Orbital mechanics

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Orbital mechanics or astrodynamics are ballistic applications and celestial mechanics for practical problems with the motion of rockets and other spacecraft. The movement of these objects is usually calculated from Newton's laws of motion and Newton's universal law of gravity. This is the core discipline in the design and control of space missions. Celestial mechanics treats the wider dynamics of the system's orbitals under the influence of gravity, including spaceships and natural astronomical bodies such as star systems, planets, moons and comets. Orbital mechanics focuses on the spacecraft's trajectory, including orbital maneuvering, airplane orbit changes, and interplanetary transfers, and used by mission planners to predict the results of propulsive maneuvers. General relativity is a more precise theory than Newton's law for counting orbits, and sometimes necessary for greater accuracy or in high gravity situations (such as orbits close to the Sun).

Video Orbital mechanics

History

Until the rise of space travel in the 20th century, there is little difference between orbital and sky mechanics. At the time of Sputnik, the field was called 'space dynamics'. The basic technique, such as that used to solve Keplerian problems (determine the position as a function of time), is therefore the same in both fields. Furthermore, the history of the field is almost entirely divided.

Johannes Kepler was the first to successfully model the planet's orbit to a high degree of accuracy, publishing its laws in 1605. Isaac Newton published the law of heavenly movement more generally in the first edition of PhilosophiÃÆ'Â| Naturalis Principia Mathematica (1687) which provides a method to find the orbit of the body following the parabolic path of the three observations. It was used by Edmund Halley to create the orbit of various comets, including those bearing his name. Newton's method of successive approximation was formalized into analytical methods by Euler in 1744, whose work was in turn generalizable to elliptical and hyperbolic orbits by Lambert in 1761-1777.

Another milestone in the determination of orbit was Carl Friedrich Gauss's help in the "recovery" of Ceres dwarf planet in 1801. Gauss's method was able to use only three observations (in the form of rising tides and right declination), to find six orbital elements that completely depict orbits. The theory of orbit determination has then been developed to the point where today is applied in a GPS receiver as well as tracking and cataloging the recently observed minor planet.

Maps Orbital mechanics

Practical techniques

The rule of thumb

The following practical rules are useful for situations predicted by classical mechanics under the assumptions of astrodynamic standards outlined under the rules. Specific examples discussed are satellites orbiting the planet, but the rule of thumb can also be applied to other situations, such as small body orbits around stars like the Sun.

• The laws of the Kepler planetary movement:
  • Orbits are elliptical, with a heavier body on an elliptical focus. This particular case is a circular orbit (circle is a special case of ellipse) with a planet in the center.
  • The line drawn from the planet to the satellite sweeps the same area at the same time no matter what part of the orbit is measured.
  • The square of the satellite orbital period is proportional to the average cube the distance from the planet.
• Without applying force (like firing a rocket engine), the period and shape of the satellite orbit will not change.
• Satellites in low orbits (or lower parts of elliptical orbits) move more rapidly in relation to planetary surfaces than satellites in higher orbits (or higher parts of elliptical orbits), because stronger gravitational attraction is closer to the planet.
• If the impulse is applied only to one point in the satellite orbit, it will return to the same point in each of the next orbits, although the rest of the path will change. Thus one can not move from one circular orbit to another with only one brief impulse.
• From a circular orbit, the thrust is applied in the opposite direction to the satellite movement turning the orbit into an ellipse; the satellite will descend and reach the lowest orbital point (periapse) at 180 degrees from the firing point; then it will rise again. Push applied in the direction of satellite movement creates an elliptical orbit with the highest point (apoapse) 180 degrees from the firing point.

The consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the same circular orbit and want to dock, unless they are very close, the carrier can not just fire the engine to go faster. This will change the shape of its orbit, causing it to gain height and actually slowing down relative to the leading craft, losing its target. Meeting rooms prior to docking usually take some precise machine discharges in some orbital periods that take hours or even days to complete.

For the degree that the astrodynamic standard assumption does not apply, the actual trajectory will vary from the calculated. For example, simple atmospheric drag is another complicating factor for objects in low Earth orbit. This rule of thumb is clearly inaccurate when describing two or more bodies with similar masses, such as binary star systems (see n-body problem). Celestial Mechanics uses more general rules that apply to a wider range of situations. Kepler's law of planetary motion, which can be derived mathematically from Newton's law, applies only in describing the motion of two gravitational objects in the absence of a non-gravitational force; they also explain the parabolic and hyperbolic trajectories. Near the big objects like stars, the difference between classical mechanics and general relativity is also important.

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The law of astrodynamics

The basic law of astrodynamics is Newton's universal law of gravitation and Newton's laws of motion, while the fundamental mathematical tool is a differential calculus.

Any orbit and trajectory outside the atmosphere can basically be recovered, that is, in a time-time function reversed. The speed is reversed and the acceleration is the same, including those caused by rocket bursts. So if the rocket bursts toward speed, in reverse case it's opposite to speed. Of course in the case of a rocket burst there is no full reversal of events, the same two ways delta-v is used and the same mass ratio holds.

Standard assumptions in astrodynamics include non-interference from outside bodies, ignored mass for one body, and other negligible forces (such as from solar wind, atmospheric drag, etc.). More accurate calculations can be made without this simplification assumption, but they are more complicated. The increased accuracy is often not enough to make the difference in the calculations become valuable.

Kepler's law of planetary motion can be derived from Newton's law, when it is assumed that the orbiting body is only subject to the gravitational force of the central contractor. When a thrust engine or propulsion force is present, Newton's law is still valid, but Kepler's law is invalid. When the thrust stops, the resulting orbit will be different but again will be explained by Kepler's law. These three laws are:

1. The orbit of each planet is an ellipse with the sun in one focus.
2. The line connecting planets and the sun sweeps the same area over the same time interval.
3. The square of the planet's orbital period is directly proportional to the cube of the semi-major axis of the orbit.

Running speed

Rumus untuk kecephala melarikan dice dengan mudah diturunkan sebagai berikut. Energi spesifik (energi per satuan massa) dari setiap kendaraan ruang angkasa terdiri dari segunda komponen, energi potensial spesifik dan energi kinetik spesifik. Energi potensial spesifik yang terkait dengan planet massa M diberikan oleh

${\ displaystyle \ epsilon _ {p} = - {\ frac {GM} {r}} \,} Â Â$

sedangkan energi kinetik spesifik dari suatu objek diberikan oleh

${\ displaystyle \ epsilon _ {k} = {\ frac {v2} {2}} \,} Â Â$

Karena to energize a dilemma,

${\ displaystyle \ epsilon = \ epsilon _ {k} \ epsilon p, \!} Â Â$

give total orbital energy to the third

${\ displaystyle \ epsilon = {\ frac {v2}} {2}} - {\ frac {GM} {r}} \,} Â Â$

tidak bergantung pada jarak, ${\ displaystyle r}  ÂÂ$ , dari pusat badan pusat ke kendaraan ruang angkasa yang dimaksud. Oleh karena itu, objek dapat mencapai tak terbatas ${\ displaystyle r}  ÂÂ$ hanya jika kuantitas ini tidak negatif, yang berarti

${\ displaystyle v \ geq {\ sqrt {\ frac {2GM} {r}}}.}  ÂÂ$

The escape velocity from the Earth's surface is about 11 km/sec, but it is not enough to send the body to an infinite distance due to the gravitational pull of the Sun. To avoid the Solar System from a location at a distance from the Sun equals the Sun-Earth distance, but not close to Earth, it requires a speed of about 42 km/sec, but there will be a "credit section" for the Earth's orbital velocity. for spacecraft launched from Earth, if further acceleration (because of the propulsion system) takes them in the same direction as Earth's journey in its orbit.

The formula for free orbit

Orbit adalah bagian berbentuk kerucut, sehingga rumus untuk jarak tubuh untuk sudut tertentu sesuai dengan rumus untuk kurva dalam koordinat polar, yaitu:

${\ displaystyle r = {\ frac {p} {1 e \ cos \ theta}}} Â Â$

${\ displaystyle \ mu = G (m1 m2) \,} Â Â$

${\ displaystyle p = h ^ 2/\ mu \,} Â Â$

${\ displaystyle \ mu} Â Â$ disebut parameter gravitasi. ${\ displaystyle m_ {1}} Â$ dan ${\ displaystyle m_ {2}} Â Â$ adalah kumpulan objek 1 dan 2, dan adalah momentum sudut spesifik objek 2 berkenaan dengan objek 1. Parameter ${\ displaystyle \ theta} Â Â$ dikenal sebagai anomali sejati, ${\ displaystyle p} Â Â$ adalah rektus semi-latus, sementara ${\ displaystyle e} Â$ adalah exentrisitas orbital, semua dapat diperoleh dari berbagai bentuk dari enam elemen orbital independen.

Orbit lingkaran

All bound orbits in which gravity of a central body dominates are ellipses in nature. This particular case is a circular orbit, which is an elliptical zero ellipse. The formula for body velocity in a circular orbit at a distance r from the center of mass gravity M can be derived as follows -

Centrifugal acceleration corresponds to acceleration due to gravity.

Jadi, ${\ displaystyle v ^ {2}/r = GM/r2} Â Â$

Karena itu,

${\ displaystyle \ v = {\ sqrt {{\ frac {GM} {r}} \}}} Â Â$

di mana ${\ displaystyle G}  ÂÂ$ adalah konstanta gravitasi, sama dengan

6,673 84 ÃÆ'— 10 ^-11 m ³ /(kg s ² )

Untuk menggunakan formula ini dengan benar, unit harus konsisten; misalnya, ${\ displaystyle M}  ÂÂ$ harus dalam kilogram, dan ${\ displaystyle r}  ÂÂ$ harus dalam satuan meter. Jawabannya akan dalam meter per detik.

Quantity $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂGÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; ÂMÂ\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â$ are often termed standard gravity parameters, which have different values ​​for every planet or month in the Solar System.

Setelah kecepted orbital melingkar diketahui, kecepatane lepas mudah ditemuk denially menagonic denially acceded to give 2:

${\ displaystyle \ v = {\ sqrt {2}} {\ sqrt {{\ frac {GM} {r}} \}} = {\ sqrt {{\ frac {2GM} {r}} \}}.} Â Â$

Untuk melepaskan dice dari gravitasi, energi kinetik setidaknya harus sesuai dengan energi potencialial negatif. Jadi, ${\ displaystyle 1/2mv2 = GMm/r} Â Â$ dan oleh karena itu,

${\ displaystyle v = {\ sqrt {{\ frac {2GM} {r}} \}}.} Â Â$

Orbit elips

Jika ${\ displaystyle 0 & lt; e & lt; 1}  ÂÂ$ , maka penyebut persamaan orbit gratis bervariasi dengan anomali sejati ${\ displaystyle \ theta}  ÂÂ$ , tetapi tetap positif, tidak pernah menjadi nol. Oleh karena itu, vektor posisi relatif tetap dibatasi, memiliki magnitudo terkecilnya pada periapsis ${\ displaystyle r_ {p}}  ÂÂ$ , yang diberikan oleh:

${\ displaystyle r_ {p} = {\ frac {p} {1 e}}}  ÂÂ$

Nilai maximum ${\ displaystyle r} Â Â$ tercapai ketika ${\ displaystyle \ theta = 180 ^ {\ circ}} Â Â$ . Titik ini disebut apoapsis, dan koordinat radialnya, dinotasikan ${\ displaystyle r_ {a}} Â Â$ , adalah

${\ displaystyle r_ {a} = {\ frac {p} {1-e}}} Â Â$

Biarkan ${\ displaystyle 2a}  ÂÂ$ adalah jarak yang diukur sepanjang garis apse dari periapsis ${\ displaystyle P}  ÂÂ$ ke apoapsis ${\ displaystyle A}  ÂÂ$ , seperti yang digambarkan dalam persamaan di bawah ini:

${\ displaystyle 2a = r_ {p} r_ {a}}  ÂÂ$

Mustapatkan's messenger team goes on, kita mendapatkan:

${\ displaystyle a = {\ frac {p} {1-e2}}} Â Â$

a adalah sumbu semimajor dari elips. Memecahkan untuk ${\ displaystyle p}  ÂÂ$ , dan mengganti hasilnya dalam rumus kurva irisan kerucut di atas, kita mendapatkan:

${\ displaystyle r = {\ frac {a (1-e ^ {2})} {1 e \ cos \ theta}}}  ÂÂ$

Periode orbit

Di bawah asumsi standar periode orbital ( ${\ displaystyle T \, \!}  ÂÂ$ ) dari badan yang berjalan di sepanjang orbit elips dapat dihitung sebagai:

${\ displaystyle T = 2 \ pi {\ sqrt {a ^ {3} \ over {\ mu}}}}  ÂÂ$

dimana:

• ${\ displaystyle \ mu \,}  ÂÂ$ adalah parameter gravitasi standar,
• ${\ displaystyle a \, \!}  ÂÂ$ adalah panjang sumbu semi-mayor.

Kesimpulan:

• Periode orbital sama dengan orbit lingkaran dengan radius orbit sama dengan sumbu semi-mayor ( ${\ displaystyle a \, \!}  ÂÂ$ ),
• Untuk sumbu semi-utama tertentu, periode orbit tidak bergantung pada eksentrisitas (Lihat juga: hukum ketiga Kepler).

Velocity

Di bawah asumsi standar kecepatan orbital ( ${\ displaystyle v \,}  ÂÂ$ ) dari badan yang melakukan perjalanan sepanjang orbit elips dapat dihitung dari persamaan Vis-viva sebagai:

${\ displaystyle v = {\ sqrt {\ mu \ left ({2 \ over {r}} - {1 \ over {a}} \ right)}}}  ÂÂ$

dimana:

• ${\ displaystyle \ mu \,}  ÂÂ$ adalah parameter gravitasi standar,
• ${\ displaystyle r \,}  ÂÂ$ adalah jarak antara badan yang mengorbit.
• ${\ displaystyle a \, \!}  ÂÂ$ adalah panjang sumbu semi-mayor.

Persamaan kecepat untuk lintasan hiperbolik memmall ${\ displaystyle {1 \ over {a}}} Â Â$ , atau itu sama dengan konvensi yang dalam hal ini a negatif.

Energi

Di bawah asumsi standar, energi orbital tertentu ( ${\ displaystyle \ epsilon \,}  ÂÂ$ ) dari orbit elips negatif dan persamaan konservasi energi orbital (persamaan Vis-viva) untuk orbit ini dapat mengambil bentuk:

${\ displaystyle {v ^ {2} \ over {2}} - {\ mu \ over {r}} = - {\ mu \ over {2a}} = \ epsilon & lt; 0}  ÂÂ$

dimana:

• ${\ displaystyle v \,}  ÂÂ$ adalah kecepatan dari tubuh yang mengorbit,
• ${\ displaystyle r \,}  ÂÂ$ adalah jarak dari tubuh yang mengorbit dari pusat massa dari badan pusat,