Physics
Orbital Period
The orbital period refers to the time it takes for an object to complete one orbit around another object. In the context of celestial bodies, such as planets or moons, the orbital period is a fundamental characteristic that determines their motion and position in space. It is typically measured in units of time, such as days, years, or seconds.
Written by Perlego with AI-assistance
Related key terms
1 of 5
7 Key excerpts on "Orbital Period"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
Hence velocities are the same and Orbital Periods are halved. In all these cases of scaling. if densities are multiplied by 4, times are halved; if velocities are doubled, forces are multiplied by 16. These properties are illustrated in the formula (derived from the formula for the Orbital Period) for an elliptical orbit with semi-major axis a , of a small body around a spherical body with radius r and average density σ, where T is the Orbital Period. Orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used (derived from Newton's laws of motion and New-ton's law of universal gravitation). There are many different ways to mathematically describe the same orbit, but certain schemes each consisting of a set of six parameters are commonly used in astronomy and orbital mechanics. A real orbit (and its elements) changes over time due to gravitational perturbations by other objects and the effects of relativity. A Keplerian orbit is merely a mathematical approximation at a particular time. Required parameters Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit. This is because the problem contains six degrees of freedom. These correspond to the three spatial dimensions which define position (the x , y , z in a Cartesian coordinate system), plus the velocity in each of these dimensions. These can be described as orbital state vectors, but this is often an inconvenient way to represent an orbit, which is why Keplerian elements (described below) are commonly used instead. Sometimes the epoch is considered a seventh orbital parameter, rather than part of the reference frame.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
So the result is Choosing the axis of the coordinate system such that , and inserting , gives: ________________________ WORLD TECHNOLOGIES ________________________ If this is the equation of an ellipse and illustrates Kepler's first law. Deriving Kepler's third law In the special case of circular orbits, which are ellipses with zero eccentricity, the relation between the radius a of the orbit and its period P can be derived relatively easily. The centripetal force of circular motion is proportional to a / P 2 , and it is provided by the gravitational force, which is proportional to 1/ a 2 . Hence, which is Kepler's third law for the special case. In the general case of elliptical orbits, the derivation is more complicated. The area of the planetary orbit ellipse is The areal speed of the radius vector sweeping the orbit area is where The period of the orbit is satisfying implying Kepler's third law ________________________ WORLD TECHNOLOGIES ________________________ Secular variations of the planetary orbits The Secular Variations of the Planetary Orbits is a concept describing alleged long-term changes (secular variation) in the orbits of the planets Mercury to Neptune. Several attempts have from time to time been undertaken to analyze and predict such deviations from ordinary satellite orbits. One such semi-analytic theory (French: Variations Séculaires des Orbites Planétaires , abbreviated as VSOP ) was developed and is maintained (updating it with the results of the latest and most accurate measurements) by the scientists at the Bureau des Longitudes in Paris, France. The first version, VSOP82, computed only the orbital elements at any moment. An updated version, VSOP87, besides providing improved accuracy, computed the positions of the planets directly, as well as their orbital elements, at any moment. An additional theory emerged 1999-2004 from compiling planetary observations by Russian astronomer Yuri B.
eBook - ePubSatellite Technology
Principles and Applications
- Anil K. Maini, Varsha Agrawal(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
A circular orbit with radius r is assumed. Remember that a circular orbit is only a special case of an elliptical orbit with both the semi-major axis and semi-minor axis equal to the radius. Equating the gravitational force with the centrifugal force gives (2.14) Replacing by ωr in the above equation gives (2.15) which gives ω 2 = Gm 1 / r 3. Substituting ω = 2 π / T gives (2.16) This can also be written as (2.17) The above equation holds good for elliptical orbits provided r is replaced by the semi-major axis a. This gives the expression for the time period of an elliptical orbit as (2.18) 2.3 Orbital Parameters The satellite orbit, which in general is elliptical, is characterized by a number of parameters. These not only include the geometrical parameters of the orbit but also parameters that define its orientation with respect to Earth. The orbital elements and parameters will be discussed in the following paragraphs: Ascending and descending nodes Equinoxes Solstices Apogee Perigee Eccentricity Semi-major axis Right ascension of the ascending node Inclination Argument of the perigee True anomaly of the satellite Angles defining the direction of the satellite Ascending and descending nodes. The satellite orbit cuts the equatorial plane at two points: the first, called the descending node (N1), where the satellite passes from the northern hemisphere to the southern hemisphere, and the second, called the ascending node (N2), where the satellite passes from the southern hemisphere to the northern hemisphere (Figure 2.9). Equinoxes. The inclination of the equatorial plane of Earth with respect to the direction of the sun, defined by the angle formed by the line joining the centre of the Earth and the sun with the Earth's equatorial plane follows a sinusoidal variation and completes one cycle of sinusoidal variation over a period of 365 days (Figure 2.10). The sinusoidal variation of the angle of inclination is defined by (2.19) where T is 365 days
eBook - ePubDescribing Motion
The Physical World
- Robert Lambourne(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
3 ), so the planet moves faster when closer to the Sun.Kepler’s third law The square of the Orbital Period of each planet is proportional to the cube of its semimajor axis.The period of a planetary orbit is the time taken for a planet to complete one full circuit of the Sun. In the case of the Earth it is one year. Kepler’s third law tells us that there is a single constant, let’s call it K, such that for any body that orbits the Sun with period T and semimajor axis a= K . (3.66)T 2a 3Kepler’s third law is illustrated in Figure 3.39 for three hypothetical planets with periods T 1 , T 2 , T 3 , and semimajor axes a 1 , a 2 , a 3 . Since the constant K in Equation 3.66 is the same for all three bodies, we can write=T 1 2a 1 3=T 2 2a 2 3. (3.67)T 3 2a 3 3Figure 3.39 An illustration of Kepler’s third law. Three hypothetical planets have periods T 1 , T 2 , T 3 , and semimajor axes a 1 , a 2 , a 3 . In each case T 2 /a 3 will have the same constant value. This means that T ∞ a 3/2 and, in the case of circular orbits, implies that the orbital speed decreases as the square root of the distance from the Sun increases, so the distance travelled in a fixed time also decreases.Table 3.2 gives the semimajor axes and periods for the planets. Using these data it is easy to confirm Kepler’s third law. This is the subject of Example 3.3 .Table 3.2 Orbital data for the planets.Example 3.3Using the data given in Table 3.2 , plot a graph of T against a 3/2 . Does your result confirm Kepler’s third law?Solution
The required graph is shown in Figure 3.40 . As can be seen, to a good approximation, the graph is a straight line. This shows thatT ∝If these quantities are proportional then their squares will also be proportional i.e.a.3 / 2T 2∝a 3.This confirms Kepler’s third law. (The reasons for plotting T against a 3/2 rather than T 2 against a 3
- Rudolf Kurth(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
Because the fundamental ideas are shown more clearly, we prefer our own definition here. 3. We return to the problem of determining the period of a planet, let us say Mars, from the observations. We set out here the assumptions: (i) The Earth and Mars move around the Sun in closed orbits which both lie in the same fixed plane through the Sun. (In reality, the orbital planes of the Earth and Mars are different, but then, instead of the actual motion of Mars, we can consider its projection on the Earth's orbital plane. Even the assump-tion that the orbital curve itself lies in a plane is unnecessary.) (ii) The radius vector Sun-projection of planet and the radius vector Sun-Earth are one-valued functions of the direction. (iii) The angular velocity of the projection of the planet relative to the Sun is a one-valued function of the direction relative to the Sun. In other words, the angular velocity is a periodic function n(t) of the time t, whose period P coincides with the sidereal period, i.e. with the length of time between two consecutive passages of the planet through the same point of its orbit (relative to the Sun and the fixed axes in space). Now at some time Mars will be in opposition to the Sun—i.e. the Sun, the Earth and Mars are situated, in that order, on a THE KINETICS OF A SINGLE PLANET 3 straight line. At the next moment this will no longer be true. However, oppositions will continually recur, let us say at the times t lt t 2 , These times were known to Kepler from obser-vations over a long interval. We now distinguish quantities concerning the Earth and Mars by the suffixes E and M respectively. Then observations at the time of opposition tell us: 0
eBook - PDFSatellite Geodesy
Foundations, Methods, and Applications
- Günter Seeber(Author)
- 2008(Publication Date)
- De Gruyter(Publisher)
66 3 Satellite Orbital Motion In mathematical formulation this means for different planets P i with periods of revo-lution U i , mean motions: n i = 2 π/U i , (3.9) and semi-major axes a i : a 3 i U 2 i = C 2 4 π 2 . (3.10) C is a constant for the planetary system. Inserting (3.9) into (3.10) gives the commonly used expression a 3 i · n 2 i = C 2 . (3.11) This law was found empirically by Kepler because it approximates very well the motion of the large planets. However, a completely different numerical value for C 2 is obtained for the system of Jovian moons. Therefore a more general relation is useful a 3 U 2 = k 2 4 π 2 (M + m), (3.12) where k is a universal constant and M, m are the respective masses. Using (3.12) it is possible to determine the masses of celestial bodies. Kepler’s laws describe the simplest form of motion of celestial bodies under the assumption that no external perturbing forces are present, and that the respective masses can be considered to be point masses or homogeneous bodies with spherical mass distribution. For the motion of an artificial Earth satellite these assumptions are only valid in a first approximation. Keplerian orbits, consequently, can only be used as a simple reference orbit and they give only qualitative information on the kind of motion. Kepler himself was convinced that his three, empirically found, laws followed a more general law. This more general law was formulated by Isaac Newton (1643–1727) in the form of the Law of Gravitation . 3.1.2 Newtonian Mechanics, Two-Body Problem 3.1.2.1 Equation of Motion In the first book of “Principia” Newton introduced his three laws of motion: 1. Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by an external impressed force. 2. The rate of change of momentum of the body is proportional to the force im-pressed and is in the same direction in which the force acts.
eBook - PDF- Scott Tremaine(Author)
- 2023(Publication Date)
- Princeton University Press(Publisher)
If the particle passes through periapsis at t = t 0 , the dimensionless variable ` = 2⇡ t − t 0 P = n(t − t 0 ) (1.45) is called the mean anomaly. Notice that the mean anomaly equals the true anomaly f when f = 0, ⇡, 2⇡, . . . but not at other phases unless the orbit is circular; similarly, ` and f always lie in the same semicircle (0 to ⇡, ⇡ to 2⇡, and so on). 7 The relation n = 2⇡P holds because Kepler orbits are closed—that is, they return to the same point once per orbit. In more general spherical potentials we must distinguish the radial period, the time between successive periapsis passages, from the azimuthal period 2⇡n. For example, in a harmonic potential Φ(r) = 1 2 ! 2 r 2 the radial period is ⇡! but the azimuthal period is 2⇡!. Smaller differences between the radial and azimuthal period arise in perturbed Kepler systems such as multi-planet systems or satellites orbiting a flattened planet (§1.8.3). For the Earth the radial period is called the anomalistic year, while the azimuthal period of 365.256 363 d is the sidereal year. The anomalistic year is longer than the sidereal year by 0.003 27 d. When we use the term “year” in this book, we refer to the Julian year of exactly 365.25 d (§1.5). 1.3. MOTION IN THE KEPLER ORBIT 13 The position and velocity of a particle in the orbital plane at a given time are determined by four orbital elements: two specify the size and shape of the orbit, which we can take to be e and a (or e and n, q and Q, L and E, and so forth); one specifies the orientation of the line of apsides ($); and one specifies the location or phase of the particle in its orbit (f , `, or t 0 ). The trajectory [r(t), (t)] can be derived by solving the differential equation (1.20) for r(t), then (1.16) for (t). However, there is a simpler method. First consider bound orbits.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
