods to determine an exomoon's sense of orbital motion (SOM), one with respect to the planet's circumstellar orbit and one with respect to the planetary rotation. Our simulations show that the required measurements will be possible with the European Extremely Large Telescope (E-ELT). The first method relies on mutual planet-moon events during stellar transits. Eclipses with the moon passing behind (in front of) the planet will be late (early) with regard to the moon's mean orbital period due to the finite speed of light. This "transit timing dichotomy" (TTD) determines an exomoon's SOM with respect to the circumstellar motion. For the ten largest moons in the solar system, TTDs range between 2 and 12 s. The E-ELT will enable such measurements for Earth-sized moons around nearby stars. The second method measures distortions in the IR spectrum of the rotating giant planet when it is transited by its moon. This Rossiter-McLaughlin effect (RME) in the planetary spectrum reveals the angle between the planetary equator and the moon's circumplanetary orbital plane, and therefore unveils the moon's SOM with respect to the planet's rotation. A reasonably large moon transiting a directly imaged planet like beta Pic b causes an RME amplitude of almost 100 m/s, about twice the stellar RME amplitude of the transiting exoplanet HD209458b. Both new methods can be used to probe the origin of exomoons, that is, whether they are regular or irregular in nature.Comment: accepted by ApJ Letters, 6 pages, 5 figures (2 color