The Dark Side of the Moon

“I'll see you on the dark side of the moon.” Folks with a little astronomy knowledge cringe when they hear these Pink Floyd lyrics. They will patiently explain there is no dark side of the moon. The moon turns a revolution over about 4 weeks. The far side as well as the near side see two weeks of darkness as well as two weeks of sunshine.

But one side is darker. Since the moon is tide locked, the far side never sees earthlight. On the other hand, someone standing on the moon’s nearside will always see earth hovering in the same region of the sky.

Viewed from the earth’s surface, both the sun and the moon subtend about half a degree. The moon’s albedo is .12, meaning it reflects about 12% of the sunlight that hits it. The moon is nearly as dark as charcoal, it only looks bright against the black void of space when our eyes have adjusted to the night’s darkness. Even the above graphic exaggerates the moon's brightness -- the sun is about 100,000 times brighter than the moon.

Viewed from the moon’s surface, the sun subtends half a degree (just as when seen from earth). But earth subtends about 2 degrees. Moreover the earth reflects about 2.5 times more light than the moon, having an albedo of around .3.

Above is a photo taken by NASA's DSCOVR satellite as the moon passed in front of the earth. The Deep Space Climate Observatory is one million miles from the earth, lieing between the earth and sun.

The larger apparent diameter and higher albedo means the earth seen from the moon is about 34 times brighter than the moon seen from earth.

The Bright Side

Let's imagine an astronaut standing at the moon's closest point to the earth. Not far from Mösting A Crater, 0 degrees latitude, 0 degree longitude. From this location, the Earth always hovers directly overhead.

Here is our astronaut pointing his iPhone straight up to snap a picture of the earth. In the foreground an iPad displays the picture he snaps:

It is sunrise. The astronaut sees a half-earth. The long shadows stretching west aren't wholly dark, they are lit by the half-earth above.

As the sun climbs towards high noon, earth is a waning crescent.
   

At noon the earth is at it's dimmest being a very thin crescent or a new-earth. But the moon remains well lit because it's high noon. Except on rare occasions when the sun passes behind the earth.

As the sun sinks towards the horizon, earth is a waxing crescent.

At sunset the waxing crescent has grown to a half-earth. The long shadows stretching east are lightened by the half-earth above.

As the sun sinks deeper behind the horizon, earth is waxing gibbous.

At midnight the astronaut sees a full-earth. This full earth is 34 times brighter than the full moon earthlings see.

From midnight to sunrise, the astronaut sees a waning gibbous earth. At sunrise we're back to where we started.

To The Dark Side

The astronaut hops in his buggy and starts driving east. As he drives closer to the far side, the earth sinks toward the horizon. When the earth is near the horizon it's possible for the sun to be below the horizon when the earth is a dim thin crescent. Even so, the astronaut enjoys strong earthlight for most the night.

When the astronaut drives over into the far side, there's no earthlight. On the far side, it's a deep stygian blackness during the two weeks from sunset to sunrise.

The far side is dark in another sense. Earth is a bright radio source. The far side of the moon is always shadowed from earth's radio noise. Radio astronomers salivate at the thought of a radio telescope under the far side's dark skies.

So you see, the Pink Floyd lyrics make some sense even if you're not under the influence.

Mini Solar Systems

Edit as of 6-23-2016. Many of our gas giant moons seem to have have internal liquid oceans. Liquid water  suggests regions with comfortable temperatures. These strata might also have human friendly pressures. There would certainly be lots of in situ water and organic compounds. I am becoming more interested in icey moons as potential homes for humans.

Originally I had pointed to Jupiter and Saturn suggesting similar moon systems in other star systems would make good science fiction settings. But perhaps the gas giant moons within our own system could provide such a setting. A moon need not reside within the "Goldilocks Zone" in order to accommodate humans.

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Most pulp science fiction of yesteryear relies on fast paced story lines that take place over a short time. Not plausible in our solar system where Hohmann launch windows are years apart and trip times between planets are months to years.

A setting Retro Rockets suggests is a mini solar system where trip times and time between launch windows are on the order of days instead of months or years. The "mini solar system" proposed is a gas giant with a family of moons, all orbiting in a star's habitable zone.

This is a plausible setting in my opinion. This spreadsheet shows travel between the moons of Jupiter or Saturn can occur at a good pace. The interval between launch windows is called synodic period.

The gas giants in our solar system have respectable families of moons and many are a comparable size to Mars and Mercury. Here's a graphic comparing some gas giant moons to rocky bodies in our inner solar system:

Retrorockets notes that while mini-solar systems allow a story with an exciting tempo, delta v (needed change in velocity) is still high. But a setting with much less delta V is plausible.

Many of the gas giant moons  in our solar system are tidally locked with the planet they orbit. That is, they always present the same face to the orbiting planet. From the surface of a tide-locked moon, the planet-moon L1 and L2 regions remain in the same part of the sky, much like geosynchronous satellites appear to hover motionless when viewed from the earth's surface. For tide-locked moons, L1 and L2 are possible centers for a space elevator.

Between two moons there exists an elliptical transfer orbit whose apoapsis angular velocity (ω) matches that of the upper moon and whose periapsis ω matches the angular velocity of the lower moon. If the moons are nearly co-planar, trips can be made between the moon's elevators with very little delta V. Here's an illustration showing tide-locked moons Phobos and Deimos:

Expressions for transfer ellipse's eccentricity, apoapsis, periapsis are shown above. They can be generalized to any pair of tide-locked, coplanar moons.

Transfer ellipses between Saturn moon beanstalks:

Tranfer ellipses between Galilean Moon beanstalks:

Something to watch out for is the planet-moon L1 and L2 locations. If L1 and L2 aren't well below the departure arrival point on the beanstalk, the influence of the moon's gravity might substantially alter the shape of the transfer orbit. In the case of Jupiter's and Saturn's moons, the L1 & L2s are well below the tether tops.

Another thing to watch out for is gas giant rings. The chunks of ice in Saturn's rings might well be a debris field that would quickly cut some of these beanstalks.

It is a convention to label a tide-locked moons closest point as having 0 degrees latitude and 0 degrees longitude. For a civilization evolving on a tide-locked moon, I would predict religious significance being attached to 0º, 0º point. A viewer standing at this location will see the gas giant hovering in the sky's zenith. The far and near points will gain additional military and commercial importance when they anchor beanstalks going through L1 and L2.

Our earth globe has non-arbitrary features: the north pole, south pole, equator, tropic of Cancer and Capricorn and the arctic and antarctic circles. Cartographers of tidelocked moons will have additional non-arbitrary markings: A band separating the near side from the far side. I'd also expect a circle containing the near and far points as well as the north and south poles.  A simplified globe would look like a spherical octahedron:

Here is a painting I had done of Gielo (Giant In Earth Like Orbit) and Elm (Earth Like Moon):

A very interesting setting with lots of possibilities. I hope science fiction writers will do stories of habitable moons orbiting a gas giant.

Deboning the Porkchop Plot

Changing direction causes ΔV (change in velocity), often more than a change in speed. Compare the velocity vectors below. When going the same direction, the difference is 1 km/s. When at right angles the difference is 5 km/s. We know this from driving in traffic. Two cars going almost the same speed hit each other. If they’re in the same lane going the same direction, it’s a mild bump. If one car runs a red light and T-bones a car in cross traffic, the impact is serious: 

This is the strength of a Hohmann transfer orbit. Velocity vectors are pointing the same direction at departure as well as destination. No direction change is needed, only a speed change:

Note the Hohmann transfer path moves 180 degrees about the sun:

A Hohmann transfer assumes the departure and destination orbits are co-planar. But what if the destination orbit is inclined?

Orbit Planes and Spherical Trigonometry

A plane passing through a sphere’s center cuts the sphere along a great circle. A group of planes all sharing a common point can be represented as great circles on a sphere. Since every orbit about the sun is a conic section having the sun as a focus, each orbital plane shares the sun as a common point. Representing the orbital planes as great circles is convenient. There are already a lot of theorems in spherical trigonometry which gives us a suite of tools for looking at angles between orbital planes.

The shortest path (or geodesic) along a spherical surface between two points is an arc of a great circle. If we set the sphere’s radius to be 1, the arc length is also the angular separation in radians.

A familar group of great circles are the longitude lines on a globe. The equator is the only great circle among the latitude lines. All the longitude lines are great circles passing through the poles.

Let’s use the equatorial great circle to represent the departure plane. Recall the Hohmann transfer moves 180 degrees about the center. In this illustration, latitude and longitude for departure and destination is (0º, 0º) and (7º, 180º). The only great circle connecting these points is a polar orbit nearly 90º from the departure and destination planes! Big plane changes at departure and destination destroys the virtue of a Hohmann orbit.

I’ve also tried to demonstrate this in this video:

The big delta V needed for large plane changes makes the ridge in a porkchop plot:

(image courtesy NASA)

Porkchop plots are drawn by doing iterations of various Lambert Space Triangles. Lambert iterations give polar transfer orbits when departure and destination longitudes differ by 180º.

Does this mean Hohmann transfers are no good if the destination orbit’s inclined? No, the big plane changes can be avoided with a mid course plane change. Here is a broken plane transfer where a plane change burn is done at the ascending node:

The line where the destination and departure planes intersect form the ascending and descending nodes. Starting in the departure plane and doing a plane change at the node avoids the two major plane changes. The departure and destination planes differ by an angle called i, for inclination.

Changing a vector by an angle i takes dv of v * 2 * sin(i/2).

The Vis Viva Equation tells us v = sqrt(μ(2/r - 1/a)). So v ranges from sqrt (μ((1-e)/(a(1+e)))) at aphelion to sqrt (μ((1+e)/(a(1-e)))) at perihelion. Let's look at a Ceres transfer orbit. An ellipse with a 1.88 a.u. semi major axis and eccentricity .47 will have speeds ranging from 36 km/s (at perihelion) to 13 km/s (at apohelion). Inclination's about 10.6 degrees. So plane change ranges from 36 km/s   * 2 * sin(10º/2) to 13 km/s   * 2 * sin(10º/2) or from 6.7 to 2.4 km/s. Is the a 2.4 km/s plane change at aphelion the best we can do?  No, it's possible to have less plane change expense.

Launch is at the perihelion of an outbound Hohmann orbit. If the launch coincides with a node, the entire plane change can be done during earth departure or at arrival. Then the delta V entails a speed change as well as a direction change. Doing a single plane change/speed change burn saves delta V as shown by this diagram:

Law of cosines tells us for a triangle a, b, c, a² + b² - 2ab cos(i) = c². In this case, i is the angle between a and b and c is the delta V needed from the combined plane change and speed change.

At aphelion, a combined speed change/plane change only costs .76 km/s more than the speed change alone.

When launching deep in earth’s gravity well, we enjoy an Oberth benefit. Ceres' gravity well lends a little Oberth benefit at the destination. If the line of nodes coincides with transfer orbit's line of apsides, plane change can cost as little as .52 km/s extra.

This indicates as much plane change as possible should be made at departure and arrival. What sort of plane changes should we make to minimize the angle of the midcourse plane change?

The fattest part of an orange slice is right in the middle:

The angular separation at launch has to be some part of the orange slice. To minimize the angle between transfer plane and destination plane, the angular separation at launch should be in the middle. Having the transfer plane intersect the destination plane 90º from launch minimizes plane change angle.

An object on an elliptical path moves slower as it moves further from the sun, so doing plane changes further out are cheaper. The 90º from launch is a minimum. There will be a larger plane change angle 100 degrees from launch, but velocity will be slower. Also plane change lessens as flight path angle increases. I hope to talk about this more when I have time.

But for now I believe this shows that the Lambert iterations greatly exaggerates plane change expense for a Hohmann path where departure and destination points are 180º degrees apart. Most of that plane change expense can be eliminated by choosing a good place to do a midcourse plane change.

A PDF on Broken Plane Maneuvers Fernando Abilleria of NASA Jet Propulsion Laboratory