Is the SOI a spherical region or a oblate-spheroid-shaped region?

Is the SOI a spherical region or a oblate-spheroid-shaped region?

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The Wikipedia article on Sphere of influence states that:

"A sphere of influence (SOI) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body." (Emphasis added.)

It then gives an expression for the radius of the sphere of influence.

Is the SOI a spherical region or a oblate-spheroid-shaped region? If it is an oblate-spheroid-shaped region, then why?

Is the SOI a spherical region or a oblate-spheroid-shaped region?

The sphere of influence is neither a sphere nor an oblate spheroid. It is a surface with no name. An approximation of this surface is

$$left(frac r R ight)^{10}(1+3cos^2 heta) = left( frac m M ight)^4$$

This is neither a sphere nor an oblate spheroid, and this is but an approximation. Thefull expression is an absolute mess. Dropping the factor of $(1+3cos^2 heta)^{1/10}$ (which is close to one) yields

$$r = left( frac m M ight)^{2/5} R$$

Tada! The equation of a sphere!

The true surface is defined in terms of two ratios. Consider two gravitating bodies, call them body A and body B. From the perspective of an inertial frame, the acceleration of a tiny test mass toward these two bodies is given by Newton's law of gravitation. These two bodies accelerate toward one another as well, so a frame based at the origin of either body is non-inertial.

From the perspective of a frame at the center of body A, the acceleration of the test mass is the inertial frame acceleration of the test mass toward body A plus the inertial frame acceleration of the test mass toward body B less the acceleration of body A toward body B. Denote the acceleration of the test mass toward body A as the primary acceleration and the difference between the inertial frame accelerations of the test mass and body A toward body B as the disturbing acceleration. Finally, define $Q_A$ as the ratio of these two. Now do the same for a frame with origin at the center of body B. The sphere of influence is that surface where $Q_A = Q_B$.

Is the SOI a spherical region or a oblate-spheroid-shaped region? - Astronomy

Can you go to the other side of a black hole? As in, is a black hole a sphere that you can "go to the other side of," or if you orbited it from lightyears away, could you go "around" it?

It is, in fact, possible to orbit a black hole. You do not even have to be light years outside of it. You simply have to be outside the event horizon, the distance at which everything, even light, falls into the black hole. For a normal-sized black hole, between fifty and seventy miles is a safe distance to orbit.

A black hole is a sphere in the sense that everything that goes within its Schwarzschild radius (the distance from the center of the black hole to the event horizon) cannot escape its gravity. Thus, there is a dark sphere around the infinitely dense center, or singularity, from which nothing can escape.

There is a supermassive black hole at the center of the Milky Way, and we orbit this black hole approximately every 230 million years.

Edit by Michael Lam on August 21, 2015: To answer the title question, if a black hole is rotating, then it will be shaped as an oblate spheroid, slightly larger around the equator than in the direction of the poles. However, the equations of general relativity tell us that rather than having one radius, the location of the event horizon, there are two important radii, the spherical event horizon on the inside, and the oblate spheroidal exterior surface. The region in between the two is called the ergosphere, where particles cannot remain at rest and objects can still escape the black hole. Such a black hole looks like this:

As in the non-rotating case, no particle entering the event horizon can escape.

Is the SOI a spherical region or a oblate-spheroid-shaped region? - Astronomy

Are other bodies in the solar system (such as the Moon and Pluto) spheres, or are they flat disks?

All objects in the solar system are three-dimensional, just like things on the surface of Earth. Furthermore, most bodies larger than hundreds of kilometers across are spherical. They are not perfect spheres, as the radius varies gradually. The typical shape (including those of Earth, Moon, and Pluto) is a oblate spheroid: a squashed sphere.

The simplest evidence for a spherical Moon:

  1. During solar eclipses, the Sun's shadow is always nearly circular. The only geometric object that can yield a near-circular eclipse in any orientation is a spheroid.
  2. The terminator of the Moon (the boundary between the day-side and night-side) as viewed from Earth is always arc-shaped. Only spheroids can show such an edge in any orientation.

Theoretical reasons for a spherical Moon: The lowest gravitational potential energy of a system of particles is achieved when they form a sphere as opposed to a disk. However, it is possible for smaller collections of particles to withstand the force of gravity with counteracting forces (mostly electromagnetic forces that give rise to chemical bonds) and aggregate into non-spherical shapes. This is why many smaller objects like asteroids and even Mars's two moons (Phobos and Deimos) are shaped like lumpy boulders.

Modern evidence for a spherical Moon: Data (such as images from orbit, characteristics of the satellite orbits, Moon's gravity field, and images on the Moon's surface) from the lunar missions such as the Apollo, Clementine, Zond, Lunar Prospector, and upcoming data from Kaguya (completed missions are summarized online). The simplest proof from such data is that the Moon looks like a disk when viewed from any point in the orbit - only a spheroid, not a disk, can appear so.

Pluto (first imaged at close range in 2015 by NASA's New Horizons mission) is also a spheroid. Even before New Horizons, we knew that Pluto is certainly massive enough (

0.2% of Earth's mass the Moon is

1% of Earth's mass) to be a spheroid due to self-gravitation.

This page was last updated on February 10, 2016.

About the Author

Suniti Karunatillake

After learning the ropes in physics at Wabash College, IN, Suniti Karunatillake enrolled in the Department of Physics as a doctoral candidate in Aug, 2001. However, the call of the planets, instilled in childhood by Carl Sagan's documentaries and Arthur C. Clarke's novels, was too strong to keep him anchored there. Suniti was apprenticed with Steve Squyres to become a planetary explorer. He mostly plays with data from the Mars Odyssey Gamma Ray Spectrometer and the Mars Exploration Rovers for his thesis project on Martian surface geochemistry, but often relies on the synergy of numerous remote sensing and surface missions to realize the story of Mars. He now works at Stonybrook.

Monday, June 20, 2016

Olli - Olly Oxen Free Autonomy Hits The Streets

I'm symbolic. Image Credit: Local Motors (2016)

Olli - Olly Oxen Free Autonomy Hits The Streets

Local Motors, in partnership with computer giant IBM through Watson, introduces an organic small group/mass autonomous transportation solution perfect for most any community.

No oxen were used in this creation - Actually, "Olly Olly Oxen Free" is a catchphrase used in such children's games as hide and seek, capture the flag, or kick the can to indicate that players who are hiding can come out into the open without losing the game, that the position of the sides in a game has changed, or, alternatively, that the game is entirely over (ht: Wikipedia).

In this case the Olli is the name given to a driverless/autonomous vehicle that seats 12 people which through its computer partner, IBM's Watson, can interact with passengers and navigate the streets and deliver the people riding inside to their destination.

The game of driverless/autonomous vehicles has changed with the Olli concept because this application does not currently partner with a software services company like Google or Yahoo that are focused on individual transportation pods thus removing any pursuit of happiness from the process of actual driving.

This excerpted and edited from Electric Cars Report -

Local Motors Debuts First Self-driving Vehicle to Tap the Power of IBM Watson IoT

Local Motors, the creator of the world’s first 3D-printed cars, today introduced the first self-driving vehicle to integrate the advanced cognitive computing capabilities of IBM Watson.

The vehicle, dubbed ‘Olli,’ was unveiled during the grand opening of a new Local Motors facility in National Harbor, MD, and transported Local Motors CEO and co-founder John B. Rogers, Jr. along with vehicle designer Edgar Sarmiento from the Local Motors co-creation community into the new facility.

Olli exterior. Image Credit: Local Motors (2016)

The electric vehicle, which can carry up to 12 people, is equipped with some of the world’s most advanced vehicle technology, including IBM Watson Internet of Things (IoT) for Automotive, to improve the passenger experience and allow natural interaction with the vehicle.

Olli is the first vehicle to utilize the cloud-based cognitive computing capability of IBM Watson IoT to analyze and learn from high volumes of transportation data, produced by more than 30 sensors embedded throughout the vehicle.
Furthermore, the platform leverages four Watson developer APIs — Speech to Text, Natural Language Classifier, Entity Extraction and Text to Speech — to enable seamless interactions between the vehicle and passengers.

Olli interior. Image Credit: Local Motors (2016)

Passengers will be able to interact conversationally with Olli while traveling from point A to point B, discussing topics about how the vehicle works, where they are going, and why Olli is making specific driving decisions. Watson empowers Olli to understand and respond to passengers’ questions as they enter the vehicle, including about destinations (“Olli, can you take me downtown?”) or specific vehicle functions (“how does this feature work?” or even “are we there yet?”).

Passengers can also ask for recommendations on local destinations such as popular restaurants or historical sites based on analysis of personal preferences.
It's a beautiful Father's Day in #NationalHarbor! Catch up with CEO @johnbrogers & #meetolli- our latest innovation. Image Credit: Local Motors via @localmotors

As part of Olli’s debut, Local Motors officially opened its new National Harbor facility in Maryland to serve as a public place where co-creation can flourish and vehicle technologies can rapidly advance. The company’s 3D-printed cars are on display, along with a large-scale 3D printer and an interactive co-creative experience that showcases what the future of the nation’s capital might look like.
Olli features a 15 kWh battery pack powering a 20 kW continuous, 30 kW max electric motor that delivers 125 N·m of torque. Maximum speed is 20 km/h (12 mph) and all-electric range is 58 km (32.4 miles). Olli is equipped with 2 Velodyne VLP16 LiDAR units, 2 IBEO ScaLa laser scanners, 2 ZED optical cameras and an Ellipse N GPS.

The very first Olli will remain in National Harbor this summer, and the public will be able to interact with it during select times over the next several months.
[Reference Here]

Now, riding around on this Oblate Spheroid, we say "Olli Olli Oxen Free" - to indicate that players who are hiding can come out into the open without losing the game, that the position of the sides in a game has changed, or, alternatively, that the game is entirely over - community driverless/autonomous transportation has arrived with a form factor and application that allows everyone the pursuit of happiness while pushing the technology envelope of possibilities forward.

TAGS: Local Motors, Watson, Cloud Computing, Internet of Things, IoT, driverless, autonomous, vehicle, Olli, 3D-printed cars, 3D printer, electric motor, ZED optical camera, Ellipse N GPS, Speech to Text, Natural Language Classifier, Entity Extraction, Text to Speech, National Harbor, Maryland

Influence And Conformity

Open with on-stage re-enactment/illustration of this scenario (see below) � ask individuals in your church to come:

A few years ago psychologist Ruth W. Berenda and her associates carried out an interesting experiment with teenagers designed to show how a person handled group pressure. The plan was simple. They brought groups of ten adolescents into a room for a test. Subsequently, each group of ten was instructed to raise their hands when the teacher pointed to the longest line on three separate charts. What one person in the group did not know was that nine of the others in the room had been instructed ahead of time to vote for the second-longest line. Regardless of the instructions they heard, once they were all together in the group, the nine were not to vote for the longest line, but rather vote for the next to the longest line. The experiment began with nine teen-agers voting for the wrong line. The stooge would typically glance around, frown in confusion, and slip his hand up with the group. The instructions were repeated and the next card was raised. Time after time, the self-conscious stooge would sit there saying a short line is longer than a long line, simply because he lacked the courage to challenge the group. This remarkable conformity occurred in about 75% of the cases, and was true of small children and high-school students as well. Berenda concluded that, "Some people had rather be president than right," which is certainly an accurate assessment.

C. Swindoll, Living Above the Level of Mediocrity, p. 225.

No matter how NAME came out on this, I thought I might have a good object lesson to begin our look at influence and conformity. If NAME had gone against the tide, despite the pressure and influence of these adults of this church, it would have illustrated that despite the influence, we need not conform to the influence of those around us, especially when those around us are wrong. We do not have to conform, we�re not powerless � it is a choice.

If NAME had conformed, unable to muster the emotional energy to challenge the crowd, HE/SHE would have revealed only that HE/SHE�s quite normal. In a similar experiment by a psychologist, about 75% of those who were not in on the experiment conformed to the group.

Now this experiment was done with teenagers, but it�s a reality that even adults have to wrestle with conforming to this world.

Why else would Paul admonish in Romans 12:2 � �do not be conformed to this world.�

Why else would Proverbs tell us in chapter 4:23:

Proverbs 4:23 (NLT) Guard your heart above all else, for it determines the course of your life.

We have these admonitions and commands in scripture because we are subject to influence. And because we are subject to influence in almost every area of our lives, we must choose daily what influences we will allow into our lives, and what we will either reject, or be cautious about, or freely receive.

Here�s a definition of influence: The power to affect, control or manipulate something or someone the ability to change the development of fluctuating things such as conduct, thoughts or decisions.

There�s another interesting definition I want to look at � it relates to astrodynamics

A sphere of influence (SOI) in astrodynamics and astronomy is the spherical region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in our solar system where planets dominate the orbits of surrounding objects (such as moons), despite the presence of the much more massive (but distant) Sun

Now, let�s for just a moment consider the spiritual implications of this astrodynamic definition of �sphere of influence.�

When we�re conforming to the world, we might look at ourselves as moons. Despite the presence of a much more massive, much more powerful Son, a planet which is nearby influences or dominates our orbit, keeping us conforming to an orbit around that planet, rather than the significantly larger influence of the Son of God, our Lord and Savior Jesus Christ.

How does that look in real life? As followers of Christ, we know what the driving influence of our lives should be in this analogy � the sun, (or Son) which is the things of God, as outlined in His Word, which tells us all that we need to know about faith and practice.

But the reality is, because of the way we sometimes choose to conduct our lives, there are things, like planets in this analogy, that are closer to us, and these things often have a greater influence on our thinking, and thus on our behavior, than the purity of devotion to Christ.

4 Summary and Discussion

We introduced the superspheroidal model for computing the scattering matrix of dust aerosols. In addition to the aspect ratio, superspheroids have another shape parameter, namely, the roundness parameter, which permits more morphological variations and thus can mimic more dust particles' characteristics (such as concavity and sharp edge). Comparisons of scattering matrix elements between spheroids and superspheroids demonstrated that major characteristics of measured scattering matrix elements of dust aerosol samples (more or less) could be obtained in two ways. One involved the use of changing the aspect ratio to a value that departs the most from unity. The other way involved the use of large roundness parameters, which indicates that concave superspheroids could be practical in modeling the scattering matrix of dust aerosols. By comparing with measurements from the Amsterdam-Granada Light Scattering Database (Muñoz et al., 2000 , 2004 , 2012 Volten et al., 2000 , 2006 , 2007 ), our study shows that extreme aspect ratios (>2.0 or <0.5) for spheroids in reproducing the measurements are unnecessary for superspheroids, and an optimal range for superspheroids' roundness parameters in modeling the scattering matrix of dust aerosols was found to be [2.4, 3.0]. Further analysis showed that superspheroids with constrained roundness parameters appear to be much better than spheroids (a unity roundness parameter) even though extreme aspect ratios of spheroids are considered.


The table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun (with the exception of the Moon which is reported relative to Earth): Ώ]

Body SOI radius (10 6 km) SOI radius (body radii)
Mercury 0.112 46
Venus 0.616 102
Earth 0.929 145
Moon 0.0661 38
Mars 0.578 170
Jupiter 48.2 687
Saturn 54.5 1025
Uranus 51.9 2040
Neptune 86.8 3525

Is the SOI a spherical region or a oblate-spheroid-shaped region? - Astronomy

The differential rotation of the sun, as deduced from helioseismology, exhibits a prominent layer of radial shear near the top of the convection zone. This shearing boundary layer just below the solar surface is composed of convection possessing a broad range of length and time scales, including granulation, mesogranulation, and supergranulation. Such turbulent convection is likely to influence the dynamics of the deep convection zone in ways that are not yet fully understood. We seek to assess the effects of this near-surface shear layer through two complementary studies, one observational and the other theoretical in nature. Both deal with turbulent convection occurring on supergranular scales within the upper solar convection zone. We characterize the horizontal outflow patterns associated with solar supergranulation by individually identifying several thousand supergranules from a 45°-square field of quiet sun. This region is tracked for a duration of six days as it rotates across the disk of the sun, using full-disk (2 ' pixels) SOI-MDI images from the SOHO space-craft of line- of-sight Doppler velocity imaging the solar photosphere at a cadence of one minute. This time series represents the first study of solar supergranulation at such high combined temporal and spatial resolution over an extended period of time. The outflow cells in this region are observed to have a distribution of sizes, ranging from 14-20 Mm across, while continuously evolving on time scales of several days. Such evolution manifests itself in the form of cell merging, fragmentation, and advection, as the supergranules and their associated network of convergence lanes respond to the turbulent convection occurring a short distance below the photosphere. We have also conducted three-dimensional numerical simulations of turbulent compressible convection within thin spherical shells located near the top of the convection zone. Vigorous fluid motions possessing several length and time scales are driven by imposing the solar heat flux and differential rotation at the bottom of the domain. The convection patterns form a connected network of downflow lanes in the surface layers that break up into more plume-like structures with depth. The regions delineated by this downflow network enclose broad upflows that fragment into smaller structures near the surface. We find that a negative radial gradient of angular velocity Ω is maintained against diffusion in these simulations by the tendency for the convective motions to partially conserve their angular momentum in radial motion. This behavior suggests that similar dynamics may be responsible for the decrease of Ω with radius as deduced from helioseismology within the upper shear layer of the solar convection zone.


Earth's topographic surface is apparent with its variety of land forms and water areas. This topographic surface is generally the concern of topographers, hydrographers, and geophysicists. While it is the surface on which Earth measurements are made, mathematically modeling it while taking the irregularities into account would be extremely complicated.

The Pythagorean concept of a spherical Earth offers a simple surface that is easy to deal with mathematically. Many astronomical and navigational computations use a sphere to model the Earth as a close approximation. However, a more accurate figure is needed for measuring distances and areas on the scale beyond the purely local. Better approximations can be had by modeling the entire surface as an oblate spheroid, using spherical harmonics to approximate the geoid, or modeling a region with a best-fit reference ellipsoid.

For surveys of small areas, a planar (flat) model of Earth's surface suffices because the local topography overwhelms the curvature. Plane-table surveys are made for relatively small areas without considering the size and shape of the entire Earth. A survey of a city, for example, might be conducted this way.

By the late 1600s, serious effort was devoted to modeling the Earth as an ellipsoid, beginning with Jean Picard's measurement of a degree of arc along the Paris meridian. Improved maps and better measurement of distances and areas of national territories motivated these early attempts. Surveying instrumentation and techniques improved over the ensuing centuries. Models for the figure of the earth improved in step.

In the mid- to late 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the figure of the Earth. The primary utility of this improved accuracy was to provide geographical and gravitational data for the inertial guidance systems of ballistic missiles. This funding also drove the expansion of geoscientific disciplines, fostering the creation and growth of various geoscience departments at many universities. [1] These developments benefited many civilian pursuits as well, such as weather and communication satellite control and GPS location-finding, which would be impossible without highly accurate models for the figure of the Earth.

The models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth.

Sphere Edit

The simplest model for the shape of the entire Earth is a sphere. The Earth's radius is the distance from Earth's center to its surface, about 6,371 km (3,959 mi). While "radius" normally is a characteristic of perfect spheres, the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth".

The concept of a spherical Earth dates back to around the 6th century BC, [2] but remained a matter of philosophical speculation until the 3rd century BC. The first scientific estimation of the radius of the Earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes's measurement ranging from -1% to 15%.

The Earth is only approximately spherical, so no single value serves as its natural radius. Distances from points on the surface to the center range from 6,353 km (3,948 mi) to 6,384 km (3,967 mi). Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 km (3,959 mi). Regardless of the model, any radius falls between the polar minimum of about 6,357 km (3,950 mi) and the equatorial maximum of about 6,378 km (3,963 mi). The difference 21 km (13 mi) correspond to the polar radius being approximately 0.3% shorter than the equatorial radius.

Ellipsoid of revolution Edit

Since the Earth is flattened at the poles and bulges at the Equator, geodesy represents the figure of the Earth as an oblate spheroid. The oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis. It is the regular geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other celestial body is called a reference ellipsoid. The reference ellipsoid for Earth is called an Earth ellipsoid.

An ellipsoid of revolution is uniquely defined by two quantities. Several conventions for expressing the two quantities are used in geodesy, but they are all equivalent to and convertible with each other:

Eccentricity and flattening are different ways of expressing how squashed the ellipsoid is. When flattening appears as one of the defining quantities in geodesy, generally it is expressed by its reciprocal. For example, in the WGS 84 spheroid used by today's GPS systems, the reciprocal of the flattening 1 / f is set to be exactly 298.257 223 563 .

The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Historically, flattening was computed from grade measurements. Nowadays, geodetic networks and satellite geodesy are used. In practice, many reference ellipsoids have been developed over the centuries from different surveys. The flattening value varies slightly from one reference ellipsoid to another, reflecting local conditions and whether the reference ellipsoid is intended to model the entire Earth or only some portion of it.

A sphere has a single radius of curvature, which is simply the radius of the sphere. More complex surfaces have radii of curvature that vary over the surface. The radius of curvature describes the radius of the sphere that best approximates the surface at that point. Oblate ellipsoids have constant radius of curvature east to west along parallels, if a graticule is drawn on the surface, but varying curvature in any other direction. For an oblate ellipsoid, the polar radius of curvature r p > is larger than the equatorial

because the pole is flattened: the flatter the surface, the larger the sphere must be to approximate it. Conversely, the ellipsoid's north–south radius of curvature at the equator r e > is smaller than the polar

Geoid Edit

It was stated earlier that measurements are made on the apparent or topographic surface of the Earth and it has just been explained that computations are performed on an ellipsoid. One other surface is involved in geodetic measurement: the geoid. In geodetic surveying, the computation of the geodetic coordinates of points is commonly performed on a reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (gravitation) and the centrifugal force of the Earth's rotation. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as geoid undulations, geoid heights, or geoid separations, will be irregular as well.

The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular (see equipotential surface). The latter is particularly important because optical instruments containing gravity-reference leveling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the plumb line which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the deflection of the vertical. It has two components: an east–west and a north–south component. [3]

Other shapes Edit

The possibility that the Earth's equator is better characterized as an ellipse rather than a circle and therefore that the ellipsoid is triaxial has been a matter of scientific inquiry for many years. [4] [5] Modern technological developments have furnished new and rapid methods for data collection and, since the launch of Sputnik 1, orbital data have been used to investigate the theory of ellipticity. [3] More recent results indicate a 70-m difference between the two equatorial major and minor axes of inertia, with the larger semidiameter pointing to 15° W longitude (and also 180-degree away). [6] [7]

Pear shape Edit

A second theory, more complicated than triaxiality, proposes that observed long periodic orbital variations of the first Earth satellites indicate an additional depression at the south pole accompanied by a bulge of the same degree at the north pole. It is also contended that the northern middle latitudes were slightly flattened and the southern middle latitudes bulged in a similar amount. This concept suggested a slightly pear-shaped Earth and was the subject of much public discussion after the launch of the first artificial satellites. [3] U.S. Vanguard 1 satellite data from 1958 confirms that the southern equatorial bulge is greater than that of the north, which is corroborated by the south pole's sea level being lower than that of the north. [8] Such a model had first been theorized by Christopher Columbus on his third voyage. Making observations with a quadrant, he "regularly saw the plumb line fall to the same point," instead of moving respectively to his ship, and subsequently hypothesized that the planet is pear-shaped. [9]

John A. O'Keefe and co-authors are credited with the discovery that the Earth had a significant third degree zonal spherical harmonic in its gravitational field using Vanguard 1 satellite data. [10] Based on further satellite geodesy data, Desmond King-Hele refined the estimate to a 45-m difference between north and south polar radii, owing to a 19-m "stem" rising in the north pole and a 26-m depression in the south pole. [11] [12] The polar asymmetry is small, though: it is about a thousand times smaller than the earth's flattening and even smaller than the geoidal undulation in some regions of the Earth. [13]

Modern geodesy tends to retain the ellipsoid of revolution as a reference ellipsoid and treat triaxiality and pear shape as a part of the geoid figure: they are represented by the spherical harmonic coefficients C 22 , S 22 ,S_<22>> and C 30 > , respectively, corresponding to degree and order numbers 2.2 for the triaxiality and 3.0 for the pear shape.

4. Summary

[11] A major compression of Saturn's magnetosphere took place during the Cassini SOI fly-through of Saturn's magnetosphere (Jackman et al., submitted manuscript, 2005). For the first time, we have witnessed the in situ effects that the CIR-related compression has on Saturn's magnetospheric dynamics. At ∼02:00 UT on day 184, a burst of SKR emission is observed which disrupts the existing pattern of planetary modulated emission seen both upstream of the magnetosphere and during the inbound pass [ Kurth et al., 2005 Jackman et al., submitted manuscript, 2005]. Simultaneously, inside the magnetosphere, Cassini experienced a region of depressed and variable magnetospheric field. In addition, ion and electron observations show that this occurs as the spacecraft is engulfed by a hot, tenuous plasma population. While subsequently cooling, the spacecraft remained within this plasma sheet population for the duration of the outbound pass. We have thus shown that following the shock-compression, the magnetosphere underwent a significant reconfiguration, exemplified by a relaxation of the field and an injection of hot plasma. We propose that this behaviour is indicative of a major episode of tail reconnection, triggered by the impact of the compression region on Saturn's magnetosphere, as discussed in relation to the January 2004 HST-Cassini interval by Cowley et al. [2005] .

Watch the video: Definition of Sphere of Influence (September 2022).