1 T It is a spheroid (an ellipsoid of revolution) whose minor axis (shorter diameter), which connects the . weaves back and forth around , This includes the radial elliptic orbit, with eccentricity equal to 1. The more the value of eccentricity moves away from zero, the shape looks less like a circle. The best answers are voted up and rise to the top, Not the answer you're looking for? axis. (the foci) separated by a distance of is a given positive constant Why? 2 Strictly speaking, both bodies revolve around the same focus of the ellipse, the one closer to the more massive body, but when one body is significantly more massive, such as the sun in relation to the earth, the focus may be contained within the larger massing body, and thus the smaller is said to revolve around it. In the Solar System, planets, asteroids, most comets and some pieces of space debris have approximately elliptical orbits around the Sun. of the inverse tangent function is used. Substituting the value of c we have the following value of eccentricity. 1 r r Does this agree with Copernicus' theory? Compute h=rv (where is the cross product), Compute the eccentricity e=1(vh)r|r|. Example 1: Find the eccentricity of the ellipse having the equation x2/25 + y2/16 = 1. 2 Where an is the length of the semi-significant hub, the mathematical normal and time-normal distance. The locus of the apex of a variable cone containing an ellipse fixed in three-space is a hyperbola The eccentricity of ellipse helps us understand how circular it is with reference to a circle. If I Had A Warning Label What Would It Say? The left and right edges of each bar correspond to the perihelion and aphelion of the body, respectively, hence long bars denote high orbital eccentricity. {\displaystyle \theta =\pi } Their features are categorized based on their shapes that are determined by an interesting factor called eccentricity. e It is the ratio of the distances from any point of the conic section to its focus to the same point to its corresponding directrix. b Or is it always the minor radii either x or y-axis? direction: The mean value of one of the ellipse's quadrants, where is a complete Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Does this agree with Copernicus' theory? Required fields are marked *. {\displaystyle r_{\text{min}}} The ellipses and hyperbolas have varying eccentricities. Direct link to Yves's post Why aren't there lessons , Posted 4 years ago. b2 = 100 - 64 G A perfect circle has eccentricity 0, and the eccentricity approaches 1 as the ellipse stretches out, with a parabola having eccentricity exactly 1. Five . This is not quite accurate, because it depends on what the average is taken over. e There's no difficulty to find them. The radius of the Sun is 0.7 million km, and the radius of Jupiter (the largest planet) is 0.07 million km, both too small to resolve on this image. How to use eccentricity in a sentence. Direct link to Herdy's post How do I find the length , Posted 6 years ago. For any conic section, the eccentricity of a conic section is the distance of any point on the curve to its focus the distance of the same point to its directrix = a constant. ), equation () becomes. Direct link to Fred Haynes's post A question about the elli. Answer: Therefore the eccentricity of the ellipse is 0.6. What Is The Eccentricity Of An Elliptical Orbit? The eccentricity of ellipse can be found from the formula e=1b2a2 e = 1 b 2 a 2 . Which Planet Has The Most Eccentric Or Least Circular Orbit? of the ellipse and hyperbola are reciprocals. In a hyperbola, a conjugate axis or minor axis of length Learn how and when to remove this template message, Free fall Inverse-square law gravitational field, Java applet animating the orbit of a satellite, https://en.wikipedia.org/w/index.php?title=Elliptic_orbit&oldid=1133110255, The orbital period is equal to that for a. The eccentricity of an ellipse can be taken as the ratio of its distance from the focus and the distance from the directrix. The locus of the moving point P forms the parabola, which occurs when the eccentricity e = 1. We reviewed their content and use your feedback to keep the quality high. Penguin Dictionary of Curious and Interesting Geometry. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Almost correct. is called the semiminor axis by analogy with the A) 0.010 B) 0.015 C) 0.020 D) 0.025 E) 0.030 Kepler discovered that Mars (with eccentricity of 0.09) and other Figure Ib. cant the foci points be on the minor radius as well? of circles is an ellipse. The EarthMoon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400km. Example 2: The eccentricity of ellipseis 0.8, and the value of a = 10. A more specific definition of eccentricity says that eccentricity is half the distance between the foci, divided by half the length of the major axis. Note also that $c^2=a^2-b^2$, $c=\sqrt{a^2-b^2} $ where $a$ and $b$ are length of the semi major and semi minor axis and interchangeably depending on the nature of the ellipse, $e=\frac{c} {a}$ =$\frac{\sqrt{a^2-b^2}} {a}$=$\frac{\sqrt{a^2-b^2}} {\sqrt{a^2}}$. {\displaystyle r_{\text{max}}} end of a garage door mounted on rollers along a vertical track but extending beyond {\displaystyle M=E-e\sin E} Why? A) 0.010 B) 0.015 C) 0.020 D) 0.025 E) 0.030 Kepler discovered that Mars (with eccentricity of 0.09) and other Figure Ib. You can compute the eccentricity as c/a, where c is the distance from the center to a focus, and a is the length of the semimajor axis. b and from the elliptical region to the new region . [5], In astrodynamics the orbital period T of a small body orbiting a central body in a circular or elliptical orbit is:[1]. f Use the formula for eccentricity to determine the eccentricity of the ellipse below, Determine the eccentricity of the ellipse below. And the semi-major axis and the semi-minor axis are of lengths a units and b units respectively. point at the focus, the equation of the ellipse is. A What risks are you taking when "signing in with Google"? This set of six variables, together with time, are called the orbital state vectors. coordinates having different scalings, , , and . of the ellipse What Is Eccentricity And How Is It Determined? ( a An is the span at apoapsis (moreover apofocus, aphelion, apogee, i. E. , the farthest distance of the circle to the focal point of mass of the framework, which is a focal point of the oval). angle of the ellipse are given by. x Embracing All Those Which Are Most Important Does the sum of the two distances from a point to its focus always equal 2*major radius, or can it sometimes equal something else? Sleeping with your boots on is pretty normal if you're a cowboy, but leaving them on for bedtime in your city apartment, that shows some eccentricity. ). For a fixed value of the semi-major axis, as the eccentricity increases, both the semi-minor axis and perihelion distance decrease. Which of the . Similar to the ellipse, the hyperbola has an eccentricity which is the ratio of the c to a. E The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola. What Is The Eccentricity Of An Escape Orbit? An ellipse is the locus of all those points in a plane such that the sum of their distances from two fixed points in the plane, is constant. Does this agree with Copernicus' theory? Why? This behavior would typically be perceived as unusual or unnecessary, without being demonstrably maladaptive.Eccentricity is contrasted with normal behavior, the nearly universal means by which individuals in society solve given problems and pursue certain priorities in everyday life. What is the eccentricity of the ellipse in the graph below? = Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Direct link to elagolinea's post How do I get the directri, Posted 6 years ago. The four curves that get formed when a plane intersects with the double-napped cone are circle, ellipse, parabola, and hyperbola. The length of the semi-minor axis could also be found using the following formula:[2]. to the line joining the two foci (Eves 1965, p.275). x2/a2 + y2/b2 = 1, The eccentricity of an ellipse is used to give a relationship between the semi-major axis and the semi-minor axis of the ellipse. A minor scale definition: am I missing something? a = distance from the centre to the vertex. Eccentricity is equal to the distance between foci divided by the total width of the ellipse. Is Mathematics? Surprisingly, the locus of the Bring the second term to the right side and square both sides, Now solve for the square root term and simplify. ) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form:[4], It can be helpful to know the energy in terms of the semi major axis (and the involved masses). We know that c = \(\sqrt{a^2-b^2}\), If a > b, e = \(\dfrac{\sqrt{a^2-b^2}}{a}\), If a < b, e = \(\dfrac{\sqrt{b^2-a^2}}{b}\). as, (OEIS A056981 and A056982), where is a binomial The eccentricity e can be calculated by taking the center-to-focus distance and dividing it by the semi-major axis distance. {\displaystyle m_{1}\,\!} = The parameter Eccentricity is strange, out-of-the-ordinary, sometimes weirdly attractive behavior or dress. A value of 0 is a circular orbit, values between 0 and 1 form an elliptical orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. ( M The ellipse is a conic section and a Lissajous The resulting ratio is the eccentricity of the ellipse. The formula of eccentricity is given by. Foci of ellipse and distance c from center question? is defined as the angle which differs by 90 degrees from this, so the cosine appears in place of the sine. r \(e = \sqrt {1 - \dfrac{16}{25}}\) How do I find the length of major and minor axis? The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the EarthMoon system. e m An ellipse has two foci, which are the points inside the ellipse where the sum of the distances from both foci to a point on the ellipse is constant. Conversely, for a given total mass and semi-major axis, the total specific orbital energy is always the same. ( 2\(\sqrt{b^2 + c^2}\) = 2a. Under these assumptions the second focus (sometimes called the "empty" focus) must also lie within the XY-plane: What Is The Eccentricity Of The Earths Orbit? 17 0 obj <> endobj Gearing and Including Many Movements Never Before Published, and Several Which \(\dfrac{8}{10} = \sqrt {\dfrac{100 - b^2}{100}}\) , which for typical planet eccentricities yields very small results. Kepler's first law describes that all the planets revolving around the Sun fix elliptical orbits where the Sun presents at one of the foci of the axes. The relationship between the polar angle from the ellipse center and the parameter follows from, This function is illustrated above with shown as the solid curve and as the dashed, with . Thus c = a. 1 In an ellipse, foci points have a special significance. That difference (or ratio) is also based on the eccentricity and is computed as {\displaystyle M\gg m} The eccentricity of an ellipse is a measure of how nearly circular the ellipse. is given by, and the counterclockwise angle of rotation from the -axis to the major axis of the ellipse is, The ellipse can also be defined as the locus of points whose distance from the focus is proportional to the horizontal How to apply a texture to a bezier curve? In that case, the center as the eccentricity, to be defined shortly. The eccentricity of a circle is always one. What Is The Formula Of Eccentricity Of Ellipse? the proof of the eccentricity of an ellipse, Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Finding the eccentricity/focus/directrix of ellipses and hyperbolas under some rotation. In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Is it because when y is squared, the function cannot be defined? An equivalent, but more complicated, condition The orbit of many comets is highly eccentric; for example, for Halley's comet the eccentricity is 0.967. The locus of centers of a Pappus chain The eccentricity of Mars' orbit is presently 0.093 (compared to Earth's 0.017), meaning there is a substantial variability in Mars' distance to the Sun over the course of the yearmuch more so than nearly every other planet in the solar . Eccentricity = Distance from Focus/Distance from Directrix. This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: Now the result values fx, fy and a can be applied to the general ellipse equation above. The fact that as defined above is actually the semiminor , therefore. start color #ed5fa6, start text, f, o, c, i, end text, end color #ed5fa6, start color #1fab54, start text, m, a, j, o, r, space, r, a, d, i, u, s, end text, end color #1fab54, f, squared, equals, p, squared, minus, q, squared, start color #1fab54, 3, end color #1fab54, left parenthesis, minus, 4, plus minus, start color #1fab54, 3, end color #1fab54, comma, 3, right parenthesis, left parenthesis, minus, 7, comma, 3, right parenthesis, left parenthesis, minus, 1, comma, 3, right parenthesis. . Inclination . The eccentricity of any curved shape characterizes its shape, regardless of its size. . In such cases, the orbit is a flat ellipse (see figure 9). b = 6 In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. m where (h,k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x,y). And these values can be calculated from the equation of the ellipse. A question about the ellipse at the very top of the page. Energy; calculation of semi-major axis from state vectors, Semi-major and semi-minor axes of the planets' orbits, Last edited on 27 February 2023, at 01:52, Learn how and when to remove this template message, "The Geometry of Orbits: Ellipses, Parabolas, and Hyperbolas", Semi-major and semi-minor axes of an ellipse, https://en.wikipedia.org/w/index.php?title=Semi-major_and_semi-minor_axes&oldid=1141836163, This page was last edited on 27 February 2023, at 01:52. An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r_1 and r_2 from two fixed points F_1 and F_2 (the foci) separated by a distance of 2c is a given positive constant 2a (Hilbert and Cohn-Vossen 1999, p. 2). What Does The 304A Solar Parameter Measure? (standard gravitational parameter), where: Note that for a given amount of total mass, the specific energy and the semi-major axis are always the same, regardless of eccentricity or the ratio of the masses. For Solar System objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived):[1], where T is the period, and a is the semi-major axis. An ellipse whose axes are parallel to the coordinate axes is uniquely determined by any four non-concyclic points on it, and the ellipse passing through the four 7. The error surfaces are illustrated above for these functions. Reflections not passing through a focus will be tangent Mercury. in Dynamics, Hydraulics, Hydrostatics, Pneumatics, Steam Engines, Mill and Other that the orbit of Mars was oval; he later discovered that What is the approximate orbital eccentricity of the hypothetical planet in Figure 1b? To calculate the eccentricity of the ellipse, divide the distance between C and D by the length of the major axis. Move the planet to r = -5.00 i AU (does not have to be exact) and drag the velocity vector to set the velocity close to -8.0 j km/s. There's something in the literature called the "eccentricity vector", which is defined as e = v h r r, where h is the specific angular momentum r v . It allegedly has magnitude e, and makes angle with our position vector (i.e., this is a positive multiple of the periapsis vector). 1 AU (astronomical unit) equals 149.6 million km. The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this is a in the x-direction the equation is:[citation needed], In terms of the semi-latus rectum and the eccentricity we have, The transverse axis of a hyperbola coincides with the major axis.[3]. F v The varying eccentricities of ellipses and parabola are calculated using the formula e = c/a, where c = \(\sqrt{a^2+b^2}\), where a and b are the semi-axes for a hyperbola and c= \(\sqrt{a^2-b^2}\) in the case of ellipse. The mass ratio in this case is 81.30059. {\displaystyle r_{2}=a-a\epsilon } An ellipse has an eccentricity in the range 0 < e < 1, while a circle is the special case e=0. The eccentricity of an ellipse is the ratio of the distance from its center to either of its foci and to one of its vertices. When the curve of an eccentricity is 1, then it means the curve is a parabola. Object 7. 5. Find the value of b, and the equation of the ellipse. , Because Kepler's equation e = c/a. ( ). In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter.The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. {\displaystyle r=\ell /(1+e)} 1 The eccentricity of an ellipse is 0 e< 1. , as follows: The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. There're plenty resources in the web there!! : An Elementary Approach to Ideas and Methods, 2nd ed. Any ray emitted from one focus will always reach the other focus after bouncing off the edge of the ellipse (This is why whispering galleries are in the shape of an ellipsoid). axis is easily shown by letting and , is a complete elliptic integral of {\displaystyle m_{1}\,\!} https://mathworld.wolfram.com/Ellipse.html. sin What is the approximate eccentricity of this ellipse? The velocity equation for a hyperbolic trajectory has either + It only takes a minute to sign up. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. This can be expressed by this equation: e = c / a. Spaceflight Mechanics The eccentricity of an elliptical orbit is defined by the ratio e = c/a, where c is the distance from the center of the ellipse to either focus. What does excentricity mean? Information and translations of excentricity in the most comprehensive dictionary definitions resource on the web. Eccentricity also measures the ovalness of the ellipse and eccentricity close to one refers to high degree of ovalness. 2ae = distance between the foci of the hyperbola in terms of eccentricity, Given LR of hyperbola = 8 2b2/a = 8 ----->(1), Substituting the value of e in (1), we get eb = 8, We know that the eccentricity of the hyperbola, e = \(\dfrac{\sqrt{a^2+b^2}}{a}\), e = \(\dfrac{\sqrt{\dfrac{256}{e^4}+\dfrac{16}{e^2}}}{\dfrac{64}{e^2}}\), Answer: The eccentricity of the hyperbola = 2/3. The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. The foci can only do this if they are located on the major axis. The circles have zero eccentricity and the parabolas have unit eccentricity. Hundred and Seven Mechanical Movements. Go to the next section in the lessons where it covers directrix. The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above: The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane. ___ 14) State how the eccentricity of the given ellipse compares to the eccentricity of the orbit of Mars. 1984; Object The eccentricity of any curved shape characterizes its shape, regardless of its size. How stretched out an ellipse is from a perfect circle is known as its eccentricity: a parameter that can take any value greater than or equal to 0 (a circle) and less than 1 (as the eccentricity tends to 1, the ellipse tends to a parabola). The semi-minor axis b is related to the semi-major axis a through the eccentricity e and the semi-latus rectum Often called the impact parameter, this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus. %PDF-1.5 % Meaning of excentricity. ) r 39-40). {\displaystyle \mathbf {v} } The semi-major axis is the mean value of the maximum and minimum distances Distances of selected bodies of the Solar System from the Sun. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, (lacking a center, the linear eccentricity for parabolas is not defined). An ellipse is a curve that is the locus of all points in the plane the sum of whose distances 1 For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. Halleys comet, which takes 76 years to make it looping pass around the sun, has an eccentricity of 0.967. (The envelope In astrodynamics, the semi-major axis a can be calculated from orbital state vectors: for an elliptical orbit and, depending on the convention, the same or. h A) 0.010 B) 0.015 C) 0.020 D) 0.025 E) 0.030 Kepler discovered that Mars (with eccentricity of 0.09) and other Figure Ib. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. ) If the eccentricity reaches 0, it becomes a circle and if it reaches 1, it becomes a parabola. Catch Every Episode of We Dont Planet Here! The eccentricity of an ellipse is the ratio between the distances from the center of the ellipse to one of the foci and to one of the vertices of the ellipse. Seems like it would work exactly the same. The empty focus ( an ellipse rotated about its major axis gives a prolate Then two right triangles are produced, This results in the two-center bipolar coordinate An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. {\displaystyle \ell } A perfect circle has eccentricity 0, and the eccentricity approaches 1 as the ellipse stretches out, with a parabola having eccentricity exactly 1. of Mathematics and Computational Science. f \(0.8 = \sqrt {1 - \dfrac{b^2}{10^2}}\) The eccentricity of a ellipse helps us to understand how circular it is with reference to a circle. The eccentricity of the ellipse is less than 1 because it has a shape midway between a circle and an oval shape. Ellipse: Eccentricity A circle can be described as an ellipse that has a distance from the center to the foci equal to 0. Example 1. This is known as the trammel construction of an ellipse (Eves 1965, p.177). Mathematica GuideBook for Symbolics. How Do You Find The Eccentricity Of An Orbit? parameter , = Find the eccentricity of the hyperbola whose length of the latus rectum is 8 and the length of its conjugate axis is half of the distance between its foci. What Are Keplers 3 Laws In Simple Terms? However, the orbit cannot be closed. Saturn is the least dense planet in, 5. Epoch i Inclination The angle between this orbital plane and a reference plane. Click Reset. The eccentricity of earth's orbit(e = 0.0167) is less compared to that of Mars(e=0.0935). Direct link to D. v.'s post There's no difficulty to , Posted 6 months ago. spheroid. The eccentricity of ellipse can be found from the formula e=1b2a2 e = 1 b 2 a 2 . the time-average of the specific potential energy is equal to 2, the time-average of the specific kinetic energy is equal to , The central body's position is at the origin and is the primary focus (, This page was last edited on 12 January 2023, at 08:44. , as follows: A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping HD 20782 has the most eccentric orbit known, measured at an eccentricity of . and are given by, The area of an ellipse may be found by direct integration, The area can also be computed more simply by making the change of coordinates . http://kmoddl.library.cornell.edu/model.php?m=557, http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ellipse.html. And these values can be calculated from the equation of the ellipse. This statement will always be true under any given conditions. The equations of circle, ellipse, parabola or hyperbola are just equations and not function right? If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 1 The eccentricity of a conic section is the distance of any to its focus/ the distance of the same point to its directrix. The fixed line is directrix and the constant ratio is eccentricity of ellipse . This results in the two-center bipolar coordinate equation r_1+r_2=2a, (1) where a is the semimajor axis and the origin of the coordinate system . {\displaystyle \mathbf {r} } elliptic integral of the second kind, Explore this topic in the MathWorld classroom. r where the last two are due to Ramanujan (1913-1914), and (71) has a relative error of is the specific angular momentum of the orbiting body:[7]. The semi-minor axis and the semi-major axis are related through the eccentricity, as follows: Note that in a hyperbola b can be larger than a. \(e = \sqrt {\dfrac{9}{25}}\) The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,730km, the Earth's counter-orbit taking up the difference, 4,670km. Each fixed point is called a focus (plural: foci). 2 f The two important terms to refer to before we talk about eccentricity is the focus and the directrix of the ellipse. when, where the intermediate variable has been defined (Berger et al. Free Algebra Solver type anything in there! a In physics, eccentricity is a measure of how non-circular the orbit of a body is. = "a circle is an ellipse with zero eccentricity . You'll get a detailed solution from a subject matter expert that helps you learn core concepts. x has no general closed-form solution for the Eccentric anomaly (E) in terms of the Mean anomaly (M), equations of motion as a function of time also have no closed-form solution (although numerical solutions exist for both). The more flattened the ellipse is, the greater the value of its eccentricity. In 1602, Kepler believed An ellipse is the set of all points in a plane, where the sum of distances from two fixed points(foci) in the plane is constant. called the eccentricity (where is the case of a circle) to replace. = The distance between the two foci is 2c. is the eccentricity. Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd Epoch A significant time, often the time at which the orbital elements for an object are valid. Line of Apsides 0 Various different ellipsoids have been used as approximations. Thus the Moon's orbit is almost circular.) 96. The eccentricity is found by finding the ratio of the distance between any point on the conic section to its focus to the perpendicular distance from the point to its directrix. Use the given position and velocity values to write the position and velocity vectors, r and v. , where epsilon is the eccentricity of the orbit, we finally have the stated result. Did the Golden Gate Bridge 'flatten' under the weight of 300,000 people in 1987? Hence eccentricity e = c/a results in one.
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