Peculiar Amplitude of Earth's Orbit in Z Axis

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I'm currently building an N-body simulator for my masters project and I'm using JPL's Horizons system to compare against.

I started with just a couple orbits and everything seemed to be working nicely (After fixing many mistakes implementing a Fourth Order Integrator) but after extending to 10+ orbits I notice that in the Z-axis the Earth's amplitude increases linearly, according to JPL, but my simulation has a constant amplitude. JPL's result is shown in red in the bottom left:

What is causing this? Currently I only have the Sun, Earth, Moon and Jupiter in the system, but I didn't even realise the amplitude increased like this? What am I missing or got wrong?

(Also, the graph only shows 3 simulated orbits, and 80 JPL orbits)

I can reproduce your JPL Z coordinate plot if I use Earth's heliocentric position relative to the J2000 ecliptic. Naturally the amplitude has a minimum around the year 2000:

Relative to the ecliptic of date, which accounts for precession, the heliocentric Z coordinate of the Earth-Moon barycenter is much smaller:

The IAU 2006 precession model has two components, detailed in Capitaine et al. 2003:

• precession of the ecliptic, a 47"/century change in the plane of Earth's orbit
• precession of the equator, a 1.4°/century change in the plane and axis of Earth's rotation

Fifty years of ecliptic precession amount to 23.5" or 0.000114 radian, and the first plot is consistent with that. The classical precession of the equinoxes (the intersection of the two planes) is a combination of both components.

Williams 1994 examines Earth's equatorial precession (dominated by the Sun and Moon) and finds that Venus's effect is larger than Jupiter's. When looking at fluctuations in Earth's year length, I noticed a cycle similar to Venus's synodic period. I expect that adding Venus to your system would reproduce some ecliptic precession.

What path does the Earth take

Actually, I'd advise against looking for that, without first steeling oneself against crackpottery. The most popular one I know of is 1. somewhat inaccurate, 2. loaded with nonsense annotations (paraphrasing: see earth does not circle around the sun, but rather it's a 'vortex'! therefore scientific cabal ivory tower we're being lied to etc.).

As for the question, it's all a matter of choosing a reference frame (i.e. with respect to what you want to describe the motion) and adding all component motions together.
For the reference frame, on the largest scale, since cosmic expansion was mentioned, I'd choose the CMBR-rest frame. It's a family of frames in which the radiation coming at us from the earliest universe looks roughly the same in every direction. It's not a bad way to think of it as 'how is the Earth moving w/r to the average distribution of matter in the universe'.

1. So you start with Earth going around the Sun in somewhat wobbly, nearly circular ellipses. It does so at 30km/s.

2. Add to that the motion around the galactic centre. Here, we need a few steps:
- include the roughly circular orbit due to the bulk rotation of the galaxy. This changes the circular path into a helical one, where the distance between the 'bends' in the helix are approx. 40 times the its radius (so it's a pretty elongated one). But the helix is angled backwards - like a slinky toy leaning to one side. This is due to the plane of the solar system being angled at approx 60° w/r to the direction of its motion.
Kinda like this:

(the picture has the +z axis pointing towards the galactic south)
- finally, the remaining motion due to interactions with local stellar neighbourhood should be added (i.e., the remaining 'peculiar motion'). This further changes the circular base orbit into a somewhat elliptical one.
The peculiar motions are subject to local interactions, and are likely to chaotically change over time.

3. Adding to that motion of the Milky Way w/r to the Local Group of galaxies - this means mostly just the hurtling towards the Andromeda galaxy.
This changes the wobbly, elliptical orbit around the galactic centre into another leaning, helical one. This one looks very squashed, since MW moves towards Andromeda almost edge-on, with inclination of only 20°

The distance between bends of this helix is approx 3 times its radius. It's much more tightly wound than the previous one.

4. Finally, add the motion of the Local Group w/r to the CMBR-rest frame. This is pretty much a straight line in the direction of the Pump (Antlia) constellation.
This changes direction of the last helix like so (rather crudely eyeballed and drawn, I know, but mostly preserves proportions):

You need to imagine the already twisted helix from 3 as going towards Andromeda changing direction towards the 'net motion' one. Think of it, again, as a slinky toy with one end at the origin, and the other being dragged from the one arrow to the other. The trick is to preserve the angle of the bends as you do so.
So this step makes the helix from 3 more elongated and twisted in yet another way.

It is worth keeping in mind that these are the motions that can be thought of as representing reality only in this particular moment in time or in rough terms. As millions of years go by, and the stars and galaxies dance around each other in their gravitational ballet, there will come deviations and largely unpredictable changes.

Now, can you imagine all these component motions happening at the same time? If so, good for you. :)
Personally, I tend to stick to the geocentric frame most of the time, else I get dizzy every time the Sun rises over the horizon.

Actually, I'd advise against looking for that, without first steeling oneself against crackpottery. The most popular one I know of is 1. somewhat inaccurate, 2. loaded with nonsense annotations (paraphrasing: see earth does not circle around the sun, but rather it's a 'vortex'! therefore scientific cabal ivory tower we're being lied to etc.).

As for the question, it's all a matter of choosing a reference frame (i.e. with respect to what you want to describe the motion) and adding all component motions together.
For the reference frame, on the largest scale, since cosmic expansion was mentioned, I'd choose the CMBR-rest frame. It's a family of frames in which the radiation coming at us from the earliest universe looks roughly the same in every direction. It's not a bad way to think of it as 'how is the Earth moving w/r to the average distribution of matter in the universe'.

1. So you start with Earth going around the Sun in somewhat wobbly, nearly circular ellipses. It does so at 30km/s.

2. Add to that the motion around the galactic centre. Here, we need a few steps:
- include the roughly circular orbit due to the bulk rotation of the galaxy. This changes the circular path into a helical one, where the distance between the 'bends' in the helix are approx. 40 times the its radius (so it's a pretty elongated one). But the helix is angled backwards - like a slinky toy leaning to one side. This is due to the plane of the solar system being angled at approx 60° w/r to the direction of its motion.
Kinda like this:
View attachment 193714
For added complexity, this alignment does not change as the Sun orbits the galaxy - the helix gets slowly squashed sideways, then relaxed, then squashed again.
- if we looked at this elongated, bent helix from afar, it'd be just a line. This line has to bend around the galactic centre to form a circle. The radius of curvature is so large, however, that it would not be noticeable at the scale where you can resolve the helix.
- additionally, it needs to follow a sinusoidal path perpendicular to the plane of rotation (plane of the galaxy). This is due to peculiar motion in the plane-normal direction taking it slightly above the disc, where mass in the disc pulls it back, and the Sun overshoots in the opposite direction, moving below the disc. And so on. I don't remember the estimated frequency of those oscillations, but it's probably on the order of a dozen per orbit. Again, only really noticeable when you zoom out from the helix.
The two above look like this:
View attachment 193715
(the picture has the +z axis pointing towards the galactic south)
- finally, the remaining motion due to interactions with local stellar neighbourhood should be added (i.e., the remaining 'peculiar motion'). This further changes the circular base orbit into a somewhat elliptical one.
The peculiar motions are subject to local interactions, and are likely to chaotically change over time.

3. Adding to that motion of the Milky Way w/r to the Local Group of galaxies - this means mostly just the hurtling towards the Andromeda galaxy.
This changes the wobbly, elliptical orbit around the galactic centre into another leaning, helical one. This one looks very squashed, since MW moves towards Andromeda almost edge-on, with inclination of only 20°
View attachment 113456
The distance between bends of this helix is approx 3 times its radius. It's much more tightly wound than the previous one.

4. Finally, add the motion of the Local Group w/r to the CMBR-rest frame. This is pretty much a straight line in the direction of the Pump (Antlia) constellation.
This changes direction of the last helix like so (rather crudely eyeballed and drawn, I know, but mostly preserves proportions): View attachment 113465
You need to imagine the already twisted helix from 3 as going towards Andromeda changing direction towards the 'net motion' one. Think of it, again, as a slinky toy with one end at the origin, and the other being dragged from the one arrow to the other. The trick is to preserve the angle of the bends as you do so.
So this step makes the helix from 3 more elongated and twisted in yet another way.

It is worth keeping in mind that these are the motions that can be thought of as representing reality only in this particular moment in time or in rough terms. As millions of years go by, and the stars and galaxies dance around each other in their gravitational ballet, there will come deviations and largely unpredictable changes.

Now, can you imagine all these component motions happening at the same time? If so, good for you. :)
Personally, I tend to stick to the geocentric frame most of the time, else I get dizzy every time the Sun rises over the horizon.

At present the celestial sky has been mapped in considerable detail for every major wavelength band, except for the ultra-long radiowave band. A space-based interferometer consisting of a swarm of satellites would make it possible to map the celestial sources of 0.1–10 MHz radiation. Such a mission concept called the Orbiting Low Frequency Array (OLFAR) is currently undergoing a feasibility study. This paper presents an analysis of possible operational orbits for the OLFAR satellites.

The strategy for OLFAR is to let the satellites drift freely after release into initial orbits. The design of the swarm's reference orbit is primarily motivated by the need for a low radio-noise environment. This results in lunar orbits being main candidates. The design of the initial swarm configuration is primarily motivated by the need for uvw-space coverage. This quantity expresses the variation of lengths and orientations of the satellite relative position vectors over time.

Numerical simulations give strong indications that the required uvw-coverage can be met within 1 year of operations with a number of satellites ranging between 25 and 100. A key conclusion is that the orbital behavior of a swarm (characterized by the absence of continuous formation control) is well suited for ultra-long wavelength radio astronomy.

Modern Astrodynamics

1.15 Lagrangian VOP—conservative forces

The VOP method is a formulation of the equations of motion that are well-suited to perturbed, dynamical systems. The concept is based on the premise that we can use the solution for the unperturbed system to represent the solution of the perturbed system, provided that we can generalize the constants in the solution to be time-varying parameters. The unperturbed system is the two-body system, and it represents a collection of formulas that provide the position and velocity vectors at a desired time. Remember, these formulas depend only on the six orbital elements and time. In principle, however, we could use any set of constants of the unperturbed motion, including the initial position and velocity vectors. Time is related to the equations of motion through the conversions of mean, eccentric, and true anomaly .

The general theory for finding the rates of change of the osculating elements is known as the Lagrange planetary equations of motion, or simply the Lagrangian VOP, and is attributed to Lagrange because he was the first person to obtain these equations for all six orbital elements. He was concerned with the small disturbances on planetary motion about the Sun due to the gravitational attraction of the planets. He chose to model the disturbing acceleration due to this conservative perturbation as the gradient of a potential function. From Vallado [ 15 ],

3 Cosmic-Ray Data and Analyses

3.1 Global Muon Detector Network

The GMDN, which is designed for accurate observation of the GCR anisotropy, comprises four multidirectional muon detectors, “Nagoya” in Japan, “Hobart” in Australia, “Kuwait City” in Kuwait, and “São Martinho da Serra” in Brazil, recording muon count rates in 60 directional channels viewing almost the entire sky around Earth. Basic characteristics of directional channels of the GMDN are also available in the Table S1. The median rigidity (Pm) of primary GCRs recorded by the GMDN, which we calculate by using the response function of the atmospheric muons to the primary GCRs given by numerical solutions of the hadronic cascade in the atmosphere (Murakami et al., 1979 ), ranges from about 50 GV for the vertical directional channel to about 100 GV for the most inclined directional channel, while the asymptotic viewing directions (corrected for geomagnetic bending of cosmic-ray orbits) at Pm covers the asymptotic viewing latitude (λasymp) from 72°N to 77°S. The representative Pm of the entire GMDN is about 60 GV.

3.2 Derivation of the GCR Density and Anisotropy

We analyze the percent deviation of the 10-min muon count rate Ii,j(t) from an average over 27 days between August 12 and September 7, 2018 in the j-th directional channel of the i-th detector (i = 1 for Nagoya, i = 2 for Hobart, i = 3 for Kuwait, and i = 4 for São Martinho da Serra) in the GMDN at universal time t, after correcting for local atmospheric pressure and temperature effects. For our correction method of the atmospheric effects using the on-site measurement of pressure and the mass weighted temperature from the vertical profile of the atmospheric temperature provided by the Global Data Assimilation System (GDAS) of the National Center for Environmental Prediction, readers can refer to Mendonça et al. ( 2016 ).

Since the observed temporal variation of Ii,j(t) at the universal time t includes contributions from variations of the GCR density (or ominidirectional intensity) I0(t) and anisotropy vector ξ(t), it is necessary to analyze each contribution separately. An accurate analysis of I0(t) and ξ(t) is possible only with global observations using multidirectional detectors. For such analyses, we model Ii,j(t) in terms of I0(t) and three components () of ξ(t) in a geocentric (GEO) coordinate system, as (1) where ti is the local time in hours at the i-th detector, , , and are coupling coefficients which relate (or “couple”) the observed intensity in each directional channel with the cosmic ray density and anisotropy in space and ω = π/12. In the GEO coordinate system, we set the x-axis to the antisunward direction in the equatorial plane, the z-axis to the geographical north perpendicular to the equatorial plane and the y-axis completing the right-handed coordinate system. The coupling coefficients in Equation 1 are calculated by using the response function of the atmospheric muon intensity to primary GCRs (Murakami et al., 1979 ) and given in the Table S1. Note that the anisotropy vector ξ(t) in Equation 1 is defined to direct opposite to the GCR streaming, pointing toward the upstream direction of the streaming (see also Equation 6 in the next section). We derive the best-fit set of four parameters by solving the following linear equations. (2) (3)

with σci,j denoting the count rate error of Ii,j(t). The best-fit anisotropy vector ξ GEO (t) in the GEO coordinate system is then transformed to ξ GSE (t) in the GSE coordinate system for comparisons with the solar wind and IMF data.

Equation 1 does not include contributions from the second order anisotropy such as the bidirectional counter-streaming sometimes observed in the MFR in MeV electron/ion intensities. We also performed best-fit analyses adding five more best-fit parameters in Equation 1 necessary to express the second order anisotropy and actually found an enhancement of the second order anisotropy in the MFR. However, we verified that the inclusion of the second order anisotropy does not change the obtained I0(t) and ξ(t) significantly keeping conclusions of the present study unchanged. In this study, therefore, we analyze only I0(t) and ξ(t) derived from Equation 1. We will present our analyses and discussion of the second-order anisotropy elsewhere.

3.3 Derivation of the Spatial Gradient of GCR Density

(4) (5) where κ is the diffusion tensor and C is the Compton-Getting (CG) factor denoted by with an assumption of U proportional to pγ with the power-law index γ = 2.7. The diffusion and drift anisotropy ξ D is given as (6)

where v is the speed of GCR particle, which is approximately equal to the speed of light c, and G = U/U is the spatial gradient of GCR density.

(7) (8) (9) where is the Larmor radius of particles with rigidity P in magnetic field B(t) and and are components of ξ w (t) parallel and perpendicular to B(t), respectively (Kozai et al., 2016 ). α and α in Equation 9 are mean-free-paths of parallel and perpendicular diffusions, respectively, normalized by RL(t), as (10) (11)

According to current understanding that GCRs at neutron monitor and muon detector energies are in the “weak-scattering” regime (Bieber et al., 2004 ), we assume λ(t) ≪ λ(t). Following models widely used in the study of the large-scale GCR transport in the heliosphere (Miyake et al., 2017 Wibberenz et al., 1998 ), we assume constant α = 0.36 for a period outside the MFR in this study. We also assume λ = 1.9 AU for the entire period. For 60 GV cosmic rays in |B(t)| ∼ 5 nT average magnetic field, RL is 0.27 AU resulting in λ = 0.096 AU and α = 7.2. For a period inside the MFR where the magnetic field is exceptionally strong, we use a constant λ = 0.010 AU without changing λ. Note that this λ was obtained as an upper limit by Munakata et al. ( 2006 ).

We are aware that our ad hoc assumptions of λ(t) and λ(t) above are difficult to validate directly from observations. However, it will be shown in the next section that G(t) derived from the observed anisotropy ξ w (t) in Equation 9 is significantly dominated by the contribution from the drift anisotropy represented by the last term on the right-hand side of Equation 9 and is insensitive to our ad-hoc assumptions of λ(t) and λ(t).

References

Sidereal and tropical year, solstices and equinoxes: webexhibits calendar page

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2020 Mar 26. s.11

Your Z-axis and X-axis are actually the same, the axis of the Earth stay fixed in space while Earth orbits the Sun, resulting in an alternating pattern.

• The Sun is zig-zaging over the sky in circles most places at the Earth, if you are close to the equator, the amplitude is high.
• All places on Earth would have the Sun in zenith at least once a year.
• All places except for the equator is going to have a dark season.
• Midnight Sun at least once a year in all locations, except for the equator that has two seasons of day-and-night twilight.
• Actually more overall solar insolation of the poles than the equator, making them the warmest region of Earth.
• Almost all stars are visible from all places on Earth during the year, not just limited to a southern or northern hemisphere.

Illustration of the Sun's motion across the sky:

In other news, the magnetic field is screwed up, so we are going to have aurora and cancer everywhere.

10000 years), the magnetic dipole is parallel to the axis of rotation. Sure, there's would be a difference to the current magnetic field, but it likely would still protect us from the solar wind. $endgroup$ &ndash user15078 Feb 4 '16 at 7:53

Tilting the Earth's axis by 90 degrees will have the same effect on the climate as Uranus, with alternating poles having extended "summers" and "winters" while the equator will be in twilight during this time. In the "spring" and "fall", the poles will be dark while the equatorial regions will be in sunlight.

Overall, because of the size and orbital period of the Earth, I think this will suppress the formation of ice caps and polar regions. Since there will be sunlight 3/4 of the year striking both poles, the period of total darkness will be rather short, and the Earth's atmosphere and oceanic currents will probably provide enough heat to keep the region simply covered in snow rather than ice caps.

The other two answers address the "rolling" scenario quite well, but I'd like to point out one glaring error you made in your reasoning: the Earth isn't rotating along the Y-axis. It's rotating on an angle.

Now, of course, you know that. But the important point is that this causes the seasons. If Earth was rotating exactly around the Y-axis (with respect to the orbital plane), there would be no difference in climate due to orbital position.

As you let the rotation axis fall away from the Y-axis, you get more and more seasonal weather, up to the point where the rotation is aligned with the orbital plane (it doesn't matter in which direction - that's just an offset to when each season starts, but the effect is the same).

For additional points, changing the rotation rate can give you more to work with. If a day takes half a year, it's going to have a huge effect on the seasonal variations - it might be interesting to explore a scenario where the rotation is a bit more tilted and slowed down. Venusian weather and climate is pretty interesting, in part due to its slow rotation and massive atmosphere while the planet's day is longer than its year, the clouds make the trip around in measly four Earth days.

Astronomical Glossary

Aberration : the apparent angular displacement of the observed position of a celestial object from its geometric position, caused by the finite velocity of light in combination with the motions of the observer and of the observed object. (See aberration, planetary).

Aberration, annual : the component of stellar aberration (see aberration, stellar) resulting from the motion of the Earth about the Sun.

Aberration, diurnal : the component of stellar aberration (see aberration, stellar) resulting from the observer's diurnal motion about the center of the Earth.

Aberration, E-terms of : terms of annual aberration (see aberration, annual) depending on the eccentricity and longitude of perihelion (see longitude of pericenter) of the Earth.

Aberration, elliptic : see aberration, E-terms of.

Aberration, planetary : the apparent angular displacement of the observed position of a celestial body produced by motion of the observer (see aberration, stellar) and the actual motion of the observed object.

Aberration, secular : the component of stellar aberration (see aberration, stellar) resulting from the essentially uniform and rectilinear motion of the entire solar system in space. Secular aberration is usually disregarded.

Aberration, stellar : the apparent angular displacement of the observed position of a celestial body resulting from the motion of the observer. Stellar aberration is divided into diurnal, annual, and secular components. (See aberration, diurnal: aberration, annual: aberration, secular).

Altitude : the angular distance of a celestial body above or below the horizon, measured along the great circle passing through the body and the zenith. Altitude is 90 ° minus zenith distance.

Anomaly : angular measurement of a body in its orbit from its perihelion.

Aphelion : the point in a planetary orbit that is at the greatest distance from the Sun.

Apogee : the point at which a body in orbit around the Earth reaches its farthest distance from the Earth.

Apparent place : the position on a celestial sphere, centered at the Earth, determined by removing from the directly observed position of a celestial body the effects that depend on the topocentric location of the observer: i.e., refraction, diurnal aberration (see aberration, diurnal) and geocentric (diurnal) parallax. Thus the position at which the object would actually be seen from the center of the Earth, displaced by planetary aberration (except the diurnal part - see aberration, planetary aberration, diurnal) and referred to the true equator and equinox.

Apparent solar time : the measure of time based on the diurnal motion of the true Sun. The rate of diurnal motion undergoes seasonal variation because of the obliquity of the ecliptic and because of the eccentricity of the Earth's orbit. Additional small variations result from irregularities in the rotation of the Earth on it axis.

Aspect : the apparent position of any of the planets or the Moon relative to the Sun, as seen from Earth.

Astrometric ephemeris : an ephemeris of a solar system body in which the tabulated positions are essentially comparable to catalog mean places of stars at a standard epoch. An astrometric position is obtained by adding to the geometric position, computed from gravitational theory, the correction for light-time. Prior to 1984, the E-terms of annual aberration (see aberration, annual aberration, E-terms of) were also added to the geometric position.

Astronomical coordinates : the longitude and latitude of a point on the Earth relative to the geoid. These coordinates are influenced by local gravity anomalies. (See zenith longitude, terrestrial latitude, terrestrial).

Astronomical unit (a.u.) : the radius of a circular orbit in which a body of negligible mass, and free of perturbations, would revolve around the Sun in 2 p /k days, where k is the Gaussian gravitational constant. This is slightly less than the semimajor axis of the Earth's orbit.

Atomic second : see second, Systeme International.

Augmentation : the amount by which the apparent semidiameter of a celestial body, as observed from the surface of the Earth, is greater than the semidiameter that would be observed from the center of the Earth.

Azimuth : the angular distance measured clockwise along the horizon from a specified reference point (usually north) to the intersection with the great circle drawn from the zenith through a body on the celestial sphere.

Barycenter : the center of mass of a system of bodies e.g., the center of mass of the solar system or the Earth-Moon system.

Barycentric Dynamical Time (TDB) : the independent argument of ephemerides and equations of motion that are referred to the barycenter of the solar system. A family of time scales results from the transformation by various theories and metrics of relativistic theories of Terrestrial dynamical Time (TDT). TDB differs from TDT only by periodic variations. In the terminology of the general theory of relativity. TDB may be considered to be a coordinate time. (See dynamical time.)

Brilliancy : for Mercury and Venus the quantity ks 2 /r 2 , where k=0.5 (1+cos i), i is the phase angle, s is the apparent semidiameter, and r is the heliocentric distance.

Calendar : a system of reckoning time in which days are enumerated according to their position in cyclic patterns.

Catalog equinox : the intersection of the hour circle of zero right ascension of a star catalog with the celestial equator. (See dynamical equinox equator.)

Celestial ephemeris pole : the reference pole for nutation and polar motion: the axis of figure for the mean surface of a model Earth in which the free motion has zero amplitude. This pole has no nearly-diurnal nutation with respect to a space-fixed or Earth-fixed coordinate system.

Celestial equator : the plane perpendicular to the celestial ephemeris pole. Colloquially, the projection onto the celestial sphere of the Earth's equator. (See mean equator and equinox: true equator and equinox.)

Celestial pole : either of the two points projected onto the celestial sphere by the extension of the Earth's axis of rotation to infinity.

Celestial sphere : an imaginary sphere of arbitrary radius upon which celestial bodies may be considered to be located. As circumstances require, the celestial sphere may be centered at the observer, at the Earth's center, or at any other location.

Conjunction : the phenomenon in which two bodies have the same apparent celestial longitude (see longitude, celestial) or right ascension as viewed from a third body. Conjunctions are usually tabulated as geocentric phenomena. For Mercury and Venus, geocentric inferior conjunction occurs when the planet is between the Earth and Sun, and superior conjunction occurs when the Sun is between the planet and Earth.

Constellation : a grouping of stars, usually with pictorial or mythical associations, that serves to identify an area of the celestial sphere. Also one of the precisely defined areas of the celestial sphere, associated with a grouping of stars, that the International Astronomical Union has designated as a constellation.

Coordinated Universal Time (UTC) : the time scale available from broadcast time signals. UTC differs from TAI (see International Atomic Time) by an integral number of seconds: it is maintained within ± 0.90 second of UT1 (see Universal Time) by the introduction of one second steps (leap seconds). (See leap second).

Culmination : passage of a celestial object across the observer's meridian also called "meridian passage". More precisely, culmination is the passage through the point of greatest altitude in the diurnal path. Upper culmination (also called "culmination above pole" for circumpolar stars and the Moon) or transit is the crossing closer to the observer's zenith. Lower culmination (also called "culmination below pole" for circumpolar stars and the Moon) is the crossing farther from the zenith.

Day : an interval of 86 400 SI seconds (see second, Systeme International), unless otherwise indicated.

Day numbers : quantities that facilitate hand calculations of the reduction of mean place to apparent place. Besselian day numbers depend solely on the Earth's position and motion: second-order day numbers, used in higher precision reductions, depend on the positions of both the Earth and the star.

Declination : angular distance on the celestial sphere north or south of the celestial equator. It is measured along the hour circle passing through the celestial object. Declination is usually given in combination with right ascension or hour angle.

Defect of illumination : the angular amount of the observed lunar or planetary disk that is not illuminated to an observer on the Earth.

Deflection of light : the angle by which the apparent path of a photon is altered from a straight line by the gravitational field of the Sun. The path is deflected radially away from the Sun by up to 1 ² .75 at the Sun's limb. Correction for this effect, which is independent of wavelength, is included in the reduction from mean place to apparent place.

Deflection of the vertical : the angle between the astronomical vertical and the geodetic vertical. (See zenith: astronomical coordinates: geodetic coordinates.)

Delta T ( D T) : the difference between dynamical time and Universal Time specifically the difference between Terrestrial Dynamical Time (TDT) and UT1: D T = TDT-UT1.

Direct motion : for orbital motion in the solar system, motion that is counterclockwise in the orbit as seen from the north pole of the ecliptic for an object observed on the celestial sphere, motion that is from west to east, resulting from the relative motion of the object and the Earth.

Diurnal motion : the apparent daily motion caused by the Earth's rotation, of celestial bodies across the sky from east to west.

D UT1 : the predicted value of the difference between UT1 and UTC, transmitted in code on broadcast time signals: D UT1= UT1-UTC. (See Universal Time Coordinated Universal Time.)

Dynamical equinox : the ascending node of the Earth's mean orbit on the Earth's true equator: i.e., the intersection of the ecliptic with the celestial equator at which the Sun's declination is changing from south to north. (See catalog equinox, equinox, true equator and equinox.)

Dynamical time : the family of time scales introduced in 1984 to replace ephemeris time as the independent argument of dynamical theories and ephemerides. (See Barycentric dynamical Time: Terrestrial Dynamical Time.)

Eccentric anomaly : in undisturbed elliptic motion, the angle measured at the center of the ellipse from pericenter to the point on the circumscribing auxiliary circle from which a perpendicular to the major axis would intersect the orbiting body. (See mean anomaly true anomaly.)

Eccentricity : a parameter that specifies the shape of a conic section one of the standard elements used to describe an elliptic orbit. (See elements, orbital.)

Eclipse : the obscuration of a celestial body caused by its passage through the shadow cast by another body.

Eclipse, annular : a solar eclipse (see eclipse, solar) in which the solar disk is never completely covered but is seen as an annulus or ring at maximum eclipse. An annular eclipse occurs when the apparent disk of the Moon is smaller than that of the Sun.

Eclipse, lunar : an eclipse in which the Moon passes through the shadow cast by the Earth. The eclipse may be total (the Moon passing completely through the Earth's umbra), partial (the Moon passing partially through the Earth's umbra at maximum eclipse), or penumbral (the Moon passing only through the Earth's penumbra).

Eclipse, solar : an eclipse in which the Earth passes through the shadow cast by the Moon. It may be total (observer in the Moon's umbra), partial (observer in the Moon's penumbra), or annular. (See eclipse, annular.)

Ecliptic : the mean plane of the Earth's orbit around the Sun.

Elements, Besselian : quantities tabulated for the calculation of accurate predictions of an eclipse or occultation for any point on or above the surface of the Earth.

Elements, orbital : parameters that specify the position and motion of a body in orbit. (See osculating elements: mean elements.)

Elongation, greatest : the instants when the geocentric angular distances of Mercury and Venus from the Sun are at a maximum.

Elongation (planetary) : the geocentric angle between a planet and the Sun, measured in the plane of the planet, Earth and Sun. Planetary elongations are measured from 0 ° to 180 ° , east or west of the Sun.

Elongation (satellite) : the geocentric angle between a satellite and its primary, measured in the plane of the satellite, planet and Earth. Satellite elongations are measured from 0 ° east or west of the planet.

Epact: the age of the Moon: the number of days since new moon, diminished by one day, on January 1 in the Gregorian ecclesiastical lunar cycle. (See Gregorian calendar: lunar phases.)

Ephemeris : a tabulation of the positions of a celestial object in an orderly sequence for a number of dates.

Ephemeris hour angle : an hour angle referred to the ephemeris meridian.

Ephemeris longitude : longitude (see longitude, terrestrial) measured eastward from the ephemeris meridian.

Ephemeris meridian : a fictitious meridian that rotates independently of the Earth at the uniform rate implicitly defined by Terrestrial Dynamical Time (TDT). The ephemeris meridian is 1.002 738 D T east of the Greenwhich meridian, where D T= TDT-UT1.

Ephemeris time (ET) : the time scale used prior to 1984 as the independent variable in gravitational theories of the solar system. In 1984, ET was replaced by dynamical time.

Ephemeris transit : the passage of a celestial body or point across the ephemeris meridian.

Epoch : an arbitrary fixed instant of time or date used as a chronological reference datum for calendars (see calendar), celestial reference systems, star catalogs, or orbital motions (see orbit).

Equation of center : in elliptic motion the true anomaly minus the mean anomaly. It is the difference between the actual angular position in the elliptic orbit and the position the body would have if its angular motion were uniform.

Equation of the equinoxes : the right ascension of the mean equinox (see mean equator and equinox) referred to the true equator and equinox apparent sidereal time minus mean sidereal time. (See apparent place: mean place.)

Equation of time : the hour angle of the true Sun minus the hour angle of the fictitious mean sun alternatively, apparent solar time minus mean solar time.

Equator : the great circle on the surface of a body formed by the intersection of the surface with the plane passing through the center of the body perpendicular to the axis of rotation. (See celestial equator.)

Equinox : either of the two points on the celestial sphere at which the ecliptic intersects the celestial equator also the time at which the Sun passes through either of these intersection points ie., when the apparent longitude (see apparent place longitude, celestial) of the Sun is 0 ° or 180 ° . (See catalog equinox dynamical equinox for precise usage.)

Era : a system of chronological notation reckoned from a given date.

Fictitious mean sun : an imaginary body introduced to define mean solar time essentially the name of a mathematical formula that defined mean solar time. This concept is no longer used in high precision work.

Flattening : a parameter that specifies the degree by which a planet's figure differs from that of a sphere the ratio f = (a-b)/a, where a is the equatorial radius and b is the polar radius.

Frequency : the number of cycles or complete alternations per unit time of a carrier wave, band, or oscillation.

Frequency standard : a generator whose output is used as a precise frequency reference a primary frequency standard is one whose frequency corresponds to the adopted definition of the second (see second, Systeme International), with its specified accuracy achieved without calibration of the device.

Gaussian gravitational constant (k= 0.017 202 098 95) : the constant defining the astronomical system of units of length (astronomical unit), mass (solar mass) and time (day), by means of Kepler's third law. The dimensions of k 2 are those of Newton's constant of gravitation: L 3 M -1 T -2 .

Gegenshein : faint nebulous light about 20 ° across near the ecliptic and opposite the Sun, best seen in September and October. Also called counterglow.

Geocentric : with reference to, or pertaining to, the center of the Earth.

Geocentric coordinates: the latitude and longitude of a point on the Earth's surface relative to the center of the Earth also celestial coordinates given with respect to the center of the Earth. (See zenith latitude, terrestrial, longitude, terrestrial.)

Geodetic coordinates : the latitude and longitude of a point on the Earth's surface determined from the geodetic vertical (normal to the specified spheroid). (See zenith latitude, terrestrial longitude, terrestrial.)

Geoid : an equipotential surface that coincides with mean sea level in the open ocean. On land it is the level surface that would be assumed by water in an imaginary network of frictionless channels connected to the ocean.

Geometric position : the geocentric position of an object on the celestial sphere referred to the true equator and equinox, but without the displacement due to planetary aberration. (See apparent place mean place aberration, planetary.)

Greenwich sidereal date (GSD) : the number of sidereal days elapsed at Greenwich since the beginning of the Greenwich sidereal day that was in progress at Julian date 0.0.

Greenwich sidereal day number : the integral part of the Greenwich sidereal date.

Gregorian calendar : the calendar introduced by Pope Gregory XIII in 1582 to replace the Julian calendar the calendar now used as the civil calendar in most countries. Every year that is exactly divisible by four is a leap year, except for centurial years, which must be exactly divisible by 400 to be leap years. Thus 2000 is a leap year, but 1900 and 2100 are not leap years.

Height : elevation above ground or distance upwards from a given level (especially sea level) to a fixed point. (See altitude).

Heliocentric : with reference to, or pertaining to, the center of the Sun.

Horizon : a plane perpendicular to the line from an observer to the zenith. The great circle formed by the intersection of the celestial sphere with a plane perpendicular to the line from an observer to the zenith is called the astronomical horizon.

Horizontal parallax : the difference between the topocentric and geocentric positions of an object, when the object is on the astronomical horizon.

Hour angle : angular distance on the celestial sphere measured westward along the celestial equator from the meridian to the hour circle that passes through a celestial object.

Hour circle : a great circle on the celestial sphere that passes through the celestial poles and is therefore perpendicular to the celestial equator.

Inclination : the angle between two planes or their poles usually the angle between an orbital plane and a reference plane one of the standard orbital elements (see elements, orbital) that specifies the orientation of an orbit.

International Atomic Time (TAI) : the continuous scale resulting from analyses by the Bureau International des Poids et Mesures of atomic time standards in many countries. The fundamental unit of TAI is the SI second (see second, Systeme International), and the epoch is 1958 January 1.

Invariable plane : the plane through the center of mass of the solar system perpendicular to the angular momentum vector of the solar system.

Irradiation : an optical effect of contrast that makes bright objects viewed against a dark background appear to be larger than they really are.

Julian calendar : the calendar introduced by Julius Caesar in 46 B.C. to replace the Roman calendar. In the Julian calendar a common year is defined to comprise 365 days, and every fourth year is a leap year comprising 366 days. The Julian calendar was superseded by the Gregorian calendar.

Julian date (JD) : the interval of time in days and fraction of a day since 4713 B.C. January 1, Greenwich noon, Julian proleptic calendar. In precise work the timescale, e.g., dynamical time or universal time, should be specified.

Julian date, modified (MJD) : the Julian date minus 2400000.5.

Julian day number (JD) : the integral part of the Julian date.

Julian proleptic calendar : the calendric system employing the rules of the Julian calendar, but extended and applied to dates preceding the introduction of the Julian calendar.

Julian year : a period of 365.25 days,. This period served as the basis for the Julian calendar.

Laplacian plane : for planets see invariable plane for a system of satellites, the fixed plane relative to which the vector sum of the disturbing forces has no orthogonal component.

Latitude, celestial : angular distance on the celestial sphere measured north or south of the ecliptic along the great circle passing through the poles of the ecliptic and the celestial object.

Latitude, terrestrial : angular distance on the Earth measured north or south of the equator along the meridian of a geographic location.

Leap second : a second (see second, Systeme International) added between 60s and 0s at announced times to keep UTC within 0.90 s of UT1. Generally, leap seconds are added at the end of June or December.

Librations : variations in the orientation of the Moon's surface with respect to an observer on the Earth. Physical librations are due to variations in the orientation of the Moon's rotational axis in inertial space. The much larger optical librations are due to variations in the rate of the Moon's orbital motion, the obliquity of the Moon's equator to its orbital plane, and the diurnal changes of geometric perspective of an observer on the Earth's surface.

Light, deflection of : the bending of the beam of light due to gravity. It is observable when the light from a star or planet passes a massive object such as the Sun.

Light-time : the interval of time required for light to travel from a celestial body to the Earth. During this interval the motion of the body in space causes an angular displacement of its apparent place from its geometric place (see geometric position). (See aberration, planetary.)

Light-year : the distance that light traverses in a vacuum during one year.

Limb: the apparent edge of the Sun, Moon, or a planet or any other celestial body with a detectable disc.

Limb correction : correction that must be made to the distance between the center of mass of the Moon and its limb. These corrections are due to the irregular surface of the Moon and are a function of the librations in longitude (see longitude, celestial) and latitude (see latitude, celestial) and the position angle from the central meridian.

Local sidereal time : the local hour angle of a catalog equinox.

Longitude, celestial : angular distance on the celestial sphere measured eastward along the ecliptic from the dynamical equinox to the great circle passing through the poles of the ecliptic and the celestial object.

Longitude, terrestrial : angular distance measured along the Earth's equator from the Greenwich meridian to the meridian of a geographic location.

Luminosity class : distinctions among stars of the same spectral class. (See Spectral types or classes.)

Lunar phases : cyclically recurring apparent forms of the Moon. New moon, first quarter, full moon and last quarter are defined as the times at which the excess of the apparent celestial longitude (see longitude, celestial) of the Moon over that of the Sun is 0 ° , 90 ° , 180 ° and 270 ° , respectively.

Lunation : the period of time between two consecutive new moons.

Magnitude, stellar : a measure on a logarithmic scale of the brightness of a celestial object considered as a point source.

Magnitude of a lunar eclipse : the fraction of the lunar diameter obscured by the shadow of the Earth at the greatest phase of a lunar eclipse (see eclipse, lunar), measured along the common diameter.

Magnitude of a solar eclipse : the fraction of the solar diameter obscured by the Moon at the greatest phase of a solar eclipse (see eclipse, solar), measured along the common diameter.

Mean anomaly : in undisturbed elliptic motion, the product of the mean motion of an orbiting body and the interval of time since the body passed pericenter. Thus the mean anomaly is the angle from pericenter of a hypothetical body moving with a constant angular speed that is equal to the mean motion. (See true anomaly eccentric anomaly.)

Mean distance : the semimajor axis of an elliptic orbit.

Mean elements : elements of an adopted reference orbit (see elements, orbital) that approximates the actual, perturbed orbit. Mean elements may serve as the basis for calculating perturbations.

Mean equator and equinox : the celestial reference system determined by ignoring small variations of short period in the motions of the celestial equator. Thus the mean equator and equinox are affected only by precession. Positions in star catalogs are normally referred to the mean catalog equator and equinox (see catalog equinox) of a standard epoch.

Mean motion : in undisturbed elliptic motion, the constant angular speed required for a body to complete one revolution in an orbit of a specified semimajor axis.

Mean place : the geocentric position, referred to the mean equator and equinox of a standard epoch, of an object on the celestial sphere centered at the Sun. A mean place is determined by removing from the directly observed position the effects of refraction, geocentric and stellar parallax, and stellar aberration (see aberration, stellar), and by referring the coordinates to the mean equator and equinox of a standard epoch. In compiling star catalogs it has been the practice not to remove the secular part of stellar aberration (see aberration, secular). Prior to 1984, it was additionally the practice not to remove the elliptic part of annual aberration (see aberration, annual aberration, E-terms of).

Mean solar time : a measure of time based conceptually on the diurnal motion of the fictitious mean sun, under the assumption that the Earth's rate of rotation is constant.

Meridian : a great circle passing through the celestial poles and through the zenith of any location on Earth. For planetary observations a meridian is half the great circle passing through the planet's poles and through any location on the planet.

Month : the period of one complete synodic or sidereal revolution of the Moon around the Earth also a calendrical unit that approximates the period of revolution.

Moonrise, moonset : the times at which the apparent upper limb of the Moon is on the astronomical horizon: i.e., when the true zenith distance, referred to the center of the Earth, of the central point of the disk is 90 ° 34 ¢ + s - p , where s is the Moon's semidiameter, p is the horizontal parallax, and 34 ¢ is the adopted value of horizontal refraction.

Nadir : the point on the celestial sphere diametrically opposite to the zenith.

Node : either of the points on the celestial sphere at which the plane of an orbit intersects a reference plane. The position of a node is one of the standard orbital elements (see elements, orbital) used to specify the orientation of an orbit.

Nutation : the short-period oscillations in the motion of the pole of rotation of a freely rotating body that is undergoing torque from external gravitational forces. Nutation of the Earth's pole is discussed in terms of components in obliquity and longitude (see longitude, celestial.)

Obliquity : in general the angle between the equatorial and orbital planes of a body or, equivalently, between the rotational and orbital poles. For the Earth the obliquity of the ecliptic is the angle between the planes of the equator and the ecliptic.

Occultation : the obscuration of one celestial body by another of greater apparent diameter especially the passage of the Moon in front of a star or planet, or the disappearance of a satellite behind the disk of its primary. If the primary source of illumination of a reflecting body is cut off by the occultation, the phenomenon is also called an eclipse. The occultation of the Sun by the Moon is a solar eclipse (see eclipse, solar.)

Opposition : a configuration of the Sun. Earth and a planet in which the apparent geocentric longitude (see longitude, celestial) of the planet differs by 180 ° from the apparent geocentric longitude of the Sun.

Orbit : the path in space followed by a celestial body.

Osculating elements : a set of parameters (see elements, orbital) that specifies the instantaneous position and velocity of a celestial body in its perturbed orbit. Osculating elements describe the unperturbed (two-body) orbit that the body would follow if perturbations were to cease instantaneously.

Parallax : the difference in apparent direction of an object as seen from two different locations conversely, the angle at the object that is subtended by the line joining two designated points. Geocentric (diurnal) parallax is the difference in direction between a topocentric observation and a hypothetical geocentric observation. Heliocentric or annual parallax is the difference between hypothetical geocentric and heliocentric observations it is the angle subtended at the observed object by the semimajor axis of the Earth's orbit. (See also horizontal parallax.)

Parsec : the distance at which one astronomical unit subtends an angle of one second of arc equivalently the distance to an object having an annual parallax of one second of arc.

Penumbra : the portion of a shadow in which light from an extended source is partially but not completely cut off by an intervening body the area of partial shadow surrounding the umbra.

Pericenter : the point in an orbit that is nearest to the center of force. (See perigee perihelion).

Perigee : the point at which a body in orbit around the Earth most closely approaches the Earth. Perigee is sometimes used with reference to the apparent orbit of the Sun around the Earth.

Perihelion : the point at which a body in orbit around the Sun most closely approaches the Sun.

Period : the interval of time required to complete one revolution in an orbit or one cycle of a periodic phenomenon, such as a cycle of phases. (See phase.)

Perturbations : deviations between the actual orbit of a celestial body and an assumed reference orbit also the forces that cause deviations between the actual and reference orbits. Perturbations, according to the first meaning, are usually calculated as quantities to be added to the coordinates of the reference orbit to obtain the precise coordinates.

Phase : the ratio of the illuminated area of the apparent disk of a celestial body to the area of the entire apparent disk taken as a circle. For the Moon phase designations (see lunar phases) are defined by specific configurations of the Sun, Earth and Moon. For eclipses, phase designations (total, partial, penumbral, etc.) provide general descriptions of the phenomena. (See eclipse, solar eclipse, annular eclipse, lunar.)

Phase angle : the angle measured at the center of an illuminated body between the light source and the observer.

Photometry : a measurement of the intensity of light usually specified for a specific frequency range.

Planetocentric coordinates : coordinates for general use, where the z-axis is the mean axis of rotation, the x-axis is the intersection of the planetary equator (normal to the z-axis through the center of mass) and an arbitrary prime meridian, and the y-axis completes a right-hand coordinate system. Longitude (see longitude, celestial) of a point is measured positive to the prime meridian as defined by rotational elements. Latitude (see latitude, celestial) of a point is the angle between the planetary equator and a line to the center of mass. The radius is measured from the center of mass to the surface point.

Planetographic coordinates : coordinates for cartographic purposes dependent on an equipotential surface as a reference surface. Longitude (see longitude, celestial) of a point is measured in the direction opposite to the rotation (positive to the west for direct rotation) from the cartographic position of the prime meridian defined by a clearly observable surface feature. Latitude (see latitude, celestial) of a point is the angle between the planetary equator (normal to the z-axis and through the center of mass) and normal to the reference surface at the point. The height of a point is specified as the distance above a point with the same longitude and latitude on the reference surface.

Polar motion : the irregularly varying motion of the Earth's pole of rotation with respect to the Earth's crust. (See celestial ephemeris pole.)

Precession : the uniformly progressing motion of the pole of rotation of a freely rotating body undergoing torque from external gravitational forces. In the case of the Earth, the component of precession caused by the Sun and Moon acting on the Earth's equatorial bulge is called lunisolar precession the component caused by the action of the planets is called planetary precession. The sum of lunisolar and planetary precession is called general precession. (See nutation.)

Proper motion : the projection onto the celestial sphere of the space motion of a star relative to the solar system thus the transverse component of the space motion of a star with respect to the solar system. Proper motion is usually tabulated in star catalogs as changes in right ascension and declination per year or century.

Quadrature : a configuration in which two celestial bodies have apparent longitudes (see longitude, celestial) that differ by 90 ° as viewed from a third body. Quadratures are usually tabulated with respect to the Sun as viewed from the center of the Earth.

Radial velocity : the rate of change of the distance to an object.

Refraction, astronomical : the change in direction of travel (bending) of a light ray as it passes obliquely through the atmosphere. As a result of refraction the observed altitude of a celestial object is greater than its geometric altitude. The amount of refraction depends on the altitude of the object and on atmospheric conditions.

Retrograde motion : for orbital motion in the solar system, motion that is clockwise in the orbit as seen from the north pole of the ecliptic for an object observed on the celestial sphere, motion that is from east to west, resulting from the relative motion of the object and the Earth. (See direct motion.)

Right ascension : angular distance on the celestial sphere measured eastward along the celestial equator from the equinox to the hour circle passing through the celestial object. Right ascension is usually given in combination with declination.

Saber's Beads : detached points of light seen along the limb of very young and old lunar crescents. The necklace of staggered brightness peaks near New Moon is reminiscent of the moments before and after a total solar eclipse.

Satellite : natural body revolving around a planet.

Satellite, artificial : device launched into a closed orbit around the Earth, another planet, the Sun, etc.

Second, Systeme International (SI): the duration of 9 192 631 770 cycles of radiation corresponding to the transition between two hyperfine levels of the ground state of cesium 133.

Selenocentric : with reference to, or pertaining to, the center of the Moon.

Semidiameter : the angle at the observer subtended by the equatorial radius of the Sun, Moon or a planet.

Semimajor axis : half the length of the major axis of an ellipse a standard element used to describe an elliptical orbit (see elements, orbital.)

Sidereal day : the interval of time between two consecutive transits of the catalog equinox. (See sidereal time.)

Sidereal hour angle : angular distance on the celestial sphere measured westward along the celestial equator from the catalog equinox to the hour circle passing through the celestial object. It is equal to 360 ° minus right ascension in degrees.

Sidereal time : the measure of time defined by the apparent diurnal motion of the catalog equinox hence a measure of the rotation of the Earth with respect to the stars rather than the Sun.

Solstice : either of the two points on the ecliptic at which the apparent longitude (see longitude, celestial) of the Sun is 90 ° or 270 ° also the time at which the Sun is at either point.

Spectral types or classes : catagorization of stars according to their spectra, primarily due to differing temperatures of the stellar atmosphere. From hottest to coolest, the spectral types are O, B, A, F, G, K and M.

Standard epoch : a date and time that specifies the reference system to which celestial coordinates are referred. Prior to 1984 coordinates of star catalogs were commonly referred to the mean equator and equinox of the beginning of a Besselian year (see year, Besselian). Beginning with 1984 the Julian year has been used, as denoted by the prefix J, e.g., J2000.0.

Stationary point (of a planet) : the position at which the rate of change of the apparent right ascension (see apparent place) of a planet is momentarily zero.

Sunrise, sunset : the times at which the apparent upper limb of the Sun is on the astronomical horizon i.e., when the true zenith distance, referred to the center of the Earth, of the central point of the disk is 90 ° 50 ¢ , based on adopted values of 34 ¢ for horizontal refraction and 16 ¢ for the Sun's semidiameter.

Surface brightness (of a planet): the visual magnitude of an average square arc-second area of the illuminated portion of the apparent disk.

Synodic period : for planets, the mean interval of time between successive conjunctions of a pair of planets, as observed from the Sun for satellites, the mean interval between successive conjunctions of a satellite with the Sun, as observed from the satellite's primary.

Synodic time : pertaining to successive conjunctions successive returns of a planet to the same aspect as determined by Earth.

Terrestrial Dynamical Time (TDT) : the independent argument for apparent geocentric ephemerides. At 1977 January 1d00h00m00sTAI, the value of TDT was exactly 1977 January 1.0003725 d. The unit of TDT is 86 400 SI seconds at mean sea level. For practical purposes TDT= TAI + 32.184 s. (See Barycentric Dynamical Time dynamical time International Atomic Time.)

Terminator : the boundary between the illuminated and dark areas of the apparent disk of the Moon, a planet or a planetary satellite.

Topocentric : with reference to, or pertaining to, a point on the surface of the Earth.

Transit : the passage of the apparent center of the disk of a celestial object across a meridian: also the passage of one celestial body in front of another of greater apparent diameter (e.g., the passage of Mercury or Venus across the Sun or Jupiter’s satellites across its disk) however, the passage of the Moon in front of the larger apparent Sun is called an annular eclipse (see eclipse, annular). The passage of a body's shadow across another body is called a shadow transit however, the passage of the Moon's shadow across the Earth is called a solar eclipse. (See eclipse, solar.)

True anomaly : the angle, measured at the focus nearest the pericenter of an elliptical orbit, between the pericenter and the radius vector from the focus to the orbiting body one of the standard orbital elements (see elements, orbital). (See also eccentric anomaly mean anomaly.)

True equator and equinox : the celestial coordinate system determined by the instantaneous positions of the celestial equator and ecliptic. The motion of this system is due to the progressive effect of precession and the short-term, periodic variations of nutation. (See mean equator and equinox.)

Twilight : the interval of time preceding sunrise and following sunset (see sunrise, sunset) during which the sky is partially illuminated. Civil twilight comprises the interval when the zenith distance, referred to the center of the Earth, of the central point of the Sun's disk is between 90 ° 50 ¢ and 96 ° , nautical twilight comprises the interval from 96 ° to 102 ° , astronomical twilight comprises the interval from 102 ° to 108 ° .

Umbra : the portion of a shadow cone in which none of the light from an extended light source (ignoring refraction) can be observed.

Universal Time (UT) : a measure of time that conforms, within a close approximation, to the mean diurnal motion of the Sun and serves as the basis of all civil timekeeping. UT is formally defined by a mathematical formula as a function of sidereal time. Thus UT is determined from observations of the diurnal motions of the stars. The time scale determined directly from such observations is designated UT0: it is slightly dependent on the place of observation. When UT0 is corrected for the shift in longitude (see longitude, terrestrial) of the observing station caused by polar motion, the time scale UT1 is obtained.

Vernal equinox : the ascending node of the ecliptic on the celestial equator also the time at which the apparent longitude (see apparent place longitude, celestial) of the Sun is 0 ° . (See equinox.)

Vertical : apparent direction of gravity at the point of observation (normal to the plane of a free level surface.)

Week : an arbitrary period of days, usually seven days approximately equal to the number of days counted between the four phases of the Moon. (See lunar phases.)

Year : a period of time based on the revolution of the Earth around the Sun. The calendar year (see Gregorian calendar) is an approximation to the tropical year (see year, tropical). The anomalistic year is the mean interval between successive passages of the Earth through perihelion. The sidereal year is the mean period of revolution with respect to the background stars. (See Julian year: year, Besselian.)

Year, Besselian : the period of one complete revolution in right ascension of the fictitious mean sun, as defined by Newcomb. The beginning of a Besselian year, traditionally used as standard epoch, is denoted by the suffix ".0". Since 1984 standard epochs have been defined by the Julian year rather that the Besselian year. For distinction, the beginning of the Besselian year is now identified by the prefix B (e.g., B1950.0).

Year, tropical : the period of one complete revolution of the mean longitude of the sun with respect to the dynamical equinox. The tropical year is longer than the Besselian year (see year, Besselian) by 0.148T s, where T is centuries from B1900.0.

Zenith : in general, the point directly overhead on the celestial sphere. The astronomical zenith is the extension to infinity of a plumb line. The geocentric zenith is defined by the line from the center of the Earth through the observer. The geodetic zenith is the normal to the geodetic ellipsoid at the observer's location. (See deflection of the vertical.)

Zenith distance : angular distance on the celestial sphere measured along the great circle from the zenith to the celestial object. Zenith distance is 90 ° minus altitude.

Zodiacal light : a nebulous light seen in the east before twilight and in the west after twilight. It is triangular in shape along the ecliptic with the base on the horizon and its apex at varying altitudes. It is best seen in middle latitudes (see latitude, terrestrial) on spring evenings and autumn mornings.

* Note : This glossary is taken from The Astronomical Almanac (1994), M1-M13, after requesting a permission.

Peculiar Amplitude of Earth's Orbit in Z Axis - Astronomy

Department of Aerospace Engineering, Karunya University, Coimbatore, India

Received 20 July 2016 accepted 11 September 2016 published 14 September 2016

This paper deals with generation of halo orbits in the three-dimensional photogravitational restricted three-body problem, where the more massive primary is considered as the source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. Both the terms due to oblateness of the smaller primary are considered. Numerical as well as analytical solutions are obtained around the Lagrangian point L1, which lies between the primaries, of the Sun-Earth system. A comparison with the real time flight data of SOHO mission is made. Inclusion of oblateness of the smaller primary can improve the accuracy. Due to the effect of radiation pressure and oblateness, the size and the orbital period of the halo orbit around L1 are found to increase.

Halo Orbits, Photogravitational Restricted Three-Body Problem, Oblateness, Lindstedt-Poincaré Method, Lagrangian Point, SOHO

The subject of halo orbits in restricted three-body problem (RTBP) has received considerable attention in the last four decades. A halo orbit is a periodic, three-dimensional orbitnear the collinear Lagrangian points in the three-body problem. Analytical solutions to generate the halo orbits around any of the collinear points can be obtained. Farquhar first used the name “halo” in his doctoral thesis [1] and proposed an idea for placing a satellite in an orbit around L2 of the Earth-Moon RTBP. Had this idea been implemented, one could continuously view the Earth and the dark side of the moon at the same time. A communication link satellite is not necessary, if satellites are placed around Lagrangian points of the respective systems. Breakwell & Brown [2] generated halo orbits for the Earth-Moon system. Richardson [3] contributed significantly to obtain the halo orbits, by finding an analytical solution using Lindstedt-Poincaré method. A good amount of research has been carried out till date in this interesting area of halo orbits. Some of the important contributions are by Howell [4] , Howell and Pernicka [5] , Folta and Richon [6] and Howell et al. [7] .

The classical model of the restricted three-body problem are relatively less accurate for studying the motion of any Sun-planet system as they do not account for the effect of the perturbing forces such as oblateness of planets, cosmic rays, magnetic field, radiation pressure etc. Cosmic rays are immensely high-energy radiation, mainly originating outside the solar system. They produce showers of secondary particles that penetrate and impact the Earth’s atmosphere and sometimes even reach the surface. They interact with gaseous and other matter at high altitudes and produce secondary radiation. The combination of both contributes to the space radiation environment. In addition to the particles originating from the Sun are particles from other stars and heavy ion sources such as nova and supernova in our galaxy and beyond. In interplanetary space these ionizing particles constitute the major radiation threat. These particles are influenced by planetary or Earth’s magnetic field to form radiation belts, which in Earth’s case are known as Van Allen Radiation belts, containing trapped electrons in the outer belt and protons in the inner belt. The composition and intensity of the radiation varies significantly with the trajectory of a space vehicle. Anomalies in communication satellite operation have been caused by the unexpected triggering of digital circuits by the cosmic rays. In our present study, we restrict ourselves with the solar radiation and oblateness of the planet.

Radzievskii [8] was the first one to study the effect of solar radiation pressure. He found out that the maximum force due to the radiation pressure acts in the radial direction, given by

where q is defined in terms of particle radius a, density δ and radiation pressure efficiency factor x as

/>(c.g.s. units)

/>.

/>is a variable, depending upon the nature of the third body (satellite). The value of q can be considered as a constant, if the fluctuations in the beam of solar radiation and the effect of planet’s shadow are neglected. Using the model of [1] , Dutt and Sharma [9] studied periodic orbits in the Sun-Mars system using the numerical technique of Poincare surface of sections and found out more than 74 periodic orbits.

Sharma and Subba Rao [10] introduced the oblateness of the more massive primary in the three-dimensional restricted three-body problem. It has two terms, one with a z term in the numerator. Sharma [11] studied the periodic orbits around the Lagrangian points in the planar RTBP by considering Sun as source of radiation and smaller primary as an oblate spheroid with its equatorial plane coincident with the plane of motion. Tiwary and Kushvah [12] followed the model of Sharma [11] to study the halo orbits around the Lagrangian points L1 and L2 analytically. However, they did not consider the z term in oblateness in their study. In the present work, we have considered both the terms due to oblateness of the smaller primary in the photogravitational restricted three-body problem to study the halo orbits analytically as well as numerically around L1.

The first satellite placed in the halo orbit at Sun-Earth L1 point was International Sun-Earth Explorer-3 (ISEE-3), launched in 1978. Solar Heliospheric Observatory (SOHO), launched in 1975 by NASA succeeded ISEE-3.We have taken data of the path of the SOHO mission from the mission website over a period of January-June 2008, to validate our analytical and numerical solutions.

2. Circular Restricted Three-Body Problem

The circular RTBP consists of two primary masses revolving in circular orbits around their centre of mass under the influence of their mutual gravity. The third body of infinitesimal mass moves under the gravitational effect of these two primaries (Figure 1). RTBP has five equilibrium points, called Lagrangian or libration points. These points are the points of zero velocities and an object placed in these points remains there. Out of the five Lagrangian points, three are collinear (L1, L2, L3) and the other two points (L4, L5) form equilateral triangles with the primaries. Although the Lagrangian point is just a point in empty space, its peculiar characteristic is that

Figure 1 . Three-dimensional restricted three-body problem.

it can be orbited. Halo orbits are three-dimensional orbits around the collinear points.

The equations of motion for the RTBP (Szebehely [13] ) including radiation pressure and oblateness of the smaller primary is written in accordance with Sharma & Subba Rao [10] , Sharma [11] as

/>(1)

/>(2)

/>(3)

/>, />

/>, where m1 and m2 are masses of larger and smaller primary respectively.

The perturbed mean motion, n of the primaries due to oblateness is given by

AE, AP being the dimensional equatorial and polar radii of the smaller primary and R is the distance between the primaries. The two terms occurring in Ω due to oblateness of the smaller primary were introduced by Sharma and Subba Rao [10] .

3. Computation of Halo Orbits

For the computation of the halo orbits, the origin is transferred to the Lagrangian points L1 and L2. The transformation is given by

The equation of motion can be written as

/>, />

The upper sign in the above equations depicts the Lagrangian point L1 and the lower sign corresponds to L2. The distance between these Lagrangian points and the smaller primary is considered to be the normalized unit as in Koon et al. [14] and Tiwary and Kushvah [12] .

The usage of Legendre polynomials can result in some computational advantages, when non-linear terms are considered. The non-linear terms are expanded by using the following formula given in Koon et al. [14] :

The above formula is used for expanding the non-linear terms in the equations of motion. The equations of motion after substituting the values of the non-linear terms and by some algebraic manipulations by defining a new variable cm after expanding up to m = 2 become

/>(4)

/>(5)

/>(6)

Neglecting the higher-order terms in Equations ((4)-(6)), we get

/>(7)

/>(8)

/>(9)

It is clear that the z-axis solution obtained by putting X = Y = 0 does not depend upon X and Y and c2 > 0. Hence we can conclude that the motion in Z-direction is simple harmonic. The motion in XY-plane is coupled.

A fourth degree polynomial is obtained which gives two real and two imaginary roots as eigenvalues:

The solution of the linearized Equations ((7)-(9)), as in Thurman and Worfolk [15] , is

/>are arbitrary constants. Since we are concentrating on constructing a halo orbit, which is periodic, we consider />.

Frequency and amplitude terms are introduced to find solution through Lindstedt-Poincaré method. The solution of the linearized equations is written in terms of the amplitudes (Ax and Az) and the phases (In-plane phase, ϕ and out-of-plane phase, ψ) and the frequencies (λ and />), with an assumption that />, as

For halo orbits, the amplitudes />and />are constrained by a non-linear algebraic relationship given by Richardson [3] :

where l1 and l2 depend upon the roots of the characteristic equation of the linear equation. The correction term />arises due to the addition of frequency term in Equation (9).

Hence, any halo orbit can be characterized by specifying a particular out-of-plane amplitude />of the solution to linearized equations of motion. Both analytical and numerical methods employ this scheme. From the above expression, we can find the minimum permissible value of Ax to form the halo orbit (Az > 0).

For halo orbits, the phases ϕ and ψ are related as

When Ax is greater than certain value, the third-order solution bifurcates. This bifurcation is manifested through the phase-angle constraint. The solution branches are obtained according to the value of m. For m = 1, Az is positive and we have the northern halo (z > 0). For m = 3, Az is negative and we have the southern halo (z < 0).

3.3. Lindstedt-Poincaré Method

Lindstedt-Poincaré method involves successive adjustments of the frequencies to avoid secular terms and provides approximate periodic solution. The equations of motion with non-linear terms up to third-order approximation as in Richardson [3] and Thurman and Worfolk [15] are

A new independent variable τ = ωt is introduced, where />refers to />.

The values of />are chosen in such a way that the secular terms are removed with successive approximations. Most of the secular terms are removed by the following assumptions:

The coefficients />and />are given in Appendix. The equations of motion are

/>(10)

/>(11)

/>(12)

We continue the perturbation analysis by assuming the solutions of the form

/>(13)

/>(14)

/>(15)

where />is a small parameter.

Substituting Equations ((13)-(15)) in Equations ((10)-(12)), and equating the coefficients of the same order of />, we get the first-, second- and the third-order equations, respectively.

The first-order equations are obtained by taking the coefficients of the term />. These are

The periodic solution to the above equations is

/>(16)

/>(17)

/>(18)

where />.

3.3.2. Second-Order Equations

Coefficients of the term />provide the second-order equations

To remove the secular terms, we set ω1 = 0. The particular solution of the second-order equations are obtained with the help of Maxima software as

/>(19)

/>(20)

/>(21)

The coefficients are given in the Appendix.

By collecting the coefficients of />, we get the third-order equations

From the above equations, it is not possible to remove the secular terms by setting ω2 = 0.

Hence, the phase relation is used to remove the secular terms.

The solution of the third-order equation is obtained as

(22)

(23)

(24)

The coefficients are given in the Appendix. Thus, the third-order analytical solution is obtained.

The mapping, and , will remove from all the equations. We now combine the solutions up to third-order to get the final solution as

The coefficients are given in the Appendix.

4. Analytical Construction of Orbit

The input parameters to generate a halo orbit include mass ratio (μ), mean distance between the two primaries, radiation pressure (q or ε), oblateness coefficient (A2), amplitude in z-direction (Az). Once all the input parameters are given, the co-ordinates for the halo orbit are computed.

4.2. Halo Orbit for Classical Case

The halo orbits are generated with the following data for Sun-Earth L1:

Mass ratio = 0.000003, mean distance = 149,600,000 km, q = 1, A2 = 0.

The amplitude in z-direction is taken to be Az= 110,000 km (Amplitude of the ISEE-3 mission around the Sun-Earth L1 point). Figures 2-5 show the different views of the halo orbits for the classical case.

4.3. Halo Orbit for Different Values of Radiation Pressure

The halo orbits are generated for different values of radiation pressurein the vicinity of L1. Figure 6 shows the variation in orbit for different values of radiation pressure. The orbital period increases and the orbits increase in size with the increase in the radiation pressure of the more massive primary.

Table 1 shows the change in the distance of the Lagrangian point L1 due to radiation pressure. It is noticed that the Lagrangian point L1 moves towards the more massive primary with increase in radiation pressure. It is also noticed that L1 moves further towards the more massive primary for q < 0.8.

5. Numerical Computation of Halo Orbits

The third-order analytical solution provides a good initial estimate to compute the halo orbits numerically. Analytical approximation must be combined with a suitable numerical method to obtain accurate halo orbits around the Lagrangian points. The method of differential correction is a powerful application of Newton’s method that employs the State Transition Matrix (STM) to solve various kinds of boundary value problems. In order to propagate an orbit in RTBP, the equations of motion are numerically integrated by using adaptive fourth-order Runge-Kutta method, until the desired orbit is obtained.

The initial guess for finding the numerical solution is taken from the analytic solution. Since the halo orbits are symmetric about xz-plane (y = 0), and they intersect this plane perpendicularly i.e., the initial state vector takes the form

The final state vector which lies on the same plane, takes the form as given below and crosses the xz-plane perpendicularly:

Figure 2 . X vs Y for Sun-Earth system.

Figure 3 . Y vs Z for Sun-Earth system.

Then the orbit will be periodic with period The state transition matrix at can be used to adjust the initial values of a nearby periodic orbit. The equations of motion are integrated until y changes sign. Then the step size is reduced and the integration goes forward again. This is repeated until y becomes almost zero, and the time at this point is defined to be. The tolerance is considered to be around 10 −12 . The orbit is considered periodic if and are nearly zero at. If this is not the case, and can be reduced by correcting two of the three initial conditions and then integrate the equations of motion again.

5.1. Numerically Generated Halo Orbits

The numerically generated orbits which are more accurate, are usually considered for mission purposes. Figure 7

Figure 4 . X vs Z for sun-earth system.

Figure 5 . 3D view for sun-earth system.

Figure 6 . Analytically generated halo orbits [A2 = 0].

Figure 7 . Numerically generated halo orbits for different values of q [A2 = 0].

Table 1 . Variation in the location of Lagrangian point L1 with radiation pressure q.

shows the numerically generated orbit due to variation in radiation pressure. The numerically generated obits show a close resemblance to the analytically generated orbits. Figure 8 shows the comparison between them.

The analytical solution which includes the radiation pressure of more massive primary and oblateness of the smaller primary can improve the accuracy over the existing solution as given by Tiwary and Kushvah [12] .

The time period of SOHO orbit was around 179 days and its dimensions were approximately 4.3 × 2.7 × 3.7 m. The computed value of radiation pressure is roughly 0.99997. The flight data of SOHO mission from January to June 2008 was taken from the mission website and is plotted. From Figure 9 and Figure 10, it is clear that the addition of the extra term due to oblateness shows improvement in the orbit computation. However, the deviation in the orbit is due to the actual scenario, where the real satellite in space undergoes the effect of various other perturbations.

Figure 8 . Comparison of numerically and analytically generated halo orbits.

Figure 9 . Comparison of numerically generated halo orbit with SOHO Mission orbit.

Figure 10 . Comparison of numerically generated halo orbit with SOHO mission orbit (enlarged view).

Figure 11 . Effect of adding the new term on halo orbit at L1 with q = 1, A2 = 0.000001.

5.2. Improvement over the Existing Results

It can be seen from the Figure 11 that the results show improvement in orbit computation due to the addition of the new term due to oblateness.

The time period of the orbit increases by 30 minutes for q = 1 and A2 = 0.000001 as given in Table 2. Thus, it is clear that the addition of the new term in the potential function has helped in predicting the orbit closer to the real time condition.

5.3. Perturbing Effects on the Orbit

The effects due to radiation pressure and oblateness can be seen from Figure 12. It is seen that with increase in radiation pressure as well as oblateness, the orbit moves towards the source of radiation i.e. the more massive primary. The oblateness coefficient are taken as A2 = 0, 0.000001, 0.000002 and the radiation pressure, q = 1, 0.99, 0.98. With increase in perturbing forces, it is seen that the orbit moves towards the more massive primary.

Since the amplitude Az is bounded by amplitude constraint, the variation in it also affects the size and time period of the orbit, which can be seen in Figure 13. The time period of the orbit also changes with the perturbations and the amplitude Az. For the halo around L1, with increase in radiation pressure and oblateness, the time period increases. It can be noticed in Figure 14 and Table 3.

The amplitudes about the z-axis, Az was taken in a range of 80,000 to 140,000 km. There is a minimum Ax for the feasibility of a halo orbit, when we fix the amplitude Az.

The analytical solution obtained using Lindstedt-Poincaré method provides good initial estimate to generate halo orbits around the collinear point L1 with numerical integration in the photogravitational restricted three-body

Table 2 . Effect of new term on time period of the halo orbit.

Figure 12 . Variation in orbit due to perturbations around L1 (μ = 0.000003).

Figure 13 . Variation in orbit due to Az (μ = 0.000003).

Figure 14 . Time period due to q and Amplitude, Az around L1 (μ = 0.000003).

problem, when the smaller primary is considered as an oblate spheroid with its equatorial plane coincident with the plane of motion. A comparison with the orbital data of SOHO mission is carried out. The radiation pressure and oblateness are found to play significant role in improving the accuracy of halo orbits computation. In the vicinity of L1, it is noticed that with increase in radiation pressure, the size of halo orbit increases with the increase in the orbital period. The results for L1 coincide with the results obtained by Eapen and Sharma [16] . The Lagrangian point L1 moves towards the more massive primary with perturbations and the orbit tends to bend towards the smaller primary with the increase in oblateness. The future aspects of the project involve designing

Table 3 . Time period of the halo orbit around L1 (μ = 0.000003).

interplanetary trajectories from Earth to Mars with the help of halo orbits. These trajectories can reduce the cost of the interplanetary and deep space missions.

Prithiviraj Chidambararaj,Ram Krishan Sharma, (2016) Halo Orbits around Sun-Earth L1 in Photogravitational Restricted Three-Body Problem with Oblateness of Smaller Primary. International Journal of Astronomy and Astrophysics,06,293-311. doi: 10.4236/ijaa.2016.63025

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Coefficients for the second and third-order equations and solution

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Passing of Charged Particles Through Matter

Ilya Obodovskiy , in Radiation , 2019

5.3.1 Role of the Particle Velocity

With the maximum probability, the transfer of energy to ionization and excitation occurs at a particle velocity of the order of the orbital velocity of electrons in an atom. At a lower speed, even if the particle has enough energy to ionize or excite, it flies past the atom too slowly to pass to the atomic electron the energy needed to move to a higher level or to enter a free state. Approaching the atom, the particle slowly deforms its electron shell and just as slowly disappears. The shell gradually regains its previous structure as the particle is removed—the electronic state of the atom does not change. Such a slow quasistatic effect, in which the system remains at any time in a state close to mechanical equilibrium, is called adiabatic.

A model representation illustrating the velocity effect can be a vertical cylindrical spring with a ball mounted on top. The ball imitates the electron and the spring the Coulomb interaction with the nucleus. If the spring is compressed and then released slowly, the ball remains in place. If the compressed spring releases sharply, the ball flies off, imitating the act of ionization. Here, “fast” and “slow” correlate with the frequency of the natural oscillations of the spring.

In the case of an atom, the velocity of the perturbation must be related to the velocity of the electrons in their orbits. Electrons of different shells move on their orbits at different velocities.

When the velocity of the particle becomes less than the velocity of the orbital motion of the K electrons, the probability of ionization of them decreases sharply and, consequently, the K electrons disappear from the process and, consequently, the energy losses also decrease. With a further decrease in the velocity of the particles, the electrons of the L-shell, then the M electrons, etc., subsequently cease to participate in the inelastic processes. The particle energy corresponding to the electron velocity on the K shell can be calculated from the nonrelativistic formula

where α = 1/137 is the constant of fine structure and vB is the first Bohr velocity, Eq. (1.13) . The values of the energy of protons and alphas for some substances, calculated according to Eq. (5.38) are presented in Table 5.1 .

Table 5.1 . The Energy of Protons and Alphas, at Which the Particle Velocity Becomes Equal to the K-Electron Velocity

In the formulas for specific energy losses, the effect of low velocity on the particle's ability of ionization is considered by the term “shell corrections.”

For most particles, this behavior of the specific energy losses in this energy region is masked by the possible capture of electrons and, consequently, by a decrease in the effective charge. However, the transition through the maximum of the specific energy losses in the low-energy region is also characteristic of a muon, which does not capture electrons. This is clearly seen in the figure, which illustrates the dependence of the specific energy losses of the muon on energy (Fig. 5.4) .

For electrons, the transition through the maximum and the decrease of the specific losses with decreasing energy should not be observed because the boundary energy corresponds to an energy of the order of the ionization energy.

Thus, for ionization and excitation, the particle should not only have an energy exceeding a certain energy threshold but also a velocity larger than a certain value ∼10 8 cm/s. Macroscopic bodies cannot have such a speed. Even the bullet has a speed much lower than this threshold (5·10 4 cm/s—the average speed for gun and rifle bullets). The energy of a bullet weighing 10 g flying at a speed of 500 m/s is equal to E = Mv 2 /2 = 1.25·10 6 J = 7.8·10 24 eV = 7.8 YeV (Yotta is a metric prefix = 10 24 ). It is seen that the energy is enough to produce a fantastic amount of ion pairs, but bullets do not produce any ionization or excitation of atoms and molecules of matter in the flight. Well-known electrification by friction is not connected with the motion of bodies in any way, but it is connected only with their contact. “Electrification by friction” is a name that has only a historical origin.

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