Space Travel Under Constant 1g Acceleration

The basic principle behind every high-thrust interplanetary space probe is to accelerate briefly and then coast, following an elliptical, parabolic, or mildly hyperbolic solar trajectory to your destination, using gravity assists whenever possible. But this is very slow.

Imagine, for a moment, that we have a spacecraft that is capable of a constant 1g (“one gee” = 9.8 m/s2) acceleration. Your spacecraft accelerates for the first half of the journey, and then decelerates for the second half of the journey to allow an extended visit at your destination. A constant 1g acceleration would afford human occupants the comfort of an earthlike gravitational environment where you would not be weightless except during very brief periods during the mission. Granted such a rocket ship would require a tremendous source of power, far beyond what today’s chemical rockets can deliver, but the day will come—perhaps even in our lifetimes—when probes and people will routinely travel the solar system in just a few days. Journeys to the stars, however, will be much more difficult.

The key to tomorrow’s space propulsion systems will be fusion and, later, matter-antimatter annihilation. The fusion of hydrogen into helium provides energy E = 0.008 mc2. This may not seem like much energy, but when today’s technological hurdles are overcome, fusion reactors will produce far more energy in a manner far safer than today’s fission reactors. Matter-antimatter annihilation, on the other hand, completely converts mass into energy in the amount given by Einstein’s famous equation E = mc2. You cannot get any more energy than this out of any conceivable on-board power or propulsion system. Of course, no system is perfect, so there will be some losses that will reduce the efficiency of even the best fusion or matter-antimatter propulsion system by a few percent.

How long would it take to travel from Earth to the Moon or any of the planets in our solar system under constant 1g acceleration for the first half of the journey and constant 1g deceleration during the second half of the journey? Using the equations below, you can calculate this easily.

Keep in mind that under a constant 1g acceleration, your velocity quickly becomes so great that you can assume a straight-line trajectory from point a to point b anywhere in our solar system.

Maximum velocity is reached at the halfway point (when you stop accelerating and begin decelerating) and is given by

The energy per unit mass needed for the trip (one way) is then given by

How much fuel will you need for the journey?

hydrogen fusion into helium gives: Efusion = 0.008 mfuel c2

matter-antimatter annihilation gives: Eanti = mfuel c2

This assumes 100% of the fuel goes into propelling the spacecraft, but of course there will be energy losses and operational energy requirements which will require a greater amount of fuel than this. Moreover, we are here calculating the amount of fuel you’ll need for each kg of payload. We would need to use calculus to determine how much additional energy will be needed to accelerate the ever changing amount of fuel as well. The journey may well be analogous to the traveler not being able to carry enough water to survive crossing the desert on foot.

Now, let’s use the equations above for a journey to the nearest stars. There are currently 58 known stars within 15 light years. The nearest is the triple star system Alpha Centauri A & B and Proxima Centauri (4.3 ly), and the farthest is LHS 292 (14.9 ly).

I predict that interstellar travel will remain impractical until we figure out a way to harness the vacuum energy of spacetime itself. If we could extract energy from the medium through which we travel, we wouldn’t need to carry fuel onboard the spacecraft.

We already do something analogous to this when we perform a gravity assist maneuver. As the illustration below shows, the spacecraft “borrows” energy by infinitesimally slowing down the much more massive Jupiter in its orbit around the Sun and transferring that energy to the tiny spacecraft so that it speeds up and changes direction. When the spacecraft leaves the gravitational sphere of influence of Jupiter, it is traveling just as fast as it did when it entered it, but now the spacecraft is farther from the Sun and moving faster than it would have otherwise.


Of course, our spacecraft will be “in the middle of nowhere” traveling through interstellar space, but what if space itself has energy we can borrow?

Theory and Observation

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis F1: Philosophical choices necessarily underly cosmological theory.
Some cosmologists tend to ignore the philosophical choices underlying their theories; but simplistic or unexamined philosophical standpoints are still philosophical standpoints!

Cosmology, and indeed all human inquiry, is based on (at least) two unproven (though certainly reasonable) assumptions:

  1. The Universe exists.
  2. The human mind is at least to some degree capable of perceiving and understanding the Universe.

Any cosmological theory will have additional foundational unproven assumptions.  These are called axioms.  Ellis admonishes us to at least be aware of them, and to admit to them.

8.1 Criteria for theories
As regards criteria for a good scientific theory, typical would be the following four areas of assessment: (1) Satisfactory structure: (a) internal consistency, (b) simplicity (Ockham’s razor), and (c) aesthetic appeal (‘beauty’ or ‘elegance’); (2) Intrinsic explanatory power: (a) logical tightness, (b) scope of the theory—the ability to unify otherwise separate phenomena, and (c) probability of the theory or model with respect to some well-defined measure; (3) Extrinsic explanatory power, or relatedness: (a) connectedness to the rest of science, (b) extendability—providing a basis for further development; (4) Observational and experimental support, in terms of (a) testability: the ability to make quantitative as well as qualitative predications that can be tested; and (b) confirmation: the extent to which the theory is supported by such tests as have been made.

As you can see, a theory is not an opinion.  It must be well-supported by facts.  It must be internally consistent.  It must have explanatory power.  The Russian physicist A. I. Kitaĭgorodskiĭ (1914-1985) put it succinctly: “A first-rate theory predicts; a second-rate theory forbids, and a
third-rate theory explains after the event.”  Einstein’s special and general relativity are spectacular examples of first-rate theories.  In over 100 years of increasingly rigorous and sophisticated experiments and observations, relativity has never been proven to be incorrect.

Ellis emphasizes the importance of observational and experimental support in any scientific theory.

It is particularly the latter that characterizes a scientific theory, in contrast to other types of theories claiming to explain features of the universe and why things happen as they do.  It should be noted that these criteria are philosophical in nature in that they themselves cannot be proven to be correct by any experiment.  Rather their choice is based on past experience combined with philosophical reflection.  One could attempt to formulate criteria for good criteria for scientific theories, but of course these too would need to be philosophically justified.  The enterprise will end in infinite regress unless it is ended at some stage by a simple acceptance of a specific set of criteria.

So, even our criteria about what makes a good scientific theory rest upon axioms that cannot be proven.  But unlike religion, scientific theories never posit the existence of any supernatural entity.

Thesis F3: Conflicts will inevitably arise in applying criteria for satisfactory cosmological theories.
The thrust of much recent development has been away from observational tests toward strongly theoretically based proposals, indeed sometimes almost discounting observational tests.  At present this is being corrected by a healthy move to detailed observational analysis of the consequences of the proposed theories, marking a maturity of the subject.  However because of all the limitations in terms of observations and testing, in the cosmological context we still have to rely heavily on other criteria, and some criteria that are important in most of science may not really make sense.

String theory? Cosmic inflation?  Multiverse? If a theory is currently neither testable nor directly supported by observations, is it science, or something else?

Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.

Bonner Durchmusterung und Gaia

As our civilization and technology continue to evolve, it seems we take far too much for granted.  We neglect to consider how incredibly hard people used to work years ago to achieve results we today would pass off as almost trivial.  But history has many lessons to teach us, if only we would listen.

As an example, Prussian astronomer Friedrich Wilhelm August Argelander (1799-1875) at the age of 60 began publishing the most comprehensive star catalogue and atlas ever compiled, as of that date.  From 1852 to 1859, Argelander and his assistants carefully and accurately recorded the position and brightness of over 324,000 stars using a 3-inch (!) telescope in Bonn, Germany.  Employing the Earth’s rotation, star positions were measured as each star drifted across the eyepiece reticle in the stationary meridian telescope by carefully recording when each star crossed the line, and where along the line the crossing point was.

Stars Transiting in a Meridian Telescope

One person observed through the telescope and called off the star’s brightness as each star crossed the line, noting the exact position along the reticle on a pad with a cardboard template so that the numbers could be written down without looking away from the telescope.  A second person, the recorder, noted the exact time of reticle crossing and the brightness called out by the observer.  In this way, two people were able to record the position and brightness of every star.

Each star was observed at least twice so that any errors could be detected and corrected.  In some areas of the Milky Way, as many as 30 stars would cross the reticle each minute.  What stamina and dedication it must have taken Argelander and his staff to make over 700,000 observations in just seven years!  Argelander’s catalogue is called the Bonner Durchmusterung and is still used by astronomers even today.  It was the last major star catalogue to be produced without the aid of photography.

Like Argelander’s small meridian telescope, the European Space Agency’s Gaia astrometric space observatory is currently measuring tens of thousands of stars each minute (down to mv = 20) as they transit across a large CCD array—the modern day equivalent of an eyepiece reticle.  But instead of utilizing the Earth’s rotation period relative to the background stars of 23h56m04s, Gaia’s twin telescopes separated by exactly 106.5° sweep across the stars as Gaia rotates once every six hours.  A slight precession in Gaia’s orientation ensures that the field of view is shifted so that there is only a little overlap during the next six-hour rotation.

When Gaia completes its ongoing mission, it will have measured the positions, distances, and 3D space motions of around a billion stars, not just twice but 70 times!

Though electronic computers do most of the work these days, someone still has to program them.  Some 450 scientists and software experts are immersed in the challenging task of converting raw data into scientifically useful information.

I’d like to conclude this entry with a quotation from Albert Einstein (1879-1955), who was born and died exactly 80 years after Argelander.

Many times a day I realize how much of my outer and inner life is built upon the labors of my fellowmen, both living and dead, and how earnestly I must exert myself in order to give as much as I have received.

I love that quote.  Words to live by.