## Astrogation Room

What's in the astrogation room?

Everything needed for interplanetary navigation. Instrument to determine the ship's current trajectory and calculating devices to plot new trajectories.

There are many navigational instruments.
A periscope sextant to take navigational readings, with its azimuth ring. *(In THE REVOLT ON VENUS, this is what Roger Manning was looking through when he noticed the atomic bomb attached to the Polaris' tail)* There also might be a goniometer, which is used to measure angles. A good-sized telescope, either in a dome or with a coleostat.

*(The periscope, the telescope, or both will be equipped with a filar micrometer.)*Star trackers, star scanners, solar trackers, sun sensors, and planetary limb sensors and trackers. Inertial tracking repeaters

*(note that the inertial tracker platform will have to be manually realigned every twelve hours because it tends to drift. The star tracker is used for reference.)*. There might even be a pulsar positioning system.

In addition to navigational instruments there will be other necessary gear. There will be an incredibly precise chronometer. An integrating accelerograph (displaying elapsed time, velocity, and distance in dead-reckoning). An indicator of the spacecraft's current mass ratio. An integral audio recorder and a log book for radio messages and navigational fixes.

Not to mention lots of paper, pencils, slide rules, and ballistic calculators. Or instead a laptop computer with an *AstrogateMe ^{TM}* app.

Secondly, readouts for the ship's navigational and tactical sensors. The big radar scope. Doppler radar and radar altimeter. This might be a separate deck, if you think it is insane to have a single crewperson responsible for navigation, detection, *and* communication like in Tom Corbett Space Cadet.

Thirdly communication gear, perhaps even with something like a Morse code key for use when radio interference becomes a problem *(If this was a Metalunan ship, this is where you'd find the interociter)*.

There might be a separate communications deck, which is generally called a "radio shack", manned by a communication officer whose nickname is "Sparks." If this is a military spacecraft this might be the place for the safe containing the code book. Hit the red "incinerate" button to keep the one-time pad and Captain Midnight secret decoder ring from falling into enemy hands. On some ships this safe might be in the captain's cabin.

## Astrodome

Many of the navigational instruments might be mounted inside an "astrodome", which is a blister dome of some strong but transparent material used with a manual sextant as a back-up to the periscope. *(Note that astrodomes cause optical distortion that need a mathematical correction.)*

If there is an astrodome, the room will have alternative lighting that is all red, like a darkroom. This is to preserve night vision. It should also have a retractable shield. This is to preserve * day* vision in case the rotation of the ship moves the eye-destroying fury of the Sun into view. The shield is not only useful to keep sunlight out, but to keep the atmosphere in, in case the astrodome is breached or shattered. Not to mention protecting the astrodome from melting if the ship does some aerobraking in a planetary atmosphere.

If the ship spins on its axis for artificial gravity, it might be a good idea to locate the astrodome in the nose of the ship, *i.e.,* at the center of the axis of rotation. A tiny room with the astrodome in it could be counter-spun. So while the ship was spinning, the room would be stationary, freeing the astrogator from the difficulty of making observations of a sky that is madly spinning about. The problem is that if this is a nuclear powered ship, the docking port has to be on the nose. It is possible to rig in a coleostat a shutter that is synchronized with the spin of the ship. This will provide a stroboscopic but steady image if you cannot counter-spin the astrodome.

If the ship is advanced enough to have an actual centrifuge, instead of spinning the entire ship, things will be easier. Just make sure the astrodome is on the stationary part of the ship.

## Astrogation Calculation

Astrogators have two main jobs:

**Orbit Determination:**Knowing the spacecraft's current position and velocity, and predicting future position and velocity.**Flight Path Control:**Calculating maneuvers for the pilot to alter the spacecraft's trajectory in the desired direction.

The astrogator is responsible for offering the Captain a range of solutions for the mission the captain orders, plotting the course for the chosen solution, giving the pilot the specifications for the required maneuvers needed to implement the course, and to monitor the progress of the spacecraft along the course while calculating mid course corrections for the pilot in order to keep the ship in the groove.

Nowadays there will be no astrogator. The captain will type the desired mission into their cell phone's astrogator/pilot app and let it do all the math and ship piloting. From an author's point of view this is a disaster due to Burnside's Zeroth Law. One possible solution is to make the personnel on the spacecraft not be "crew" so much as system managers. Rick Robinson points out that you'll need a human astrogator if something drastically unexpected happens. For instance, if the unexpected arrival of a Klingon invasion fleet unexpectedly overlaps the optimal trajectory delivered by the astrogation computer.

Astrogation Glossary

**Apoapsis**In an orbit, the point of the orbit farthest from the astronomical body currently being orbited.**Periapsis**In an orbit, the point of the orbit closest to the astronomical body currently being orbited. Some like to replace the "-apsis" part with the name of the body being orbited, but that gets out of hand real quick. For example "perigee", "perihelion", "pericynthion", and zillions of other unwieldy terms.**Prograde**In the direction of the spacecraft's trajectory,*i.e.,*"forwards". Fun fact: since the trajectory is curved, prograde is actually at a tangent to the trajectory.**Retrograde**In the opposite direction of the spacecraft's trajectory,*i.e.,*"backwards". 180 degrees from Prograde.**Normal**At 90 degrees (perpendicular) to the spacecraft's orbital plane, in the orbital "North" direction (using the "right-hand rule").**Anti-normal**At 90 degrees to the spacecraft's orbital plane, in the orbital "South" direction. 180 degrees from Normal.**Radial in**In the direction of the astronomical body currently being orbited.**Radial out**In the opposite direction of the astronomical body currently being orbited. 180 degrees from Radial in.

### Choosing Trajectory

Actually calculating interplanetary trajectories is true rocket science, and beyond the scope of this website *(translation: I don't know how to do it)*. If you simply must know how, a good starting text is Fundamentals of Astrodynamics by Roger Bate, ISBN: 0486600610. The book assumes you are already well versed in calculus.

The captain of the spacecraft will ask the astrogator for a mission plan to travel from point A to point B in time T. The astrogator will determine a family of mission plans, with the current ship's delta-V capacity as the upper limit *(or the ship will not be capable of performing that mission)* and with the captain's specfied mission time as the lower limit *(or the captain will be unhappy)*. You see, a Hohmann trajectory generally uses the least delta-V, but also has the longest possible mission time, and the mission can only start on specific dates *("launch windows")* as well. By increasing the delta-V used the mission time can be reduced.

What the astrogator will do is have the navigation computer draw a pork-chop plot, which is a graph with departure times on one axis, arrival times on the other axis, and delta-V requirements drawn as contour lines in the graph. Cross out the areas of delta-V that are too high for the spacecraft, cross out the part of the graph with a mission duration that is too long, what remains are the possible missions.

If it turns out there is no possible mission inside the stated parameters, the astrogator will have to confer with the captain over what is possible.

Once the specific mission is chosen, with delta-V and duration time, the astrogator does the hard part calculating the trajectory, burn vectors, and check-points. If the SF author wants to go full Heinlein and do that, I refer them to Fundamentals of Astrodynamics or equivalent.

### Maneuvers

The given mission composed of a series of trajectories. At each point where the spacecraft makes a transition from one trajectory to another is a "maneuver". A maneuver is where the spacecraft uses a burn of its rockets to alter its vector to the new trajectory.

For each maneuver, the astrogator will calculate three maneuver parameters for the pilot:

- The Axis of Acceleration
*(where the ship's nose should be pointing during the burn)* - The required amount of Delta V
*(pilot will figure the proper engine thrust setting and burn duration for this)* - The starting time of the maneuver
*(this should happen at the mid-point of the burn duration, pilot will calculate this. Figure burn duration required for delta V, divide by 2, and subtract from astrogator-supplied maneuver time)*

These will be passed to the pilot. If the pilot finds a problem *(such as the spacecraft not possessing enough propellant reserves to create the required delta V)* the astrogator will have to frantically recalculate to fix the problem.

There are five basic maneuvers:

**Ascend to orbit**

**Change Orbit Shape**

Burning with the axis of acceleration pointed in the prograde direction ("burning prograde") will expand the size of the orbit. Burning retrograde will contract the size of the orbit. In both cases, the point on the orbit the spacecraft is currently occupying stays put, that is the center point of the orbit expansion/contraction.

Burning prograde at periapsis will raise your apoapsis (move it farther away from planet). Burning retrograde at periapsis will lower your apoapsis (move it closer to the planet). Burning prograde at apoapsis will raise your periapsis. Burning retrograde at apoapsis will lower your periapsis. If you keep in mind that the periapsis and apoapsis are always exactly opposite each other, this makes perfect sense.

You "circularize" an orbit by making the periapsis and apoapsis the same distance from the planet, *i.e.,* you make the orbital eccentricity close to zero, thus making the orbit a circle instead of some kind of egg shape.

You change orbits by altering either the periapsis or apoapsis to the new orbital distance (depending upon whether the new orbit is smaller or larger), wait until you reach that distance, then circularize the orbit into the new orbit.

Note that for a circular orbit, the larger the radius = the lower the orbital speed. So to match locations with a ship in the same orbit as you, increase your orbital radius to slow down or increase it to speed up (depending upon whether the other ship is ahead of you or behind you). Wait until other ship approaches then match its orbital radius.

Burning radial in or radial out will spin the entire orbit in place. This only has a noticable effect if the orbit is egg shaped. The orbit can only be spun a maximum of 90 degrees clockwise or counterclockwise. These burns are not used very much, since it is almost always more efficient to use prograde / retrograde burns to do the same thing.

**Match Orbital Inclination**

If your orbital plane is at a different angle with respect to the desired new orbital plane, you will have to match orbital inclination. This is the first step to making a rendezvous with a planet or docking with another spacecraft. This is also notoriously the most expensive maneuver in terms of delta V.

Where the two orbital planes cross each other are two "nodes", the ascending node and the descending node. At either of the nodes, you burn normal or anti-normal (depending upon the angle of the new orbital plane with respect to the old one, at that node). After burning an exorbitant amount of propellant, you will be in the same plane.

**Planetary Rendezvous**or

**Spacecraft Docking**

The procedure is much the same whether one is trying to leave an interplanetary trajectory to enter orbit around a planet or trying to dock to another spacecraft in orbit around the same planet as you are. In the first case the "target" is the orbit around the planet, in the second case the target is the ship one is docking to.

First match orbital inclination with the target.

Secondly change orbit shape so that your orbit is at a tangent to the target orbit, preferably at your apoaspsis or proapsis (if docking, you want the tangent to also be where the other spacecraft will be when you arrive). When making a planetary rendezvous, your spacecraft will commonly have lots of velocity that has to be gotten rid off. Often aerobraking is used to avoid having to burn lots of expensive propellant.

At the tangent point change orbit shape to match the target's orbit.

**Aerobraking**

Altering the spacecraft's trajectory so that the periapsis is inside the atmosphere of the planet being orbited. The spacecraft will slow down due to atmospheric drag.

Can be a prelude to landing, can also be used to slow the spacecraft into a capture orbit without having to expend any expensive propellant.

Warning: if the drag and/or heat from friction becomes too strong, bits of the spacecraft will be torn off or melted away. If the drag becomes monstrously strong the entire spacecraft will be shredded or melted away. If you have an astrodome, be sure to protect it by closing the retractable shield. The plasma sheath of ionized atmosphere will cut off radio communcation.

To get an idea of what the bare minimum is, we will unashamedly be taking a good look at the solution in the computer game Kerbal Space Program. Since that is a game, the designers were forced to distill the controls to the very essentials *(because the players will quickly get fed up and leave if they think the game is too complicated)*. As a matter of fact, that game is so wonderfully educational yet fun, you might be better off if you skipped this section of the website and instead started playing the Kerbal game.

In Kerbal Space Program there is a solar system map display. This displays the planets in their orbits and the ship in its trajectory *(including altitude, position and time of apoapsis and periapsis)*. To create a maneuver, the player/astrogator uses something called the "maneuver node tool."

In broad over view: player will click on the ship's trajectory to create a new maneuver node. The node has six "controls" on it. By tugging on the controls, the ship's trajectory will be bent in various directions. The player manipulates the the six controls until the desired new trajectory is created. The three components of the the maneuver will be automatically calculated (acceleration axis, delta V, and manuever start time) and displayed on the pilot's Nav Ball.

In more detail:

The position of the maneuver node determines the maneuver starting time. Basically, when the spacecraft crawling along the trajectory reaches the position of the manuever node, it is time to start the manuever.

On the maneuver node, there is one control for each of the six burn directions: prograde, retrograde, normal, anti-normal, radial in, and radial out. Selecting and dragging a given control will set the desired velocity change in that direction. Pulling the control away from the center of the node increases the velocity, pulling it closer decreases it (the equivalent of pulling the control on the opposite side of the node). One can burn in several directions at once, the control will calculate the appropriate axis of thrust and delta V so that it is the equivalent of the vector sum of all desired burns.

So between the position of the manuever node and the values for the six burn directions the acceleration axis, delta V and maneuver start time can be calculated and relayed to the pilot's nav ball.

Remember that prograde / retrograde burns are used to change orbit shape, and normal / anti-normal burns are used to change orbital inclination. Radial out / radial in burns are used to rotate the orbit, but that isn't used very much. Don't forget that normal / anti-normal burns are very expensive in terms of delta V.

### Keeping on Track

During the mission the astrogator will periodically check the spacecraft's current position, vector (speed and direction it is traveling in), and point in time to ensure that the ship is on course. Astrogators know that pilots are only human, and no maneuver is 100% perfect. And they know that astrogators are only human as well, unavoidable perturbations can creep in.

If the spacecraft is leaving the required trajectory, mid-course corrections (Trajectory Correction Maneuver or TCM) will be needed, which the astrogator will calculate. This is a vector that will correct the spacecraft into the desired trajectory.

Say Roger want's to fix the position of the ** Polaris**. From the ephemeris he knows where Terra is, and thus the Sol-Terra line. The ephemeris also tells him where Venus is, and thus the Sol-Venus line. Roger uses the periscopic sextant to measure angle A and angle B. With simple geometry the

**current position is fixed. Of course this is an approximation based on assuming that everything is in the plane of the ecliptic. If the course gets more three dimensional a third angle will be required.**

*Polaris'*The spacecraft's vector isn't quite so simple. You will have to wait a while, make a second position fix, and calculate what the vector had to be. If you are inside a solar system you can use the observed positions of the planets against the background of stars. The positions can be precalculated at a checkpoint. When that checkpoint is reached, the planet's position is measured with a telescope. If the planet is not at the calculated position, you are off-course. Currently such observations have an accuracy on the order of 5 μ-radians, or about 750 kilometers at one astronomical unit.

Currently I have no idea how to calculate what sort of delta-V requirements TCMs will need. In Proceeding of the Symposium on Manned Planetary Missions 1963/1964 they suggested that with then-current navigation gear the total delta V required for TCM on the Terra-Mars trajectory was typically about 105 m/s and the Mars-Terra trajectory would 92 m/s.

If you are close to a planet, the distance to it can be determined by radar. Further away, the filar micrometer in the periscope can be used to determine the angular size of the planet. Since the planet's diameter is known, simple trigonometry will yield the distance. A filar micrometer is an instrument mounted in a telescope. It displays two cross hairs that can be positioned with dials *(one dial rotates the micrometer, the other adjusts the distance between the two cross hairs)*. Once set, the angular separation between the two cross hairs can be read from the scale.

Astronomers and space engineers are currently working on a way to navigate a spacecraft by using pulsars, see below.

For NASA space probes, and future spacecraft operating in the civilized sections of a solar system, things are easier due to ground support. A ground installation can see the position of your spacecraft relative to that planet. The ground installation optically sees your spacecraft's right ascension and declination. The ship and the installation trade radio pulses with time stamps on them, lightspeed lag yields the distance. Two angles and a distance gives your spacecraft position in spherical coordinates, relative to the planet. The planets position is known, correct for that an you have your spacecraft's position. Doppler radar will even give you the component of your velocity normal to the planet. All this can be had if you've paid your fees to ground installation.

In a dense asteroid drift a variable-baseline stereoscopic radar could come in handy. Look through the double eyepiece and you'll see the surrounding asteroids in 3-D. Use the sweep control to pan the view fore, aft, port, or starboard. The pilot might have one of these as well. Keep in mind that there does not appear to be any "dense astroid drifts" in our solar system, outside of Saturn's rings.

(ed note: This is how they do interstellar navigation in Star Trek. Some of the equations in the diagrams appear to have errors. I have taken the liberty of correcting them. Warning: this is not to say that my corrections are in fact correct. The system described is science fictional, with "subspace beacons" sending signals faster than light, but a more realistic one would work within the solar system. It is basically a glorified GPS system.)

3.2 POSITION DETERMINATIONThe most accurate method of determining the position of a spacecraft if the inertial reference platform is unavailable is to use the subspace beacon system. This system consists of the central beacon, the beacons defining the quadrant boundaries

(the two X-axis beacons and the two Y-axis beacons), and the north and south beacons(the two Z-axis beacons). Each beacon continually transmits, on a specific frequency, its call sign followed by a code indicating the exact time the transmission was made. Since the speed of propagation of a signal through subspace is proportional to the power of the transmitter, and the power is known, the speed of the signal can be determined. By computing the time difference between when the signal was transmitted and the present time on the ship, the delay, and in turn, the distance from the transmitter, can be calculated. The first step in determining the position of the ship is to calculate the distance between the ship and all seven beacons. The two closest quadrant boundary beacons mark the edges of the quadrant wherein the ship is located. Which sectors of the quadrant the ship is in depends upon whether the north or south beacon is closer, if the north beacon is closer, the ship is in the northern sectors. Likewise, if the south beacon is closer, the ship is in the southern sectors. On rare occasions, when the distance to these two beacons is the same, the ship is on the XY plane. It should be remembered that if the distance to the central beacon is less than 90 parsecs the ship is inside the central sphere. This does not change the method of determining the position of the ship; it just means that the ship will not be in one of the quadrants.In addition to the distances

(a, b, c, r),the angles between the central beacon and the three closest beacons(A, B, C)are needed. This arrangement is shown in figure 3.2

a= distance between ship and X-axis quadrant boundary beaconb= distance between ship and Y-axis quadrant boundary beaconc= distance between ship and Z-axis quadrant boundary beaconr= distance between ship and central beaconA= angle between X-axis quadrant boundary beacon and central beaconB= angle between Y-axis quadrant boundary beacon and central beaconC= angle between Z-axis quadrant boundary beacon and central beaconX= distance between X-axis quadrant boundary beacon and central beacon(constant)Y= distance between Y-axis quadrant boundary beacon and central beacon(constant)Z= distance between Z-axis quadrant boundary beacon and central beacon(constant)x, y, z= coordinates of spacecraft(to be calculated)

Central Beacon Angles:

D= sin^{-1}(sin A × (a / X))E= sin^{-1}(sin B × (b / Y))F= sin^{-1}(sin C × (c / Z))

Position:

x= r × cos Dy= r × cos Ez= r × cos F

**Introduction to Navigation - Star Fleet Command**included in

**Star Trek Maps**by Geoffrey Mandel (1980) ISBN 0-553-01202-9

The three values calculated using the equations in the above figure are absolute values. They do not have the positive or negative direction needed to locate a ship in the proper region of the grid. The directions are found by noting which beacons were used in the calculations. The

XandYvalues take on the same direction as the quadrant boundary beacons used to determine them. TheZ-axis is positive if the North beacon was used. Conversely, theZ-axis is negative if the South beacon was closer. This information is summarized in table 3.1.

**Introduction to Navigation - Star Fleet Command**included in

**Star Trek Maps**by Geoffrey Mandel (1980) ISBN 0-553-01202-9

The next section in the Star Trek nav text is how to cope when your subspace radio is non-functional. The astrogator can use naturally occuring pulsars for navigation *(Navigator Chekov sniffs "how primitive!")*. This is more or less the system used on the 14 Pulsar Pioneer Map. Note that accuracy can be drastically decreased if one of the pulsar suffers a glitch.

You might want to use this handy table of the 14 pulsars used in the map. Hey, if it is good enough for NASA, it's good enough for you. Table 3 has each pulsar's RA (right ascension), DEC (declination) and distance in parsecs (multiply by 3.26 to convert to light years). Use this with the technique I give here to plot your very own three D star map of navigational pulsars.

Bertolomé Coll at the Observatoire de Paris in France and Albert Tarantola have proposed a system using pulsars as a GPS **for the solar system** (not for insterstellar space). They suggest using pulsars PSR J0751+1807, PSR J2322+2057, 0711-6830 and 1518+0205B. These form a rough tetrahedron centered on the Solar System. The UK’s National Physical Laboratory and the University of Leicester are working with the European Space Agency to investigate pulsar methods for spacecraft in the solar system. The Royal Astronomical Society is looking further afield at interstellar navigation.

“GPS in space”: NPL and University of Leicester bring autonomous interplanetary travel closer to realityAn accurate method for spacecraft navigation takes a leap forward today as the National Physical Laboratory (NPL) and the University of Leicester publish a paper that reveals a spacecraft’s position in space in the direction of a particular pulsar can be calculated autonomously, using a small X-ray telescope on board the craft, to an accuracy of 2km. The method uses X-rays emitted from pulsars, which can be used to work out the position of a craft in space in 3D to an accuracy of 30 km at the distance of Neptune. Pulsars are dead stars that emit radiation in the form of X-rays and other electromagnetic waves. For a certain type of pulsar, called ‘millisecond pulsars’, the pulses of radiation occur with the regularity and precision of an atomic clock and could be used much like GPS in space.

The paper, published in Experimental Astronomy, details simulations undertaken using data, such as the pulsar positions and a craft’s distance from the Sun, for a European Space Agency feasibility study of the concept. The simulations took these data and tested the concept of triangulation by pulsars with current technology (an X-ray telescope designed and developed by the University of Leicester) and position, velocity and timing analysis undertaken by NPL. This generated a list of usable pulsars and measurements of how accurately a small telescope can lock onto these pulsars and calculate a location. Although most X-ray telescopes are large and would allow higher accuracies, the team focused on technology that could be small and light enough to be developed in future as part of a practical spacecraft subsystem. The key findings are:

– At a distance of 30 astronomical units – the approximate distance of Neptune from the Earth – an accuracy of 2km or 5km can be calculated in the direction of a particular pulsar, called PSR B1937+21, by locking onto the pulsar for ten or one hours respectively

– By locking onto three pulsars, a 3D location with an accuracy of 30km can be calculated

This technique is an improvement on the current navigation methods of the ground-based Deep Space Network (DSN) and European Space Tracking (ESTRACK) network as it:

– Can be autonomous with no need for Earth contact for months or years, if an advanced atomic clock is also on the craft. ESTRACK and DSN can only track a small number of spacecraft at a time, putting a limit on the number of deep space manoeuvres they can support for different spacecraft at any one time.

– In some scenarios, can take less time to estimate a location. ESTRACK and DSN are limited by the time delay between the craft and Earth which can be up to several hours for a mission at the outer planets and even longer outside the solar system.

Dr Setnam Shemar, Senior Research Scientist, NPL, said: “Our capability to explore the solar system has increased hugely over the past few decades; missions like Rosetta and New Horizons are testament to this. Yet how these craft navigate will in future become a limiting factor to our ambitions. The cost of maintaining current large ground-based communications systems based on radio waves is high and they can only communicate with a small number of craft at a time. Using pulsars as location beacons in space, together with a space atomic clock, allows for autonomy and greater capability in the outer solar system. The use of these dead stars in one form or another has the potential to become a new method for navigating in deep space and, in time, beyond the solar system.”

Dr John Pye, Space Research Centre Manager, University of Leicester, concludes:

“Up until now, the concept of pulsar-based navigation has been seen just as that – a concept. This simulation uses technology in the real world and proves its capabilities for this task. Our X-ray telescope can be feasibly launched into space due to its low weight and small size; indeed, it will be part of a mission to Mercury in 2018. NPL’s timing analysis capability has been developed over many years due to its long heritage in atomic clocks. We are entering a new era of space exploration as we delve deeper into our solar system, and this paper lays the foundations for a potential new technology that will get us there.”

**GPS IN SPACE**from the University of Leicester

A given pulsar's signal can only be seen from certain locations, so the interstellar navigator needs a large list of pulsars to ensure that at least three on the list are visible from the ship's current location. This is because the beam from the pulsar's magenetic north pole and the beam from the magnetic south pole sweep out along the surface of a a cone centered on either the north or south rotational axis, respectively. If the ship is not on the surface of the cone, the pulsar is invisible. Keep in mind that the surface is rather thick. If you can see a pulsar from the orbit of Mercury, you will still be able to see it from the orbit of Pluto. This means astrogators who stay within the solar system can make do with a list of four pulsars.

*14 Pulsar Pioneer Map**Pulsar beam on the magnetic axis sweeps out two yellow cones centered on rotational axis. Ship must be on the surface of one of the cones in order to see the pulsar.*

Another method of determining the position of the ship can be used, if it is not possible for the spacecraft to receive subspace signals. This method uses the various pulsars located in Federation space. Each pulsar, which is actually a rapidly rotating neutron star, has a unique puise frequency which slowly decreases over time as the rotation of the star slows down. By determining the frequency of the signal received from the pulsar it is possible to identify it. Since the frequency change is linear over time, the present frequency of the pulsar can be calculated. The difference between the two frequencies tells when the signal left the pulsar and in turn the distance from the ship to the pulsar, since the signal travels at the speed of light. This distance defines the radius of a sphere with the pulsar at the center and the spacecraft located somewhere on the surface. If three widely separated pulsars are selected and the distances to them are determined, a series of intersecting spheres is produced. This arrangement is shown in figure 3.3. There is only one point where all three spheres intersect, the location of the spacecraft. To find this point the set of equations shown in figure 3.3 must be solved simultaneously. That is, a set of values for

x, y,andzmust be found that, when inserted into all three equations at the same time, causes them to balance.

x= coordinates of Pulsar A_{A}, y_{A}, z_{A}x= coordinates of Pulsar B_{B}, y_{B}, z_{B}x= coordinates of Pulsar C_{C}, y_{C}, z_{C}a= distance between spacecraft and Pulsar Ab= distance between spacecraft and Pulsar Bc= distance between spacecraft and Pulsar Cx, y, z= coordinates of spacecraft(to be calculated)

Simultaneous Equations:

- (x - x
_{A})^{2}+ (y - y_{A})^{2}+ (z - z_{A})^{2}= a^{2}- (x - x
_{B})^{2}+ (y - y_{B})^{2}+ (z - z_{B})^{2}= b^{2}- (x - x
_{C})^{2}+ (y - y_{C})^{2}+ (z - z_{C})^{2}= c^{2}

**Introduction to Navigation - Star Fleet Command**included in

**Star Trek Maps**by Geoffrey Mandel (1980) ISBN 0-553-01202-9

There is an easier way than solving simultaneous equations. Use Trilateration. The way I understand it, first you have to rotate the coordinate system so that all three sphere centers have a Z-coord of 0, sphere one's center is at the origin, and sphere two's center is on the X-axis. You then perform trilateration, and rotate the result back to the original coordinate system. I'd go into more detail, but I'm still trying to wrap my brain around the problem. A Google search on "calculation intersection three spheres" will yield all sorts of algorithms and Matlab scripts.

By *"easier"* I mean *"easier than randomly selecting x, y, and z values until you stumble over the solution."*

The section below applies mainly to interstellar travel, or situations where the start and ending locations are stationary relative to each other, and the course between is a straight line. This is not true in interplanetary travel, where the start and destination planets are moving in their orbits, and the sun's gravity bends the course into a curved line. If you want to calculate that, read the aforementioned Fundamentals of Astrodynamics.

If all that is available to the navigator is the sighting telescope, the position of the ship can still be determined. This requires the accurate identification by their spectra of several widely spaced stars and calculating the angles between them. The mathematics needed to convert these sightings into a position goes beyond the scope of this introductory manual. A copy of a more advanced text, such as Navigation Techniques (TM:300420), should be obtained if the reader is interested in this technique.

3.3 COURSE CALCULATIONSOnce you have determined the coordinates of your present position and found those of your destination from either your computer or a chart, it is possible to plot a course between these two points. A course is defined by two angles, azimuth and elevation. These angles can be referred to in either absolule or relative terms. If they are given on an absolute basis, the angles are based on the stationary grid system with a zero azimuth angle pointing along the positive X-axis and a zero elevation angle in the XY plane. If the angles are expressed on a relative basis, the present orientation of the ship is used as the zero reference. It is standard for course angles to be given in absolute terms, and a standard terminology has been developed to prevent any confusion over which version is being used. The phrase "come to course" is used when the course angles are given in absolute terms. If the word "steer" is used instead, the angles are relative. For example, "Come to course 37 mark 136" would mean to place the ship on a course with an elevation angle of 37 degrees and an azimuth angle of 136 degrees absolute.

The absolute course angles are found by using the first three equations in figure 3.4. The distance to the destination (

r) is calculated first. It is the square root of the sum of the squares of the differences in position in each of the three axes. Next, the elevation angle (E) and the azimuth angle (A) are determined. The elevation angle is the inverse sine of the difference inZ-axis positions divided by the distance to the destination. The azimuth angle is the inverse tangent of the difference inY-axis position divided by the difference inX-axis positions. These two angles become the departure angles or bearing—the direction in which the ship heads for its destination. (In most cases the arrival angle will be the same as the departure angle relative to the galactic coordinate system.) The navigator should know two other angles, the arrival angles or bearing—these are the absolute angles at which the ship will approach its destination. They are given by the last two equations in the first group on figure 3.4. To find the position of the ship at any point along its course, the second set of equations are used. They convert the departure bearing and the distance travelled into X, Y, and Z coordinates.The course found using the equations above will take you on a straight line to your destination; however, with the large number of objects in Federation space, that course may not be the safest one. It might take you through such unpleasant places as a black hole, a supernova, or the Klingon Empire. Thus, to avoid such mishaps, once the course has been plotted on the appropriate astrogation map, its track must be examined for any unusual objects.

x= present position of spacecraft_{P}, y_{P}, z_{P}(given)x= position of destination_{D}, y_{D}, z_{D}(given)r= distance between present postion and destination(to be calculated)"Come to course E_{D}mark A_{D}"E= departure angle elevation_{D}(to be calculated)A= departure angle azimuth_{D}(to be calculated)E= arrival angle elevation_{R}(to be calculated)A= arrival angle azimuth_{R}(to be calculated)r= distance currently traveled along the course towards destination_{C}(given)x= current location along the course_{C}, y_{C}, z_{C}(to be calculated)

Distance

r= sqrt((x_{D}- x_{P})^{2}+ (y_{D}- y_{P})^{2}+ (z_{D}- z_{P})^{2})

Departure Angle

E= sin_{D}^{-1}((z_{D}- z_{P}) / r)A= tan_{D}^{-1}((y_{D}- z_{P}) / (x_{D}- x_{P}))

Arrival Angle

E= E_{R}_{D}A= A_{R}_{D}- 180°

Current Position

x= x_{C}_{P}+ (r_{C}× sin E_{D}× cos A_{D})y= y_{C}_{P}+ (r_{C}× sin E_{D}× sin A_{D})z= z_{C}_{P}+ (r_{C}× cos E_{D})

**Introduction to Navigation - Star fleet Command**included in

**Star Trek Maps**by Geoffrey Mandel (1980) ISBN 0-553-01202-9

## Computers

The astrogator will be performing plenty of math in the astrogation room, with assistance from whatever level of technology they are allowed or have access to. This can range from pencil-and-paper to slide rules to analog computers to ballistic integrators to full blown electronic digital computers.

These tools will be used to calculate the maneuvers for the mission: start time, delta V, and axis of acceleration (perhaps in the form of guide star settings for a coelostats if the pilot has no gyro horizon instrument). If this is a pre-transistor ship, all the books, slide rules and whatnot should be magnetized to stick to the desk, be on tethers, under elastic straps, or otherwise restrained so they don't float around the room. *(Or turn into deadly missiles if the spacecraft has to abruptly accelerate. Spacers have a fastidious horror of unsecured objects.)* For Tom Corbett fans, the ephemeris is the functional equivalent of Roger's space charts.

*Image from Dropping The Science*

### Manual Calculation

Doing astrogation the old school way is a nightmare. They will have a current ephemeris, a book of nine-place logarithms, rulers, dividers, protractors, pads of light green Keuffel & Esser graph paper, realms of scratch paper and lots of pencils. And a pencil sharpener designed to capture every last shaving. You don't want electrically conductive bits of graphite floating into the circuitry.

A few years ago I was visited by an astronomer, young and quite brilliant. He claimed to be a longtime reader of my fiction and his conversation proved it. I was telling him about a time I needed a synergistic orbit from Earth to a 24-hour station; I told him what story it was in, he was familiar with the scene, mentioned having read the book in grammar school.

This orbit is similar in appearance to cometary interplanet transfer but is in fact a series of compromises in order to arrive in step with the space station; elapsed time is an unsmooth integral not to be found in Hudson's Manual but it can be solved by the methods used on Siacci empiricals for atmosphere ballistics: numerical integration.

I'm married to a woman who knows more math, history, and languages than I do. This should teach me humility

(and sometimes does, for a few minutes). Her brain is a great help to me professionally. I was telling this young scientist how we obtained yards of butcher paper, then each of us worked three days, independently, solved the problem and checked each other—then the answer disappeared intooneline ofoneparagraph(SPACE CADET)but the effort had been worth-while as it controlled what I could do dramatically in that sequence.Doctor Whoosis said, "But why didn't you just shove it through a computer?"

I blinked at him. Then said slowly, gently, "My dear boy—"

(I don't usually call Ph.D.'s in hardcore sciences "My dear boy"—they impress me. But this was a special case.)"My dear boy ... this was 1947."It took him some seconds to get it, then he blushed.

**EXPANDED UNIVERSE**by Robert Heinlein (1980)

Logarithms were the mathematical marvel of the age back in the 1600s. Pierre-Simon Laplace called logarithms
*"...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."*

How do logarithms help so much?

Well, if you are old enough that you were actually taught how to add, subtract, multiply, and divide using pencil and paper you probably noticed that addition was so much easier and quicker than multiplication. Here's the trick: if you take two numbers, convert each number to its corresponding logarithm, *add* the logarithms, then convert the result from a log back to a number (the antilogarithm), the result is the two numbers *multiplied*. And if you subtracted logarithms, the result was division.

John Napier popularized this method in his book *Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms)*. After that, various people published books full of logarithmic tables. I still have a small pocket book of logarithms I used in junior high in the early 1970s.

“So you move the stores and our most necessary personal belongings in here while I’m figuring out an orbit for the

Violet. We don’t want her anywhere near us, and yet we want her to be within reaching distance while we are piloting this scout ship of ours to the place where she is supposed to be in Plan X821 S.”......The conversation definitely at an end, Loring again encased himself in his space suit and set to work. For hours he labored, silently and efficiently, at transferring enough of their Earthly possessions and stores to render possible an extended period of living aboard the vessel of the Fenachrone. He had completed that task and was assembling the apparatus and equipment necessary for the rebuilding of the power plant before DuQuesne finished the long and complex computations involved in determining the direction and magnitude of the force required to give the Violet the exact trajectory he desired. The problem was finally solved and checked, however, and DuQuesne rose to his feet, closing his

book of nine-place logarithmswith a snap.

**SKYLARK OF VALERON**by E. E. "Doc" Smith (1934)

The government complex made up the greater part of the tunnel-and-vacuole system that honeycombed the subsurface of the asteroid Harmony, that had been the asteroid Perth in the time before the Civil War, before the founding of the Grand Harmony. The chill began to eat its way through his heavy brown uniform jacket; he pushed one hand into his pocket, using the other to push himself along the wall. He was a short man, barely 1.9 meters, and stocky, for a Belter. There was a quality of inevitability about him, and there had been a time when he had endured the cold better than most. But he was a career navy man, and he had spent most of his adult life on ships in space, where adequate heat was the least of their problems...

...Unconsciously he chose a route that took him through the computing center, guided by past habit while he considered the future. The past and the present surprised him as he became aware of his surroundings: of the crowded rows of young faces intent on calculation, or gaping up at his passage.

He looked toward the far corner of the chamber, almost expecting to ﬁnd his own face still bent over a slate of scribbled figures. He had worked in this room, twelve-hundred-odd megaseconds (38 years) ago, starting his career while still a boy as a computer fourth class. A computer in the oldest sense, because the

sophisticated machinery that had borne the Discans’ burden of endless computations had been lost during the Civil War. After the war,the Grand Harmony had learned the hard way that it would never survive without precise data about the constantly changing interrelationships of the major planetoids. And so they had fallen back on human computation, using the inefficient and plentiful to replace the efficient but nonexistent, as they had had to do so many times.A bright child could learn to do the simpler calculations, and so bright children were used, freeing stronger backs for heavier labor. Raul remembered sitting squeezed onto a bench with another boy and a girl, huddled together for warmth. His nose had dripped and his lips were chapped, and he had stared enviously at the back of his half-brother Djem, who was one hundred and fifty megasecs (4.7 years) older and a computer second class. The higher your rank, the closer you sat to the stove in the center of the room ... By the time Djem made first class, Raul had joined him, and been rewarded with warmth and one of the

few hand calculators that still worked.

**THE OUTCASTS OF HEAVEN BELT**by Joan Vinge (1978 )

### Slide Rules

If you take two rulers (with scales ruled off in equal intervals) and slide them edge to edge, you can use it to add or subtract numbers.

Hmmmm, what would happen if the scales were *logarithmic* instead of equal? Then the contraption can be used to easily multiply and divide. You would have a slide rule on your hands.

Slide rules are even quicker to multiply numbers, because you do not have to do the conversion to logarithms and conversion to antilogarithms. The rule handles that automatically.

Besides multiplication and division, fancy slide rules can also handle trigonometry.

Even better, if in your line of work (say, if you were an astrogator), it is possible to make a *specialized* slide rule that solves a specific complicated mathematical equation. As an example, here are the instructions to make your very own Nuclear bomb effects circular slide rule.

Airplane pilots still use E6-B flight Computers, which are part circular slide rule and part analog computers. Just in case of instrument failure.

In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable *(doesn't swell or shrink with the humidity),* strong and naturally self-lubricating. It was only later they were made of celluloid, plastic, or painted aluminium.

Of course I personally would be thrilled to have some sort of hand-waved FTL drive that has the side effect of forcing the use of slide rules. It would be so deliciously retro. I keep trying to come up with one, but so far none my inventions has been free of unwanted side effects. It's hard to think of something that will kill a computer *but not the crew*.

Late breaking news, Karl Gallagher thought of a pretty good reason to use slide rules in his novel Torchship. You can read all about it in the novel.

*Pickett slide rule.**Moon Stick. This is an innovative six-slide sliderule that calculates moon phases. It is currently available from the MoonStick company.**Iron Man uses his built-in slide rule*

*ASA E6-B flight Computer. This is in that gray area between slide rules and analog computers. It is still in production. Most pilots still have an E6-B somewhere in the bottom of their flight bag in case the digital instruments fail.**Mr. Spock prefers the Jeppesen B-1 model of E6-B. From "Who Mourns for Adonais?"*

The

Bygrave slide ruleis a slide rule named for its inventor, Captain L. G. Bygrave of the RAF. It was used in celestial navigation, primarily in aviation. Officially, it was called the A. M. L. Position Line Slide Rule (A.M.L. for Air Ministry Laboratories).It was developed in 1920 at the Air Ministry Laboratories at Kensington in London and was produced by Henry Hughes & Son Ltd of London until the mid-1930s. It solved the so-called celestial triangle accurately to about one minute of arc and quickly enough for aerial navigation. The solution of the celestial triangle used the John Napier rules for solution of square-angled spherical triangles. The slide rule was constructed as two coaxial tubes with spiral scales, like the Fuller slide rules, with yet another tube on the outside carrying the cursors.

During the Second World War, a closely related version was produced in Germany by Dennert & Pape as the HR1, MHR1 and HR2.

## Famous users

Sir Francis Chichester was a renowned aviator and yachtsman. He used a Bygrave Slide Rule as an aid to navigation during flights in the 1930s, one of which was the first solo flight from New Zealand to Australia in a Gipsy Moth biplane. He later completed a round the world cruise in his yacht Gipsy Moth IV. This was the first solo circumnavigation using the clipper route. Sir Francis Chichester wrote about these exploits in his autobiography, entitled The Lonely Sea and the Sky.

## External links

- LaPook, Gary. "Modern Bygrave Slide Rule".
- van Riet, Ronald W.M. "Position Line Slide Rules"(PDF).

**BYGRAVE SLIDE RULE**

But I had seen that article about how Pluto was coming into its "summer." So I knew it was close to the orbit of Neptune now, and would be for the rest of my life—my life expectancy in Centerville; I didn't look like a preferred risk here. That gave an easy figure — 30 astronomical units.

Acceleration problems are simple s=1/2 at

^{2}; distance equals half the acceleration times the square of elapsed time. If astrogation were that simple any sophomore could pilot a rocket ship — the complications come from gravitational fields and the fact that everything moves fourteen directions at once. But I could disregard gravitational fields and planetary motions; at the speeds a wormface ship makes neither factor matters until you are very close. I wanted a rough answer.I missed my slipstick. Dad says that anyone who can't use a slide rule is a cultural illiterate and should not be allowed to vote. Mine is a beauty — a K&E 20" Log-log Duplex Decitrig. Dad surprised me with it after I mastered a ten-inch polyphase. We ate potato soup that week — but Dad says you should always budget luxuries first. I knew where it was. Home on my desk.

No matter. I had figures, formula, pencil and paper.

first a check problem. Fats had said "Pluto," "five days," and "eight gravities."

It's a two-piece problem; accelerate for half time (and half distance); do a skew-flip and decelerate the other half time (and distance). You can't use the whole distance in the equation, as "time" appears as a square — it's a parabolic. Was Pluto in opposition? Or quadrature? Or conjunction? Nobody looks at Pluto — so why remember where it is on the ecliptic? Oh, well, the average distance was 30 A.U.s — that would give a close-enough answer. Half that distance, in feet, is: 1/2 × 30 × 93,000,000 × 5280. Eight gravities is: 8 × 32.2 ft./sec./sec. — speed increases by 258 feet per second every second up to skew-flip and decreases just as fast thereafter.

So — 1/2 × 30 × 93,000,000 × 5280 = 1/2 × 8 × 32.2 x t

^{2}— and you wind up with the time for half the trip, in seconds. Double that for full trip. Divide by 3600 to get hours; divide by 24 and you have days. On a slide rule such a problem takes forty seconds, most of it to get your decimal point correct. It's as easy as computing sales tax.It took me at least an hour and almost as long to prove it, using a different sequence — and a third time, because the answers didn't match (I had forgotten to multiply by 5280, and had "miles" on one side and "feet" on the other — a no-good way to do arithmetic) — then a fourth time because my confidence was shaken. I tell you, the slide rule is the greatest invention since girls.

But I got a proved answer. five and a half days. I was on Pluto.

(Ed note: In Slide Rule terminology: K&E is Keuffel & Esser, noted manufacturer of quality slide rules. 20 inches is twice the size and accuracy of a standard slide rule. Log-log means the rule possesses expanded logarithmic scales. Duplex means there are scales on both sides of the rule and the cursor is double sided. Decitrig means the rule possesses decimal trigometric scales.)

**HAVE SPACE SUIT - WILL TRAVEL**by Robert A. Heinlein, 1958

"Suppose that that zone actually does set up a barrier in the ether, so that it nullifies gravitation, magnetism, and all allied phenomena; so that the power bars, the attractors and repellors, cannot work through it? Then what? As well as showing me the zone of force, you might well have shown me yourself flying off into space, unable to use your power and helpless if you released the zone. No, we must know more of the fundamentals before you try even a small-scale experiment."

"Oh, bugs! You're carrying caution to extremes, Mart. What can happen? Even if gravitation should be nullified, I would rise only slowly, heading south the angle of our latitude—that's thirty-nine degrees—away from the perpendicular. I couldn't shoot off on a tangent, as some of these hop-heads have been claiming. Inertia would make me keep pace, approximately, with the earth in its rotation. I would rise slowly—only as fast as the tangent departs from the curvature of the earth's surface. I haven't figured out how fast that is, but it must be pretty slow."

"Pretty slow?" Crane smiled. "figure it out."

"All right—but I'll bet it's slower than the rise of a toy balloon." Seaton threw down the papers and picked up his slide rule, a

twenty-inch deci-trig duplex. "You'll concede that it is allowable to neglect the radial component of the orbital velocity of the earth, for a first approximation, won't you—or shall I figure that in too?""You may neglect that factor."

"All right—let's see. Radius of rotation here in Washington would be cosine latitude times equatorial radius, approximately—call it thirty-two hundred miles. Angular velocity, fifteen degrees an hour. I want secant fifteen less one times thirty-two hundred. Right? Secant equals one over cosine—um——m—one point oh three five. Then point oh three five times thirty-two hundred. Hundred and twelve miles first hour. Velocity constant with respect to sun, accelerated respecting point of departure. Ouch! You win, Mart—I'd step out! Well, how about this, then? I'll put on a suit and carry rations. Harness outside, with the same equipment I used in the test flights before we built Skylark One—plus the new stuff. Then throw on the zone, and see what happens. There can't be any jar in taking off, and with that outfit I can get back U.K. if I go clear to Jupiter!"

**SKYLARK THREE**by E. E. "Doc" Smith (1948)

Anthor pointed lightly, “I call your attention, Dr. Darell, to the plateau region among the secondary Tauian waves in the frontal lobe, which is what all these records have in common. Would you use my

Analytical Rule, sir, to check my statement?”The Analytical Rule might be considered a distant relation — as a skyscraper is to a shack — of that kindergarten toy, the Logarithmic Slide Rule. Darell used it with the wristflip of long practice. He made freehand drawings of the result and, as Anthor stated, there were featureless plateaus in frontal lobe regions where strong swings should have been expected.

**SECOND FOUNDATION**by Isaac Asimov (1953)

(ed note: A. E. van Vogt is one of the giants of science fiction authors, but in this case I have to file this under "unclear on the concept")

There was no whine of sirens, so it was not a battle alert. He put down his book, slipped into his coat, and headed for astrogation and instrument room. Several officers, including the ship's executive astrogational officer, were already there when he arrived. They nodded to him, rather curtly, but that was usual. He sat down at his desk, and took out of his pocket the tool of his trade:

a slide rule with a radio attachment which connected it with the nearest—in this case the ship's—mechanical brain.

**MISSION TO THE STARS**by A. E. van Vogt (1952)

### Nomograms

The Nomogram (Nomograph) or "Alignment Chart" was invented by the French mathematicians Massau and M. P. Ocagne in 1889. It is a set of scales printed on a piece of paper that will solve a specific equation. Given the all but one of the values for the equation, it will solve for the unknown value. A ruler or straight edge is laid across the scales at the points corresponding to the known values, and the unknown value can be read off directly.

These were very popular with engineers up to about the 1950's. They were quicker than using a slide rule, since they were pre-set for a specific equation. Engineers had entire books filled with nomograms.

They also allowed engineers to off-load some of their donkey-work to assistants and apprentices. The tedious bulk calculations were farmed out by giving each assistant a list of values, some blank paper, and a photocopy of the relevevant equation. The assistant might have shakey math skills but it doesn't take much brain power to lay a straight-edge on a diagram.

I have a tutorial on how to make your own nomographs here.

*Nomogram for the Solar System. Download here.*

As an example, you can play with my handy-dandy DeltaV nomogram. Download it, print it out, and grab a ruler or straightedge. You can also purchase an 11" x 17" poster of this nomogram at . Standard disclaimer: I constructed this nomogram but I am not a rocket scientist. There may be errors. Use at your own risk.

Say we needed a deltaV of 36,584 m/s for the Polaris, that's in between the 30 km/s and the 40 km/s tick marks on the DeltaV scale, just a bit above the mark for 35 km/s. The 1^{st} gen Gas Core drive has an exhaust velocity of 35,000 m/s, this is at the 35 km/s tick mark on the Exhaust Velocity scale *(thoughtfully labeled "NTR-GAS-Open (H2)")*. Now, lay the straightedge between the NTR-GAS-Open tick mark on the Exhaust Velocity scale and the "2" tick mark on the Mass Ratio scale. Note that it crosses the DeltaV scale at about 24 km/s, which is way below the target deltaV of 36,584 m/s.

But if you lay the straightedge between the NTR-GAS-Open tick mark and the "3" tick mark, you see it crosses the DeltaV scale above the target deltaV, so you know that a mass ratio of 3 will suffice.

The scale is a bit crude, so you cannot really read it with more accuracy than the closest 5 km/s. You'll have to do the math to get the exact figure. But the power of the nomogram is that it allows one to play with various parameters just by moving the straightedge. Once you find the parameters you like, then you actually do the math ** once**. Without the nomogram you have to do the math every single time you make a guess.

As with all nomograms of this type, given any two known parameters, it will tell you the value of the unknown parameter *(for example, if you had the mass ratio and the deltaV, it would tell you the required exhaust velocity)*.

Note that the Exhaust Velocity scale is ruled in meters per second on one side and in Specific Impulse on the other, because they are two ways of measuring the same thing. In the same way, the Mass Ratio scale is ruled in mass ratio on one side, and in *"percentage of ship mass which is propellant"* on the other.

Nomograms have an advantage over a raw mathematical equation when it comes to visualizing the range the solution resides in. The value that cannot change becomes the fixed "pivot point", and the straight edge is pivoted to see the various trade-offs. For example:

Download and print out my Transit Time Nomogram.

*Find the delta-v for the given engine and mass ratio on the delta-v nomogram.*

Let's say that our spacecraft is 1.5 ktons *(1.5 kilo-tons or 1500 metric tons)*. It has a single Gas-Core Nuclear Thermal Rocket engine *(NTR-GAS MAX)* and has a *(totally ridiculous)* mass ratio of 20. The equation for figuring a spacecraft's total DeltaV is **Δ _{v} = Ve * ln[R]**. On your pocket calculator, 98,000 * ln[20] = 98,000 * 2.9957 = 300,000 m/s =

**300 km/s**. Ideally this should be on the transit nomogram, but the blasted thing was getting crowded enough as it is. This calculation is on a separate nomogram found here.

The mission is to travel a distance of 0.4 AU *(about the distance between the Sun and the planet Mercury)*. Using a constant boost brachistochrone trajectory, how long will the ship take to travel that distance?

*Find the required engine for the given ship mass and desired acceleration on the Transit Time Nomogram.*

Examine the nomogram. On the Ship Mass scale, locate the 1.5 kton tick mark. On the Engine Type scale, locate the NTR-GAS MAX tick mark. Lay a straight-edge on the 1.5 kton and NTR-GAS MAX tick marks and examine where the edge crosses the Acceleration scale. Congratulations, you've just calculated the ship's maximum acceleration:2 meters per second per second *(m/s ^{2})*.

For your convenience, the acceleration scale is also labeled with the minimum lift off values for various planets.

So we know our ship has a maximum acceleration of 2 m/s^{2} and a maximum DeltaV of 300 km/s. As long as we stay under both of those limits we will be fine.

*Find the required delta-v and travel time for a given distance using the Transit Time Nomogram.*

On the Acceleration scale, locate the 2 m/s^{2} tick mark. On the Destination Distance scale, locate the 0.4 AU tick mark. Lay a straight-edge on the two tick marks and examine where it intersects the Transit time scale. It says that the trip will take just a bit under four days.

But wait! Check where the edge crosses the Total DeltaV scale. Uh oh, it says almost 750 km/s, and our ship can only do 300 km/s before its propellant tanks run dry. Our ship cannot do this trajectory.

*Decrease the acceleration if the required delta-v is too great.*

The key is to remember that 2 m/s^{2} is the ship's *maximum* acceleration, nothing is preventing us from throttling down the engines a bit to lower the DeltaV cost.

This is where a nomogram is superior to a calculator, in that you can visualize a range of solutions. This is the "pivot point" technique I was talking about earlier.

Pivot the straight-edge on the 0.4 AU tick mark (meaning, imagine there is a pin stuck in the nomogram at 0.4 AU that the straight-edge rotates around). Pivot it until it crosses the 300 km/s tick on the Total DeltaV scale. Now you can read the other mission values: 0.4 m/s^{2} acceleration and a trip time of a bit over a week. Since this mission has parameters that are under both the DeltaV and Acceleration limits of our ship, the ship can perform this mission *(we will assume that the ship has enough life-support to keep the crew alive for a week or so)*.

Of course, if you want to have some spare DeltaV left in your propellant tanks at the mission destination, you don't have to use it all just getting there. For instance, you can pivot around the 250 km/s DeltaV tick mark to find a good mission. You will arrive at the destination with 300 - 250 = **50 km/s** still in your tanks.

### Analog Computers

*Analog ballistic computer from naval gun turret.*

You can think of analog computers as "steampunk computers." Probably no actual steam but they will have zillions of gears and cams. It uses using tiny electric motors to drive mechanical shafts and gears. These position shafts to represent some mathematical value, and drive cams shaped to represent mathematical functions or statements. It is used to solve navigational equations.

An example is a ballistics integrator.

If you want the precise details about how to make a computer out of cams, differentials, and gears, read Basic fire Control Mechanisms, OP 1140, (1944). It is available as a free download here. Below are just some of the components.

**Bevel Gear Differential**

It adds and subtracts. The revolutions of input gear one and the revolutions of input gear two are added and spins the output gear a number of revolutions representing the total. Spinning either of the input gears counterclockwise subtracts their value.**flat Ballistic Cam**

It computes a function, such as a trigonometric sine or cosine. The shape of the cam edge encodes the function. The input gear rotates the cam. The roller on the sector follower arm is moved by the edge of the cam. The sector follower then rotates the output gear by an amount equal to the function value.**Cam**

It computes a function, such as a trigonometric sine or cosine. The groove in the cam face encodes the function. The input gear rotates the cam. The groove in the cam forces the follower pin to move back and forth along the track in the follower.**Rack Type Multiplier**

It multiplies*(duh)*. The first input gear moves the input rack. The second input gear moves the pivot arm. The multiplier pin is forced to occupy the intersection of the input rack and the pivot arm. The multiplier pin moves the output rack, which spins the output gear.**Single Cam Computing Multipier**

This is a combination of a cam and a rack multiplier. It takes one input value, computes a function on it, then multiplies it by a second input value. One input gear drives the input rack, the other input gear drives the cam.**Two Cam Computing Multiplier**

It takes one input value, computes a function on it, takes a second input value, computes a different function on it, then multiplies the two results together. One input gear drives the input rack, the other input gear drives the cam. The cams drive the input rack and the pivot arm.

*Barrel Cam. Produces a single output given two inputs. From Basic fire Control Mechanisms, OP 1140, (1944)**Basic fire Control Mechanisms, OP 1140, (1944)*

Lynn Albritton wanted a simple project to help learn how to use her CAD software. Her idea of "simple" is a bit more ambitious than mine. She designed an analog computer that calculates sine and cosine. Called an "Ideal Harmonic Transformer", she has made the blueprints available on Thingiverse so those with access to a 3D printer can make one for their very own.

### Digital Electronic Computers

In many of the Heinlein novels, computers capable of doing interplanetary navigation were not portable. Large computers would pre-compute the courses. And do emergency re-computations when they got a panicked radio message from a ship in trouble.

Remember that early computers are going to give their results by spitting out Hollerith punch cards, punched tape/ticker tape, or printed fanfold sheets. Standard CRT monitors displaying text come later, and monitors with cute graphic user interfaces (such as a maneuver node tool) come later still.

Actually there will still probably be manual equipment, in case the computer gets fried by a solar storm or the EMP from a nearby nuclear weapon detonation. A slide rule will be in a box on the hull, with a sign that says "In case of EMP, break glass."

To me, a machine was something to be mistrusted, checked before use, operated within the limits set forth in an operating manual, and coddled. Omer Astrabadi, the Mad Russian Space Jockey, lived up to his sobriquet. He approached machinery differently. I never saw him run a pre-flight inspection; he strapped into the seat, powered up, and went. I never saw him consult an operating manual; but he knew the limits of the machine. There was no question whatsoever that he was the master of it. He wasn't gentle with it, either. If it didn't do what he wanted, he wasn't afraid to coerce it with violence. Coming home on a flight with him to Dianaport to familiarize me with the

Bacobiclass deep space couriers, an APU power processor quit. Another APU assumed the load, so we didn't lose platform alignment or real-time course line computer tracking."I show you how to fix bad processors," Omer told me and took me to the equipment bay. There he grabbed two protrusions on the bulkhead, braced himself, and directed a solid kick at a panel bearing the label, "CAUTION! Only qualified personnel can repair this unit!"

"When it stops, kick it," Omer told me. "This model stops regularly. I told Ali not to buy from the lowest bidder ..."

"Omer, you might have busted something!" I complained. "We'd play hell getting back without a computer and autopilot!"

He pointed to the read-outs. The unit had picked up its load. "I must train you for commercial operations, Sandy. For years you believe what the Aerospace Force told you."

"I'm still alive because of it."

"In spite of it," Omer corrected me. "I was in

Frontovaia Aviatsiyabefore becoming cosmonaut. We kept aircraft flying under conditions you would not believe. I was taught tomakea machine do what I wanted; if it couldn't, it would tell me.""And kill you in the process."

"Only if I let it." Omer indicated the now-working APU processor. "What would you do?"

"Shut it down and go back to Ell-five on the other. Maintenance would fix it after I got back."

Omer shook his head. "We're short of maintenance people. Sandy, some day your life may depend on fixing something. Now, tell me what would happen if we lost all APU power."

"We'd lose the computer and autopilot."

"Consequences?"

"We might not get back to Ell-five."

"Aerospace Force thinking." Omer pointed to his eyes. "You have two eyes, good guidance system." He tapped his ear. "You have two ears. And you have optical instruments and a working comm unit. Three tracking stations follow us. 'Mayday' call would bring help, but we don't need it. We can astrogate by reference to Earth, Moon, and Sun. Do it." He reached out and shut down both APUs.

I'd been spoiled by high technology. But I made it back to L-5 without having to yell Mayday.

**MANNA**by Lee Correy (G. Harry Stine) 1983

### Redundency

Computers, whether analog or digital, should be of the 'I-tell-you-three-times' variety. It is actually three computers, each of which does the calculation. If operating perfectly, all three answers will be the same. If a malfunction occurs, two answers will agree and one won't. Use the answer the two agree on, which will allow you to get though the burn. *Then* fix the bad computer, pronto! If all three disagree, it's time to break out the slide rule.

Other critical instruments might be in triplicate as well. If you have one clock, you know the time. If you have two clocks, you are never quite sure, since they probably won't agree with each other. But if you have three clocks, you take a reading from the two clocks with values closest to each other, and assume that the actual time is somewhere in between.

(ed note: Nelson Brown is a research aerospace engineer, in aeronautics. Specifically flight controls for fixed wing aircraft. In the quote below he is talking about NASA technology.

Jennifer Linsky mentioned that in damage control and being a military medic, "two is one; one is none." That is, if you need a disposable syringe, take two. If you drop or lose one, you still have one.)

When it comes to flight control computers, two is none. Because if one fails then the two computers disagree, there's no way for the system to know which is right.

You need at least three flight control computers. More complex architectures may have control law software written by independent teams, or even dissimilar hardware. Diversity is robust.

It gets very, very complicated quickly. Between the flight computers are cross-channel data links (CCDL) where each computer tells the others what it observes — including faults in the other computers. There's actually a class of faults called Byzantine faults where the same fault presents different symptoms to different observers. Makes me dizzy. I think the name goes back to Byzantine generals trying to deduce which messengers were lying spies. ;-)

The different manufacturers have different architectures. Lockheed tends to use three-channel systems, while the old McDonnell Douglas fighters have 4 channels. I think Shuttle had that sort of system, plus a 5th dissimilar computer that only had capability for critical flight phases. I don't know what scenarios triggered the 5th computer to be in control of the vehicle.

‘It means that we're all dead,’ Martens answered flatly. ‘Without the computer, we're done for. It’s impossible to calculate an orbit back to Earth. It would take an army of mathematicians weeks to work it out on paper.’

‘That's ridiculous! The ship’s in perfect condition, we've plenty of food and fuel — and you tell me we're all going to die just because we can't do a few sums.’

‘A few sums!’ retorted Martens, with a trace of his old spirit. ‘A major navigational change, like the one needed to break away from the comet and put us on an orbit to Earth, involves about a hundred thousand separate calculations. Even the computer needs several minutes for the job.’

Pickett was no mathematician, but he knew enough of astronautics to understand the situation. A ship coasting through space was under the influence of many bodies. The main force controlling it was the gravity of the sun, which kept all the planets firmly chained in their orbits. But the planets themselves also tugged it this way and that, though with much feebler strength. To allow for all these conflicting tugs and pulls — above all to take advantage of them to reach a desired goal scores of millions of miles away — was a problem of fantastic complexity. He could appreciate Martens’ despair; no man could work without the tools of his trade, and no trade needed more elaborate tools than this one...

...‘This,’ said Dr Martens three days later, ’isn’t my idea of a joke.’ He gave a contemptuous glance at the flimsy structure of wire and wood that Pickett was holding in his hand.

‘I guessed you'd say that,’ Pickett replied, keeping his temper under control. ‘But please listen to me for a minute. My grandmother was Japanese, and when I was a kid she told me a story that I'd completely forgotten until this week. I think it may save our lives.

‘Sometime after the Second World War, there was a contest between an American with an electric desk calculator and a Japanese using an abacus like this. The abacus won.’

‘Then it must have been a poor desk machine, or an incompetent operator.’

‘They used the best in the US Army. But let's stop arguing. Give me a test — say a couple of three-figure numbers to multiply.’

’Oh — 856 times 437.’

Pickett’s fingers danced over the beads, sliding them up and down the wires with lightning speed. There were twelve wires in all, so that the abacus could handle numbers up to 999,999,999,999 — or could be divided into separate sections where several independent calculations could be carried out simultaneously.

‘374072,’ said Pickett, after an incredibly brief interval of time. ‘Now see how long you take to do it, with pencil and paper.’

There was a much longer delay before Martens, who like most mathematicians was poor at arithmetic, called out 375072.’ A hasty check soon confirmed that Martens had taken at least three times as long as Pickett to arrive at the wrong answer.

The astronomer's face was a study in mingled chagrin, astonishment, and curiosity.

‘Where did you learn that trick?’ he asked. ‘I thought those things could only add and subtract.’

‘Well — multiplication’s only repeated addition, isn't it? All I did was to add 856 seven times in the unit column, three times in the tens column, and four times in the hundreds column. You do the same thing when you use pencil and paper. Of course, there are some short cuts, but if you think I ’m fast, you should have seen my granduncle. He used to work in a Yokohama bank, and you couldn't see his fingers When he was going at speed. He taught me some of the tricks, but I've forgotten most of them in the last twenty years. I've only been practising for a couple of days, so I'm still pretty slow. All the same, I hope I've convinced you that there's something in my argument.’

‘You certainly have: I'm quite impressed. Can you divide just as quickly?’

‘Very nearly, when you've had enough experience.’

Martens picked up the abacus, and started flicking the beads back and forth. Then he sighed.

‘Ingenious — but it doesn't really help us. Even if it’s ten times as fast as a man with pencil and paper — which it isn’t — the computer was a million times faster.’

‘I've thought of that,’ answered Pickett, a little impatiently. (Martens had no guts — he gave up too easily. How did he think astronomers managed a hundred years ago, before there were any computers?)

‘This is what I propose — tell me if you can see any flaws in it . . .' Carefully and earnestly he detailed his plan. As he did so, Martens slowly relaxed, and presently he gave the first laugh that Pickett had heard aboard Challenger for days.

‘I want to see the skipper's face,’ said the astronomer, ‘when you tell him that we're all going back to the nursery to start playing with beads.’

There was scepticism at first, but it vanished swiftly when Pickett gave a few demonstrations. To men who had grown up in a world of electronics, the fact that a simple structure of wire and beads could perform such apparent miracles was a revelation. It was also a challenge, and because their lives depended upon it, they responded eagerly.

As soon as the engineering staff had built enough smoothly operating copies of Pickett's crude prototype, the classes began. It took only a few minutes to explain the basic principles; what required time was practice — hour after hour of it, until the fingers flew automatically across the wires and flicked the beads into the right positions without any need for conscious thought. There were some members of the crew who never acquired both accuracy and speed, even after a week of constant practice: but there were others who quickly outdistanced Pickett himself.

They dreamed counters and columns, and flicked beads in their sleep. As soon as they had passed beyond the elementary stage they were divided into teams, which then competed fiercely against each other, until they reached still higher standards of proficiency. In the end, there were men aboard Challenger who could multiply four-figure numbers on the abacus in fifteen seconds, and keep it up hour after hour.

Such work was purely mechanical; it required skill, but no intelligence. The really difficult job was Martens’, and there was little that anyone could do to help him. He had to forget all the machine-based techniques he had taken for granted, and rearrange his calculations so that they could be carried out automatically by men who had no idea of the meaning of the figures they were manipulating. He would feed them the basic data, and then they would follow the programme he had laid down. After a few hours of patient routine work, the answer would emerge from the end of the mathematical production line — provided that no mistakes had been made. And the way to guard against that was to have two independent teams working, cross-checking results at regular intervals.

‘What we’ve done,’ said Pickett into his recorder, when at last he had time to think of the audience he had never expected to speak to again, ‘is to build a computer out of human beings instead of electronic circuits. It's a few thousand times slower, can't handle many digits, and gets tired easily — but it's doing the job. Not the whole job of navigating to Earth — that’s far too complicated — but the simpler one of giving us an orbit that will bring us back into radio range. Once we've escaped from the electrical interference around us, we can radio our position and the big computers on Earth can tell us what to do next.

**INTO THE COMET**by Arthur C. Clarke (1960)