Sunday, August 01, 2010

To Ceres

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Ceres optimized.jpgThe first Asteroid Belt object discovered was named after Ceres, Roman goddess of growing plants, the harvest, and motherly love.

Ceres, the dwarf planet, turns out to be the largest known object in the Asteroid Belt (AB) between Mars and Jupiter. It is also acknowledged as the smallest known dwarf planet in the Solar System.

Someday, humankind will go there as well as many other AB objects.

EarthFile:Earth symbol.svg.983 AU1.0167 AU.017 1.0 Yr
MarsFile:Mars symbol.svg1.37 AU1.67 AU.093 1.88 Yr
CeresCeres symbol.svg2.54 AU2.987 AU.080 4.6 Yr
JupiterAstronomical symbol of Jupiter4.95 AU5.46 AU.048 11.9 Yr

Perihelion, q, is body's closest approach to Sol. 

Aphelion, Q, is orbit's furthest point from Sol.

Eccentricity, c, measures orbit's circularity.

Period, T, is orbit duration in Earth years.

Orbit of Ceres. Ceres's orbit falls between Mars and Jupiter, within the main asteroid belt, with a period (T) of 4.6 Earth years. Orbit of Ceres is moderately inclined to the ecliptic (inclination angle, i, is 10.6° compared to 7° for Mercury and 17° for Pluto) and moderately eccentric (e = 0.08 compared to 0.09 for Mars).

Rotational period of Ceres (Cererian day) is 9 hours and 4 minutes.

Physical characteristics. With a diameter of about 950 km, Ceres is by far the largest body in the Asteroid Belt (AB). By observing influence Ceres exerts on neighboring asteroids, scientists estimate the mass of Ceres as approximately 9.4 × 1020 kg which is about a third (32%) of the total AB mass, 3.0 × 1021 kg. Total AB mass is about four percent of the mass of Luna, Earth's moon; thus, Ceres is about 1% size of Luna.

Spherical Shape. Ceres' size and mass give it a nearly spherical shape (i.e. close to hydrostatic equilibrium); thus, Ceres is classified as a dwarf planet rather than an asteroid. In contrast, other large asteroids, such as 2 Pallas, 3 Juno, and 10 Hygiea, have irregular shaped bodies with lower gravity.

Internal Structure. Ceres likely has a rocky core overlain with an icy mantle. This 100 km-thick mantle (23–28 percent of Ceres by mass; 50 percent by volume) contains 200 million cubic kilometres of water, more water than all of Earth's fresh water. Such a resource could make Ceres an important port for interplanetary travelers.

Composition. Ceres appears to be geologically inactive with a surface sculpted by impacts. The presence of significant amounts of water ice in its composition raises the possibility that Ceres has or had a layer of liquid water in its interior. This hypothetical layer is often called an ocean. If such a layer of liquid water exists, it is believed to be located between the rocky core and ice mantle like that of the theorized ocean on Europa. The existence of an ocean is more likely if dissolved solutes (i.e. salts), ammonia, sulfuric acid or other antifreeze compounds are dissolved in the water.

Atmosphere. Ceres might even have a weak atmosphere and water frost on the surface. Significant surface ice is unlikely, since water ice will sublime if exposed directly to solar radiation (less than 5 AU from Sol). Thus, water vapor is even less likely; if it's there, it would be difficult to detect.


More than half the mass of the main belt is contained in the four largest objects: 1 Ceres, 4 Vesta, 2 Pallas, and 10 Hygiea. They have an average diameter of more than 400 km, while Ceres, the main belt's only dwarf planet, is about 950 km in diameter.

Longitude of the ascending node (symbol: Ω) is the angle from the origin of longitude to the direction of the ascending node. Ω is measured in plane of the ecliptic .

Argument of perihelion (symbol: ω) is is the angle between the orbit's perihelion (point of closest approach to Sol) and the orbit's ascending node (point where the body crosses the ecliptic from South to North). Angle is measured in the direction of motion.

True anomaly (symbol: ν) is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of perihelion and the current position of the body, as seen from the Sol.

A common misconception is that the AB is chock full of boulders and spacecraft is constantly weaving back and forth to avoid colliding with them. While there are thousands of AB objects, there's a lot of space in outer space, and numerous spacecraft can fly random paths through the AB with a tiny probability of hitting one. Within the belt, asteroid material is thinly distributed; as a matter of fact, multiple unmanned spacecraft now routinely cross it without incident.

Individual asteroids within the main belt are categorized by their spectra, with most falling into three basic groups: carbonaceous (C-type), silicate (S-type), and metal-rich (M-type). Such classifications will help future Enterprises determine which are best suited for mining or for other purposes.

Other regions of small solar system bodies include the centaurs, the Kuiper belt and scattered disk, and the Oort cloud.

Fifteen months later, Olbers discovered a second object in the same region, Pallas. Unlike the other known planets, the objects remained points of light even under the highest telescope magnifications, rather than resolving into discs.

However, for several decades it remained common practice to refer to these objects as planets.[5] By 1807, further investigation revealed two new objects in the region: 3 Juno and 4 Vesta.[13] The Napoleonic wars brought this first period of discovery to a close,[13] and it was not until 1845 that another object (5 Astraea) was discovered. Shortly thereafter new objects were found at an accelerating rate, and counting them among the planets became increasingly cumbersome. Eventually, they were dropped from the planet list as first suggested by Alexander von Humboldt in the early 1850s, and William Herschel's choice of nomenclature, asteroids, gradually came into common use.

Observing AB near Ceres.

Asteroid icons show relative density of objects in AB. An accurate "to scale" diagram would lots of empty space as well as a pin sized Sun. TE's g-force ship could fly numerious random paths throughout AB with no collisions. However, there are many thousands of objects available in AB which can be detected and collected for spaceborne habitat purposes.

Largest Asteroid in the Belt.
The dwarf planet Ceres is the only object in the AB large enough for its gravity to force it into a roughly spherical shape. Ceres has a much higher absolute magnitude than the other asteroids, of around 3.32, and may possess a surface layer of ice. Like a typical planet, Ceres is differentiated: it has a crust, a mantle and a core.

Vesta, too, has a differentiated interior, though it formed inside the Solar System's "snow line", and so is devoid of water; its composition is mainly of basaltic rock such as olivine.

Due to the low density of materials within the belt, the odds of a probe running into a random asteroid are now estimated at less than one in a billion.

Why go???

Resources are the reason.
Ceres is one of many thousands of asteroids between the orbits of the planets Mars and Jupiter (Asteroid Belt, AB ). AB is one of several such regions in the Solar System such as Near Earth Asteroids (NEAs) and Trojan asteroids (in Jupiter's orbit) not to mention the Kuiper Belt which probably has many comets as well.

Term: "Asteroid". Since they resemble stars, William Herschel suggested "asteroids" after the Greek "asteroeides" (star-like).

Expression "Asteroid Belt" first used by Alexander von Humboldt in Cosmos[1852] "...a belt of asteroids ... moving with planetary velocity". Other early appearances include Robert James Mann's A Guide to the Knowledge of the Heavens, "The orbits of the asteroids are placed in a wide belt of space...".

How Many? One hundred asteroids had been located by 1869. One thousand by 1923, 10,000 by 1951, 100,000 by 1982 and still counting.

What Materials are Available?AB contains following object classes:

  • C-type (carbonaceous) asteroids comprise over 75% of the AB's visible asteroids. They are carbon-rich and dominate the belt's outer regions; composition is similar to carbonaceous chondrite meteorites.
  • S-type (silicate) asteroids are more common toward the inner region of the belt (within 2.5 AU of the Sun). This type comprises about 17% of the AB population.
  • M-type (metallic) asteroids form about 10% of the total population; their spectra resembles iron-nickel. These were possibly formed from the metallic cores of collided planetoids.
  • V-type (basaltic) asteroids are very rare. These are thought to have broken from Vesta (a large asteroid) long ago.
  • Main-belt Comets?? Several objects in the outer belt show cometary activity. Perhpas some outer asteroids occasionally expose covered ice to sublimation after small impacts.

Pre-discovery. Incredibly, the discovery of Ceres was actually anticipated. Thence, the idea of an undiscovered planet between the orbits of Mars and Jupiter was proposed by astronomer Johann Daniel Titius in 1766.

In his translation of Charles Bonnet's Contemplation de la Nature, Titius added a footnote which noted a pattern in the planet distribution.

Titius-Bode's Law works well (thanks to Ceres) from Mercury to Uranus.
Axis (a)

0.387 AU



0.723 AU



1.000 AU



1.524 AU


CeresCeres symbol.svg

2.7663 AU



5.203 AU



9.537 AU



19.191 AU



30.069 AU


However, Neptune clearly breaks the pattern.

Let initial number be .4;

then, compute subsequent numbers:

.4 + (.3 x 2N)

where N = 0, 1, 2, ... .

This pattern, now known as the Titius-Bode Law, accurately accounted for the semi-major axes for then known six planets (Mercury, Venus, Earth, Mars, Jupiter and Saturn) given a "gap" between the orbits of Mars and Jupiter.

Titius: "Should the Lord Architect have left that space empty? Not at all!"

In 1772, Johann Elert Bode also noted this pattern in the semi-major axes of the then known planets; he also noted a missing planet between Mars and Jupiter. The pattern predicted a semi-major axis near 2.8 AU for the missing planet's orbit. Bode noted this in his Anleitung zur Kenntniss des gestirnten Himmels, but did not credit Titius; thus, many now call it "Bode's Law".

Bode's Law gained even more credibility in 1781, when William Herschel discovered Uranus near the predicted distance for the next body beyond Saturn; this further increased confidence in the T-B Law, and astronomers concluded a planet must exist between the orbits of Mars and Jupiter.

Eventually, the discovery of Neptune in 1846 discredited the Titius-Bode Law in the eyes of scientists, as its orbit was nowhere near the predicted position. To date, there is no scientific explanation for the law, and most astronomers consider the pattern to be purely coincidental. However, back in the late 1700's, the T-B Law was still held to be inevitable.

In 1800, astronomer Baron Franz Xaver von Zach recruited 24 colleagues into an informal club, the "Lilienthal Society". Determined to demonstrate order in the Solar System, the group became known as the "Himmelspolizei", or Celestial Police. Notable members included Herschel, British astronomer Royal Nevil Maskelyne, Charles Messier, and Heinrich Olbers. Each astronomer was assigned a 15° region of the Zodiac to search for the missing planet.

We now know that each astronomer could have found plenty of objects. However, back then, the scopes weren't as good as now, and they didn't know what to look for.

Discovery. As a matter of fact, an outsider was destined to make the discovery. On the first day of the Nineteenth Century (January 1, 1801), Ceres was discovered by an Italian Catholic Priest; an irony appreciated by those familiar with the story of Galileo Galilei, an Italian Catholic scientist wrongly imprisoned by the Inquisition for merely suggesting the nonuniformity of God's Universe.

Of course, Giuseppe Piazzi was not only a devout Catholic priest (of the Theatine order), he was also a highly educated mathematician as well as an astronomer; he was Chair of Astronomy at the University of Palermo, Sicily. He established an observatory at Palermo, now the Osservatorio Astronomico di Palermo.

Piazzi initially classified this tiny moving star-like object it as a comet, but the missing coma led him to reconsider. To astronomer Barnaba Oriani of Milan, Piazzi wrote: "I have announced this star as a comet, but since it is not accompanied by any nebulosity and, further, since its movement is so slow and rather uniform, it has occurred to me several times that it might be something better than a comet. But I have been careful not to advance this supposition to the public." On 24 January 1801, he announced his discovery via letters to both Orianni and Bode.

From his journal, Piazzi: "(Initially,) I had no doubt of its being any other than a fixed star. In the second evening, I repeated my observations, and having found that it did not correspond either in time or in distance from the zenith with the former observation, I began to entertain some doubts of its accuracy. I conceived afterwards a great suspicion that it might be a new star. The evening of the third, my suspicion was converted into certainty, being assured it was not a fixed star. Nevertheless before I made it known, I waited till the evening of the fourth, when I had the satisfaction to see it had moved at the same rate as on the preceding days."

Unfortunately, Ceres soon moved too close to the Sun's glare for other astronomers to confirm Piazzi's observations. Piazzi was able to record the object's changing positions during 19 observations made over 42 days. On February 12, the object disappeared in the glare of the sun, and could no longer be observed. During the whole period, the object’s total motion made an arc of only 9° on the celestial sphere. Orbital prediction methods then available were unable to accurately predict when/where Ceres would reemerge out of the Sun's glare.

How go???

Step 1: Flight profile to Mars.
Described in detail in previous chapter, To Mars.

Aries at Opposition
every 758 days
Shortest distance from Earth to Mars (.5AU); thus, quickest time.

Flight Profile Range: 0.5 AUFlight Profile Range: 1.0 AU
.5 AU.25AU1 day864 km/s1 dy2 dy
1.0AU.5AU1.41 dy1,222 km/s1.41 dy2.8 dy
Givend/2√(2(d/2)/g)g * tAcc √(d/g)2√(d/g)
864 km/s

tA+ tD
In general, a g-force profile involves following:

  • Acceleration phase: Constantly accelerate from departure to midpoint to simulate Earth surface gravity as well as gain considerable speed; so much speed, the ship must slow down to orbital velocity required for destination.
  • Midpoint Velocity (VM) is also the Maximum Velocity because g-force ship must start slowing down midway between departure and destination; this must be done at midway to maintain constant g-force both speeding and slowing.
  • Deceleration phase: Constantly decelerate from midpoint to destination to decrease speed and continue to simulate Earth surface gravity.

Thus, TE assumes for initial g-force flights to Mars, that maximum 1-way range will be .5 AU, the minimum distance ever achieved between Earth and Mars during their orbits.

Since opposition happens only once every two years, above 2 day profile (d = 0.5 AU) gets us to Mars only once every 2 years, a very infrequent opportunity.
However, this chapter, To Ceres, assumes technology advances a few years after the initial flights to Mars. Thus, TE assumes increased range of 1.0 AU. An immediate benefit from this extended range: able to access Mars during much greater portion of Mars's orbit. (See details.)

From Mars, prepare for AB exploitation via following tasks.
1. Mankind establishes habitats on Mars near Phobos and Deimos.
2. Spend several orbits gathering data about most lucrative targets in AB.

During this preparation, Ceres might prove useful as a reference point to describe locations of other AB objects.

Step 2: Extended Flight profile to Ceres and Friends.

G-force profile is extended by adding a "Cruise" phase between Acceleration and Deceleration.

Cruise phase accomodates difference between maximum range and actual distance to destination by putting g-force propulsion on stand-by and traveling at constant velocity.

In following example, 1.4 days of g-force acceleration brings ship to an enormous velocity (over 1,200 km/sec); thus, the zero gravity c0ndition will last only for a few hours. Of course, with propulsion on hold, the fuel consumption is zero.

Extended Flight Profile Example
Example: assume max g-force range: 1.0 AU;
however, destination is 1.3 AU.



0-g Cruise

+ Decel=DestTtl




+ dDec=


.5 AU .3 AU .5 AU 1.3 AU
RMax/2 Dest-RMax RMax/2 Σ




+ tDec=


1.414 dy 0.424 dy 1.414 dy 3.252 dy
√(2dAcc/g) dCru/(g*tAcc) √(2dDec/g) Σ




+ fDec=


0.45 %GW 0 %GW 0.45 %GW 0.9 %GW
Determine total fuel Σ
Determine cruise distance (dCru) by subtracting maximum range from distance to destination. For example, if dDest = 1.3 AU, and RMax = 1.0 AU, then dCru = 1.3 AU - 1.0 AU = .3 AU.
Determine total time to destination:



dCru/(g * tAcc)

+ √(2dDec/g)= tDest

To maintain same g-force as much as possible, deceleration time/distance must equal acceleration t/d. However, g-force is turned off during the cruise phase. Fortunately, this is only for a few hours for flights from Mars to nearby AB objects.
Determine total fuel to destination tDest as described below. TE assumes that mass of g-force spacecraft can range over a wide amount; thus, TE opts to describe fuel requirements in terms of percentage gross weight (%GW) of the ship itself. Thus, regardless of ship's mass, fuel mass can easily be determined.

AB contains many thousands of targets with an infinite mix of compositions; thus, extended profile puts Mars within range of several AB objects.

Restate Step 2. Gather Data. From Mars, spend several orbits determine the most lucrative targets in AB. While new objects will likely be discovered during this phase; primary purpose is to prioritize all the accessible objects as to which objects present best value.

Step 3. Initial g-force Ships. Send g-force ships via modified profile as described. Note that g-force ships will need to minimize loiter time at AB targets because Martian home base is rapidly slipping away.
---a. First few trips will be to drop off AI (Artificial Intelligence) entities (i.e., robots) to attach to target objects and perform analytical tasks.
---b. Next set of trips will be to drop off small habitats (i.e. "Island One" type modules from O'Neill's book) with perhaps 100 people. These hardy souls will leverage data collected by AI devices and will prepare payloads for subsequent trips by g-force ships.
---c. Next set of trips will be to drop off payloads of more people and supplies and perhaps additional habitat modules and pick up payloads of returnees and large packets of asteroidal/cometal commodities for use in other habitats.
Step 4. Return payloads of asteroid materials could be transported back to Mars for immediate use by habitats Deimos and Phobos as well further carried back to Earth's vicinity for use by habitats which might be orbiting Earth, orbiting Luna, or even co-orbiting Sol in positions L-4/5 (see descriptions of habitats Alpha and Omega).

In main belt, large asteroids demonstrate rotation rates much less then expected for solid bodies of same volume. Analysis suggests that most large asteroids (d > 100 m) are most likely rubble piles formed through accumulation of debris after many collisions with other asteroids.

Thus, it should be relatively easy to collect asterioid pieces (i.e. boulders or large rocks) and transport them back to habitats at Mars, Earth, or even habitats co-habiting Earth's orbit away from Earth's positions.

Re-discovery. Due to the brevity of Piazzi's observations, existing methods were unable to compute orbit of Ceres. The renowned mathematician, Carl Friedrich Gauss, then 24 years old, developed a new method of orbit calculation to help astronomers relocate Ceres. In only a few weeks, Gauss developed new comprehensive approximation methods and determined an orbital path for Ceres ; thus, he sent his results to von Zach.

Gauss supplied a crucial “correction factor,” for first-approximation of Ceres’ position. For
some one hundred fifty years, following the publication of his Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, astronomers around the world have used Gauss's method to calculate the orbits. (How Gauss Determined The Orbit of Ceres by Tennenbaum and Director)

After three months of intense work, the Gauss's prediction turned out to be accurate within a half-degree. Gauss so streamlined the cumbersome mathematics of 18th century orbital prediction that his work (Theory of Celestial Movement) remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant and contains an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss was able to prove the method in 1809 under the assumption of normally distributed errors .

On 31 December 1801, von Zach and found Ceres near the predicted position and thus recovered it. One day later, Heinrich W. M. Olbers in Bremen confirmed this with his own observation.

Zach: "without the intelligent work and calculations of Doctor Gauss, we might not have found Ceres again."

After its orbit was better determined, it was clear this object was not a comet but more like a small planet. Coincidentally, it was also almost exactly where the Titius-Bode law predicted, (i.e., semimajor axis, a = 2.8 AU).

Ceres (bottom left), the Moon and the Earth, shown to scale. For half a century, it was classified as the eighth planet much like Pluto used to be known as the ninth planet til the end of the 20th Century.

Bode believed Ceres to be the "missing planet" between Mars and Jupiter, at 2.8 AU from the Sun. Indeed, Ceres was considered a planet in astronomy books and tables for about half a century until many more asteroids had been discovered. Ceres is now known to be by far the largest body within the Asteroid Belt; Ceres is today classified as a dwarf planet due to its size and other characteristics.

Changing Status of Ceres. Ceres was known as a planet for many years, then as asteroid number 1, and eventually, a dwarf planet.

Eventually, Ceres was acknowledged as the first of a class of many similar bodies. Thus, it was given the designation 1 Ceres under the modern system of asteroid numbering.

The 2006 debate about Pluto's status as a planet led to Ceres being considered for reclassification as a planet. To define a planet, the International Astronomical Union initially considered (6 August 2006) following two criteria:

(a) Celestial body has sufficient mass for its gravity to overcome rigid body forces and assume a hydrostatic equilibrium (nearly spherical) shape.

(b) Celestial body orbits a star, and is neither a star nor a satellite of a planet.

Since Ceres satisfies above two criteria, initial resolution would have made Ceres the fifth planet from the Sun. However, above resolution was not accepted; instead, the final resolution (24 August 2006) added following criterion:

(c) Celestial body has cleared the neighborhood around its orbit.

Criterion, (c), disqualifies Ceres as a planet because it shares its orbit with the thousands of asteroids in the main belt and is now classified as a "dwarf planet".

Further distinguishing Ceres from the asteroids is its composition. Ceres is observed to be a solid, spherical body with perhaps considerable sub-surface water. Most asteroids are collections of rubble.

Water, water, everywhere...

Thought Experiment (TE) assumes about 1% of ship's original gross weight is sufficient fuel to propel ship at g-force for almost three days.

With current technology, large aircraft easily require a third of their TOGW to fly for a very long flight (perhaps 10 hours). Thus, only 1% TOGW seems like a incredibly, great deal; however, cautious optimism compels us to wonder about the trade-off.

TE further assumes water will be the source of exhaust particles; thus, water will be superheated, then transformed into ions, accelerated to .866c then allowed to exit vessel.

This constant exhaust of water ions could have sufficient momemtum (due to very high velocity of particles as well as enhanced momentum due to relativistic growth of particle size).

Trade offs are challenging.

1. First and foremost, the implementing technology is doable but far from done. TE assumes humankind will prove equal to the technological challenge, but another challenge might even be greater.

2. : "Where will the fuel come from?" One percent is a tiny percentage, but even a tiny percentage of a huge quantity can be a lot.

Assume g-force vessel is same size as modern cruiseship; this could be about 300,000 metric Tonnes (mTs). Thus, 3 days of g-force would need about 3,000 mTs of water.

Conservative fuel plan would not count of refueling being possible; thus, another 3,000 mTs for return trip; plus another 3,000 mTs for margin and peripheral uses, (water consumption by humans and other onboard life forms, perhaps even showers, etc.)

FYI, 3,000 mTs of water is enough for one Olympic sized swimming pool.

%GW0 =0.283%/day
* t * ε

%GW0 is the % of ship's initial gross weight (at zero days of travel) which must be dedicated to fuel. (%GW0 is also known as percent TakeOff GW, %TOGW).

TE assumes the constant, .283%/day, based on previous work.

Time, t, is in days.

mr , relativistic growth factor, is further explained below:

mr =1
√(1 - d2)

Let d be the exhaust particle's speed in relation to light speed, c.

TE assumes ship's propulsion system accelerates exhaust particles to a speed of 86.6 % light speed; thus, d = .866.

At this speed, Lorentz Transforms says Einsteinian relativistic effects doubles mass of .exhaust particles. and mr = 2

mr = 2 = 1

√(mr2-1) = √3 = 1.732

%TOGW = ε * .1636%/day

Finally, TE assumes efficiency factor, ε, will account for likely loss of fuel particles due to inevitable design flaws as well as peripheral energy needs. For this case, TE arbitrarily assumes ε = 2. Actual efficiency will need to be determined and will no doubt improve as g-force technology advances.

%TOGWDay =0.283%/day
* 1 dy * 2= .327 %/day

Finally, TE concludes that %TOGW is inversely proportional to particle's exhaust speed. As particle speed increases, %TOGW decreases; thus, less fuel is needed for same day's performance.

Discoverer of Ceres

In July 1770, Piazzi became the chair of Mathematics at the University of Malta.
19 January 1787, he became Professor of Astronomy at the University of Palermo. Piazzi became acquainted with the major French and English astronomers.
Jan 1, 1801, Piazzi discovered Ceres. Piazzi initially thought this to be a good opportunity to express his gratitude to his generous sponsor, and he named it "Ceres Ferdinandea," after the Roman and Sicilian goddess of grain and his sponsor, King Ferdinand IV of Naples and Sicily. However, citizens of other countries were not so enthusiastic about the Ferdinandea part which was later dropped.
In 1817, King Ferdinand put Piazzi in charge of the completion of the Capodimonte (Naples) Observatory, naming him General Director of the Naples and Sicily Observatories.
Star cataloguing. He supervised the Palermo Catalogue of stars, 7,646 star entries with unprecedented precision, The work on this catalogue was started in 1789, enabling Piazzi and collaborators to observe the sky methodically. First edition publication was complete on 1803. Piazzi was probably working on this project when he discovered Ceres.

Spurred by the successes of discovering Ceres and his catalogue program, Piazzi conducted some research into stellar parallax measurement. One candidate, 61 Cygni, was specially appointed as a good candidate for measuring a parallax later performed by Friedrich Wilhelm Bessel.

Prince of Mathematicians
Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and scientist with remarkable influence in many fields of mathematics and science and is one of history's most influential mathematicians. Gauss called Mathematics "the Queen of Sciences."

At a young age, Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig, who sent him to the Collegium Carolinum (now Technische Universität Braunschweig) from 1792 to 1795; then, the University of Göttingen from 1795 to 1798.

In 1796 (Gauss's "golden year"), Gauss discovered a number of number theory theorums which are still appreciated today.

In 1809, Gauss published his work on the motion of planetoids disturbed by large planets, Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of Motion of the Celestial Bodies Moving in Conic Sections Around the Sun).

Gauss was a child prodigy. Of many relevant anecdotes, perhaps the most famous is from his primary school class where Schoolmaster Herr Buttner and his assistant Herr Martin Bartels tasked young Gauss's class to add all the integers from 1 to 100 in arithmetic progression.

In a misguided effort to demonstrate humility to the young Johann Gauss, Herr Buttner decided to teach the brilliant young man a lesson. Herr Buttner directed entire class to correctly summ all the integers from 1 to 100. It was expected that at 5 seconds per addition, accomplishing 100 cummulative additions would take perhaps 500 seconds (8 min + 20 sec). Given that some students are quicker then others, one would expect the exercise to take the class perhaps 10 minutes.

1+100 =101
2+99 =101
3+98 =101
50+51 =101
50*101 =5,050
S=Σ1N=N (N+1)/2
Given propensity for human error, it was very likely that after this 10 minutes, many answers would be incorrect, causing confusion and even more delay.

Herr Buttner and his assistant were greatly surprised when young Gauss determined the correct answer in 10 seconds, well before anyone else had even got beyond first few summations.

Gauss's presumed method involved pairwise addition of terms from opposite ends of the list to obtain identical intermediate sums. Thus, a total sum of integers from 1 to 100:

S = 100×(100+1)/2 =50×(101) = 5,050

Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855)
In his honor, the lunar crater Gauss and Asteroid (1,001) Gaussia. The CGS unit for magnetic induction is called a gauss in his honour.
Giuseppe Piazzi (July 16, 1746 - July 22, 1826)
In 1923, the 1000th asteroid to be numbered was named 1000 Piazzia in his honor. The lunar crater Piazzi was named after him in 1935. Recently detected crater on Ceres, has been informally named Piazzi. Star system, 61 Cygni, is sometimes called Piazzi's Flying Star .

Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World: a Novel in 2006, explores Gauss's life and work through a lens of historical fiction, contrasting it with the German explorer Alexander von Humboldt.
Things named in honour of Gauss include:

==================Bode & Titius===================

Johann Elert Bode (January 19, 1747 - November 23, 1826)
In his honor, Asteroid (998) Bodea and the Bode Moon Crater are named after him. Also, Galaxy M81 (his discovery) is popularly known as "Bode's Nebula" or "Bode's Galaxy".
Johann Daniel Titius (January 2, 1729 - December 11, 1796)
In his honor, the Asteroid (1998) Titius and the Titius lunar crater are named after him.

Sidereal Period

If a heavenly observer positioned herself well above (i.e. "north" of) the Solar System, she would see many planets orbiting in a counterclockwise direction. Given a fixed reference (example: Line of Aries from Sol to a specific point of Earth's orbit attained every year at Vernal Equinox. This line continues to a specific point in the Sagittarius Constellation.), a heavenly host member (or perhaps the SOHO satellite) would could measure all planets crossing this line in a consistent duration, or period, T.)

Computing T is a straightforward process directly from Kepler's Third Law.

T √(a3) = a3/2


k a3/2

T =

2 π (a3/μ)

k =

2 π /(μ)

Standard Gravitational Parameter, μ,
product of G, universal gravitational constant,
and MSol, mass of Sol.

μ = G * MSol=6.667x10-11N*m2
Newton (N) = kg * m /sec2; thus, Standard Gravitational Parameter, μ(= G*MSol), units reduce to m3/sec2


=13.2x1019 m3
=13.2x1010x(1,000 m)3
Convert cubic meters (m3) to cubic kilometers (km3).


=13.2 x 1010 km3
Apply factors to convert kilometers to AUs
and seconds to years.
μ = 39.275 AU3/yr2
√μ = (6.267 AU √AU)/yr
k = 2 π /√μ
k = 6.28 yr/(6.267AU3/2)
k = 1.003 yr/AU3/2

Quickly Approximate.
Input semimajor axis, a, in AUs;
quickly compute period, T, in years.

a = 1.00000011 AU


k a3/2


1.003 yr

* (1.00000011AU)3/2

1.0 Yr

a = 1.52366231 AU

k a3/2


1.003 yr

* ( 1.52366 AU)3/2

1.886 yr

a = 2.7663 AU

k a3/2


1.003 yr

*(2.7663 AU)3/2

4.615 yr

Synodic Period

Define Opposition. During opposition of Mars, Sun and Mars are directly opposite each other with Earth directly between. Thus, Sun and Mars form a straight line such that some Earth bound mortals on the dark side of Earth could observe Mars directly overhead (aka "zenith") at exactly midnight. This event (aka "Opposition") occurs at regular intervals known as Synodic Periods.

Synodic period (P) is the duration between successive oppositions.

Concentric rings with increasing velocities as rings increase radius, distance from center.

An Earthly observer sees other planets in relation to his point of view. We can compute P by comparing daily angular velocities of Earth and Mars. This differs from sidereal period (T) which is what a heavenly observer would measure while observing Earth and Mars from a fixed reference (since as line of Aries).

ωEarth = 360°/365.26 days = .999°/day


1.0 Yr

= 365.25 days
ωEarth = 360°/365.26 days = .999°/day


1.886 yr

= 688.9 days

ωMars = 360°/687.05 days = .524°/day


=ωEarth -ωMars
.475°/day =.999°/day -.524°/day


=360°/ωΔ =757.9 days


1.0 Yr

= 365.25 days


4.615 yr

= 1,685.7 days

ωCeres = 360°/1,685.7 days = .214°/day


=ωEarth -ωCeres
.790°/day =.999°/day -.214°/day


=360°/ωΔ =455.7 days
Determine Angular Velocities.
Next, determine relative angular velocity to determine Earth's synodic period for Mars.

To Earthbound observers, Ceres appears to be rotating clockwise at .79°/day, diffence between their sidereal angular velocities

In like manner, one can compute Mars's Synodic Period of Ceres. During opposition of Ceres and Mars ; Sun, Mars and Ceres form a line and Mars residents would observe Ceres directly overhead (aka "zenith") at exactly midnight (Mars local time).

Time between successive oppositions of Ceres is the synodic period of Ceres. We can compute P by comparing daily angular velocities of Mars and Ceres from sidereal orbital periods (T). Note all days refer to Earth days.

ωMars = 360°/687.05 days = .524°/day

ωCeres = 360°/1,722 days = .214°/day


=ωMars -ωCeres
.310°/day =.524°/day -.214°/day


=360°/ωΔ =1,161.3 days

Mars bound observers would see a different synodic period for Ceres. (Note: all days refer to standard Earth days.)

Every 456 days, Earth-Ceres opposition brings them to about 1.8 AU apart, as close as they'll ever get. Thus, a range of 1.8 AU would enable a spaceship to make this multi-day trip every two years. However, TE artificially assumes for this chapter that humankind technology has only expanded g-force performance envelope to increase range to max of 1.0 AU (recall previous chapter, "To Mars", TE assumed max range of only .5 AU (barely enough to reach Mars at opposition).

This expanded range does not get our g-force spacecraft to Ceres, but what does it do for us?

1. Range of 1.0 AU enables spacecraft to fly at g-force to an expanded portion of Mars's orbit. Thus, spacecraft can reach Mars from 43° prior to opposition until Mars reaches 43° past opposition.

2. Can reach nearest members of AB.

However, a range of 1.0 AU still does not get us to Ceres unless one innovates as mankind tends to do.

1. Assuming 1 way range of 1 AU means that ship can travel thru four phases describes in normal 2 way trip,

2. If available volatiles exist at Mars (hopefully in Deimos and/or Phobos which makes them very available), then ship can REFUEL at Mars and extend 1 way range for another 1.0 AU.

This greatly extended range gets us to within range of many AB bodies, but not quite Ceres. At best, this gets us to .3 AU from Ceres, still not quite enough range.

3. Unless we modify flight profile as follows:

Accelerate to the max for .5 AU, 1.414 days and achieves .707 AU/day.

Turn off propulsion and save fuel until ship reaches deceleration point .5 AU from Ceres (destination) and then apply propulsion vector to decelerate ship for 1.414 days and decrease velocity to operational speed at destination.

In the middle of this profile there will need to be some zero gravity (0-g) time. Assuming travel from Mars to Ceres at their opposition, total dist would be 1.3 AU, and ship would travel at 0-g, .707 AU/day (about 600 km/sec) for .3 AU (about 10 hours, an inconvenience but not a health hazard by any means).

Typical Flight Profile From Mars to AB Object



9.865 m/sec2


0.5 AU/dy2


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