Kepler's laws of planetary motion

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Illustration of Kepler's laws with two planetary orbits.

1. The orbits are ellipses, with foci F₁ and F₂ for Planet 1, and F₁ and F₃ for Planet 2. The Sun is at F₁.
2. The shaded areas A₁ and A₂ are equal, and are swept out in equal times by Planet 1's orbit.
3. The ratio of Planet 1's orbit time to Planet 2's is $(a_1/a_2{)}^{3/2}$.

In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law helps to establish that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the slower its orbital speed, and vice versa.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation.

A more precise historical approach is found in Astronomia nova and Epitome Astronomiae Copernicanae.

Comparison to Copernicus

Johannes Kepler's laws improved the model of Copernicus. According to Copernicus:

1. The planetary orbit is a circle with epicycles.
2. The Sun is approximately at the center of the orbit.
3. The speed of the planet in the main orbit is constant.

Despite being correct in saying that the planets revolved around the Sun, Copernicus was incorrect in defining their orbits. Introducing physical explanations for movement in space beyond just geometry, Kepler correctly defined the orbit of planets as follows:

1. The planetary orbit is not a circle with epicycles, but an ellipse.
2. The Sun is not at the center but at a focal point of the elliptical orbit.
3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed (closely linked historically with the concept of angular momentum) is constant.

The eccentricity of the orbit of the Earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the Sun parallel to the equator of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

${\displaystyle e\approx \frac{\pi }{4}\frac{186-179}{186+179}\approx 0.015,}$

which is close to the correct value (0.016710218). The accuracy of this calculation requires that the two dates chosen be along the elliptical orbit's minor axis and that the midpoints of each half be along the major axis. As the two dates chosen here are equinoxes, this will be correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.

Nomenclature

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws". The Biographical Encyclopedia of Astronomers in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler (1814) that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.

Further, the current usage of "Kepler's Second Law" is something of a misnomer. Kepler had two versions, related in a qualitative sense: the "distance law" and the "area law". The "area law" is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.

History

Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler's third law was published in 1619. Kepler had believed in the Copernican model of the Solar System, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest eccentricity of all planets except Mercury. His first law reflected this discovery.

In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter. Godefroy Wendelin also made this observation in 1643. The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his Philosophical Transactions were in its favour. As the century proceeded it became more widely accepted. The reception in Germany changed noticeably between 1688, the year in which Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.

Formulary

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

First law

The orbit of every planet is an ellipse with the sun at one of the two foci.

Kepler's first law placing the Sun at the focus of an elliptical orbit

Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = r_min and for θ = 180°, r = r_max.

Mathematically, an ellipse can be represented by the formula:

${\displaystyle r=\frac{p}{1+\epsilon \cos \theta },}$

where ${\displaystyle p}$ is the semi-latus rectum, ε is the eccentricity of the ellipse, r is the distance from the Sun to the planet, and θ is the angle to the planet's current position from its closest approach, as seen from the Sun. So (r, θ) are polar coordinates.

For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity).

At θ = 0°, perihelion, the distance is minimum

${\displaystyle r_{min}=\frac{p}{1+\epsilon }}$

At θ = 90° and at θ = 270° the distance is equal to ${\displaystyle p}$.

At θ = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

${\displaystyle r_{max}=\frac{p}{1-\epsilon }}$

The semi-major axis a is the arithmetic mean between r_min and r_max:

${\displaystyle \begin{array}{rl}a& =\frac{r_{max}+r_{min}}{2}\\ a& =\frac{p}{1-{\epsilon }^2}\end{array}}$

The semi-minor axis b is the geometric mean between r_min and r_max:

${\displaystyle \begin{array}{rl}b& =\sqrt{r_{max}{r}_{min}}\\ b& =\frac{p}{\sqrt{1-{\epsilon }^2}}\end{array}}$

The semi-latus rectum p is the harmonic mean between r_min and r_max:

${\displaystyle \begin{array}{rl}p& ={\left(\frac{r_{max}^{-1}+r_{min}^{-1}}{2}\right)}^{-1}\\ pa& =r_{max}{r}_{min}=b^2\end{array}}$

The eccentricity ε is the coefficient of variation between r_min and r_max:

${\displaystyle \epsilon =\frac{r_{max}-r_{min}}{r_{max}+r_{min}}.}$

The area of the ellipse is

${\displaystyle A=\pi ab.}$

The special case of a circle is ε = 0, resulting in r = p = r_min = r_max = a = b and A = πr².

Second law

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity.

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area. Kepler notably arrived at this law through assumptions that were either only approximately true or outright false. Nevertheless, the result of the Second Law is exactly true, as it is logically equivalent to the conservation of angular momentum, which is true for any body experiencing a radially symmetric force

In a small time ${\displaystyle dt}$ the planet sweeps out a small triangle having base line ${\displaystyle r}$ and height ${\displaystyle rd\theta }$ and area $dA=\frac{1}{2}\cdot r\cdot rd\theta$, so the constant areal velocity is

$${\displaystyle \frac{dA}{dt}=\frac{r^2}{2}\frac{d\theta }{dt}.}$$

The area enclosed by the elliptical orbit is ${\displaystyle \pi ab}$. So the period ${\displaystyle T}$ satisfies

${\displaystyle T\cdot \frac{r^2}{2}\frac{d\theta }{dt}=\pi ab}$

and the mean motion of the planet around the Sun

${\displaystyle n=\frac{2\pi }{T}}$

satisfies

${\displaystyle r^{2}d\theta =abndt.}$

And so,

$${\displaystyle \frac{dA}{dt}=\frac{abn}{2}=\frac{\pi ab}{T}.}$$

Orbits of planets with varying eccentricities.

Low	High
Planet orbiting the Sun in a circular orbit (e=0.0)	Planet orbiting the Sun in an orbit with e=0.5
Planet orbiting the Sun in an orbit with e=0.2	Planet orbiting the Sun in an orbit with e=0.8
The red ray rotates at a constant angular velocity and with the same orbital time period as the planet, ${\displaystyle T=1}$. S: Sun at the primary focus, C: Centre of ellipse, S': The secondary focus. In each case, the area of all sectors depicted is identical.

Third law

The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation. It was therefore known as the harmonic law. The original form of this law (referring to not the semi-major axis, but rather a "mean distance") holds true only for planets with small eccentricities near zero. 

Using Newton's law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force:

${\displaystyle mr{\omega }^2=G\frac{mM}{r^2}}$

Then, expressing the angular velocity ω in terms of the orbital period ${\displaystyle T}$ and then rearranging, results in Kepler's Third Law:

${\displaystyle mr{\left(\frac{2\pi }{T}\right)}^2=G\frac{mM}{r^2}\to T^2=\left(\frac{4{\pi }^2}{GM}\right)r^3\to T^2\propto r^3}$

A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius, ${\displaystyle r}$, with the semi-major axis, ${\displaystyle a}$, of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass ${\displaystyle M}$ with ${\displaystyle M+m}$. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:

${\displaystyle \frac{a^3}{T^2}=\frac{G(M+m)}{4{\pi }^2}\approx \frac{GM}{4{\pi }^2}\approx 7.496\times {10}^{-6}\frac{{\text{AU}}^3}{{\text{days}}^2}\text{ is constant}}$

where ${\displaystyle M}$ is the mass of the Sun, ${\displaystyle m}$ is the mass of the planet, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle T}$ is the orbital period and ${\displaystyle a}$ is the elliptical semi-major axis, and ${\displaystyle \text{AU}}$ is the astronomical unit, the average distance from earth to the sun.

Table

The following table shows the data used by Kepler to empirically derive his law:

Data used by Kepler (1618)

Planet	Mean distance to sun (AU)	Period (days)	$\frac{R^3}{T^2}$ (10-6 AU3/day2)
Mercury	0.389	87.77	7.64
Venus	0.724	224.70	7.52
Earth	1	365.25	7.50
Mars	1.524	686.95	7.50
Jupiter	5.20	4332.62	7.49
Saturn	9.510	10759.2	7.43

Upon finding this pattern Kepler wrote:

I first believed I was dreaming... But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.

— translated from Harmonies of the World by Kepler (1619)

Log-log plot of period T vs semi-major axis a (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that a³/T² is constant (green line)

For comparison, here are modern estimates:

Modern data

Planet	Semi-major axis (AU)	Period (days)	$\frac{R^3}{T^2}$ (10-6 AU3/day2)
Mercury	0.38710	87.9693	7.496
Venus	0.72333	224.7008	7.496
Earth	1	365.2564	7.496
Mars	1.52366	686.9796	7.495
Jupiter	5.20336	4332.8201	7.504
Saturn	9.53707	10775.599	7.498
Uranus	19.1913	30687.153	7.506
Neptune	30.0690	60190.03	7.504

Planetary acceleration

Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second laws.

1. The direction of the acceleration is towards the Sun.
2. The magnitude of the acceleration is inversely proportional to the square of the planet's distance from the Sun (the inverse square law).

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his Principia that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view. Moreover, he does not assign a cause to gravity.

Newton defined the force acting on a planet to be the product of its mass and the acceleration (see Newton's laws of motion). So:

1. Every planet is attracted towards the Sun.
2. The force acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun.

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in Newton's law of universal gravitation:

1. All bodies in the Solar System attract one another.
2. The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.

As the planets have small masses compared to that of the Sun, the orbits conform approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately. (See two-body problem.)

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

Acceleration vector

See also: Polar coordinate § Vector calculus, and Mechanics of planar particle motion

From the heliocentric point of view consider the vector to the planet ${\displaystyle \mathbf{r}=r\hat{\mathbf{r}}}$ where ${\displaystyle r}$ is the distance to the planet and ${\displaystyle \hat{\mathbf{r}}}$ is a unit vector pointing towards the planet.

$${\displaystyle \frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathit{\theta }},\frac{d\hat{\mathit{\theta }}}{dt}=\dot{\hat{\mathit{\theta }}}=-\dot{\theta }\hat{\mathbf{r}}}$$

where ${\displaystyle \hat{\mathit{\theta }}}$ is the unit vector whose direction is 90 degrees counterclockwise of ${\displaystyle \hat{\mathbf{r}}}$, and ${\displaystyle \theta }$ is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.

Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:

$${\displaystyle \begin{array}{rl}\dot{\mathbf{r}}& =\dot{r}\hat{\mathbf{r}}+r\dot{\hat{\mathbf{r}}}=\dot{r}\hat{\mathbf{r}}+r\dot{\theta }\hat{\mathit{\theta }},\\ \ddot{\mathbf{r}}& =\left(\ddot{r}\hat{\mathbf{r}}+\dot{r}\dot{\hat{\mathbf{r}}}\right)+\left(\dot{r}\dot{\theta }\hat{\mathit{\theta }}+r\ddot{\theta }\hat{\mathit{\theta }}+r\dot{\theta }\dot{\hat{\mathit{\theta }}}\right)=\left(\ddot{r}-r{\dot{\theta }}^2\right)\hat{\mathbf{r}}+\left(r\ddot{\theta }+2\dot{r}\dot{\theta }\right)\hat{\mathit{\theta }}.\end{array}}$$

So

$${\displaystyle \ddot{\mathbf{r}}=a_r\hat{\mathit{r}}+a_{\theta }\hat{\mathit{\theta }}}$$

where the radial acceleration is

$${\displaystyle a_r=\ddot{r}-r{\dot{\theta }}^2}$$

and the transversal acceleration is

$${\displaystyle a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }.}$$

Inverse square law

Kepler's second law says that

$${\displaystyle r^2\dot{\theta }=nab}$$

is constant.

The transversal acceleration ${\displaystyle a_{\theta }}$ is zero:

$${\displaystyle \frac{d\left(r^2\dot{\theta }\right)}{dt}=r\left(2\dot{r}\dot{\theta }+r\ddot{\theta }\right)=r{a}_{\theta }=0.}$$

So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.

The radial acceleration ${\displaystyle a_{\text{r}}}$ is

$${\displaystyle a_{\text{r}}=\ddot{r}-r{\dot{\theta }}^2=\ddot{r}-r{\left(\frac{nab}{r^2}\right)}^2=\ddot{r}-\frac{n^2{a}^2{b}^2}{r^3}.}$$

Kepler's first law states that the orbit is described by the equation:

$${\displaystyle \frac{p}{r}=1+\epsilon \cos (\theta ).}$$

Differentiating with respect to time

$${\displaystyle -\frac{p\dot{r}}{r^2}=-\epsilon \sin (\theta )\dot{\theta }}$$

or

$${\displaystyle p\dot{r}=nab\epsilon \sin (\theta ).}$$

Differentiating once more

$${\displaystyle p\ddot{r}=nab\epsilon \cos (\theta )\dot{\theta }=nab\epsilon \cos (\theta )\frac{nab}{r^2}=\frac{n^2{a}^2{b}^2}{r^2}\epsilon \cos (\theta ).}$$

The radial acceleration ${\displaystyle a_{\text{r}}}$ satisfies

$${\displaystyle p{a}_{\text{r}}=\frac{n^2{a}^2{b}^2}{r^2}\epsilon \cos (\theta )-p\frac{n^2{a}^2{b}^2}{r^3}=\frac{n^2{a}^2{b}^2}{r^2}\left(\epsilon \cos (\theta )-\frac{p}{r}\right).}$$

Substituting the equation of the ellipse gives

$${\displaystyle p{a}_{\text{r}}=\frac{n^2{a}^2{b}^2}{r^2}\left(\frac{p}{r}-1-\frac{p}{r}\right)=-\frac{n^2{a}^2}{r^2}{b}^2.}$$

The relation ${\displaystyle b^2=pa}$ gives the simple final result

$${\displaystyle a_{\text{r}}=-\frac{n^2{a}^3}{r^2}.}$$

This means that the acceleration vector ${\displaystyle \ddot{\mathbf{r}}}$ of any planet obeying Kepler's first and second law satisfies the inverse square law

$${\displaystyle \ddot{\mathbf{r}}=-\frac{\alpha }{r^2}\hat{\mathbf{r}}}$$

where

$${\displaystyle \alpha =n^2{a}^3}$$

is a constant, and ${\displaystyle \hat{\mathbf{r}}}$ is the unit vector pointing from the Sun towards the planet, and ${\displaystyle r}$ is the distance between the planet and the Sun.

Since mean motion ${\displaystyle n=\frac{2\pi }{T}}$ where ${\displaystyle T}$ is the period, according to Kepler's third law, ${\displaystyle \alpha }$ has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire Solar System.

The inverse square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line. (See Kepler orbit.)

Newton's law of gravitation

By Newton's second law, the gravitational force that acts on the planet is:

$${\displaystyle \mathbf{F}=m_{\text{planet}}\ddot{\mathbf{r}}=-m_{\text{planet}}\alpha r^{-2}\hat{\mathbf{r}}}$$

where ${\displaystyle m_{\text{planet}}}$ is the mass of the planet and ${\displaystyle \alpha }$ has the same value for all planets in the Solar System. According to Newton's third law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, ${\displaystyle m_{\text{Sun}}}$. So

$${\displaystyle \alpha =G{m}_{\text{Sun}}}$$

where ${\displaystyle G}$ is the gravitational constant.

The acceleration of Solar System body number i is, according to Newton's laws:

$${\displaystyle {\ddot{\mathbf{r}}}_i=G\sum _{j\ne i}{m}_j{r}_{ij}^{-2}{\hat{\mathbf{r}}}_{ij}}$$

where ${\displaystyle m_j}$ is the mass of body j, ${\displaystyle r_{ij}}$ is the distance between body i and body j, ${\displaystyle {\hat{\mathbf{r}}}_{ij}}$ is the unit vector from body i towards body j, and the vector summation is over all bodies in the Solar System, besides i itself.

In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes

$${\displaystyle {\ddot{\mathbf{r}}}_{\text{Earth}}=G{m}_{\text{Sun}}{r}_{\text{Earth},\text{Sun}}^{-2}{\hat{\mathbf{r}}}_{\text{Earth},\text{Sun}}}$$

which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.

If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes

$${\displaystyle {\ddot{\mathbf{r}}}_{\text{Moon}}=G{m}_{\text{Earth}}{r}_{\text{Moon},\text{Earth}}^{-2}{\hat{\mathbf{r}}}_{\text{Moon},\text{Earth}}}$$

So in this approximation, the Moon moves around the Earth according to Kepler's laws.

In the three-body case the accelerations are

$${\displaystyle \begin{array}{rl}{\ddot{\mathbf{r}}}_{\text{Sun}}& =G{m}_{\text{Earth}}{r}_{\text{Sun},\text{Earth}}^{-2}{\hat{\mathbf{r}}}_{\text{Sun},\text{Earth}}+G{m}_{\text{Moon}}{r}_{\text{Sun},\text{Moon}}^{-2}{\hat{\mathbf{r}}}_{\text{Sun},\text{Moon}}\\ {\ddot{\mathbf{r}}}_{\text{Earth}}& =G{m}_{\text{Sun}}{r}_{\text{Earth},\text{Sun}}^{-2}{\hat{\mathbf{r}}}_{\text{Earth},\text{Sun}}+G{m}_{\text{Moon}}{r}_{\text{Earth},\text{Moon}}^{-2}{\hat{\mathbf{r}}}_{\text{Earth},\text{Moon}}\\ {\ddot{\mathbf{r}}}_{\text{Moon}}& =G{m}_{\text{Sun}}{r}_{\text{Moon},\text{Sun}}^{-2}{\hat{\mathbf{r}}}_{\text{Moon},\text{Sun}}+G{m}_{\text{Earth}}{r}_{\text{Moon},\text{Earth}}^{-2}{\hat{\mathbf{r}}}_{\text{Moon},\text{Earth}}\end{array}}$$

These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations. (See Lunar theory.)

Position as a function of time

Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, is the following five steps:

1. Compute the mean motion n = (2π rad)/P, where P is the period.
2. Compute the mean anomaly M = nt, where t is the time since perihelion.
3. Compute the eccentric anomaly E by solving Kepler's equation:
   $${\displaystyle M=E-\epsilon \sin E,}$$

   
   where ${\displaystyle \epsilon }$ is the eccentricity.
4. Compute the true anomaly θ by solving the equation:
   $${\displaystyle (1-\epsilon ){\tan }^2\frac{\theta }{2}=(1+\epsilon ){\tan }^2\frac{E}{2}}$$

   
5. Compute the heliocentric distance r:
   $${\displaystyle r=a(1-\epsilon \cos E),}$$

   
   where ${\displaystyle a}$ is the semimajor axis.

The position polar coordinates (r,θ) can now be written as a Cartesian vector ${\displaystyle \mathbf{p}=r⟨\cos \theta ,\sin \theta ⟩}$ and the Cartesian velocity vector can then be calculated as ${\displaystyle \mathbf{v}=\frac{\sqrt{\mu a}}{r}⟨-\sin E,\sqrt{1-{\epsilon }^2}\cos E⟩}$, where ${\displaystyle \mu }$ is the standard gravitational parameter.

The important special case of circular orbit, ε = 0, gives θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.

Mean anomaly, M

Main article: Mean anomaly

Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled S and the planet P. The auxiliary circle is an aid to calculation. Line xd is perpendicular to the base and through the planet P. The shaded sectors are arranged to have equal areas by positioning of point y.

The Keplerian problem assumes an elliptical orbit and the four points:

• s the Sun (at one focus of ellipse);
• z the perihelion
• c the center of the ellipse
• p the planet

and

• ${\displaystyle a=|cz|,}$ distance between center and perihelion, the semimajor axis,
• ${\displaystyle \epsilon =\frac{|cs|}{a},}$ the eccentricity,
• ${\displaystyle b=a\sqrt{1-{\epsilon }^2},}$ the semiminor axis,
• ${\displaystyle r=|sp|,}$ the distance between Sun and planet.
• ${\displaystyle \theta =\mathrm{\angle }zsp,}$ the direction to the planet as seen from the Sun, the true anomaly.

The problem is to compute the polar coordinates (r,θ) of the planet from the time since perihelion, t.

It is solved in steps. Kepler considered the circle with the major axis as a diameter, and

• ${\displaystyle x,}$ the projection of the planet to the auxiliary circle
• ${\displaystyle y,}$ the point on the circle such that the sector areas |zcy| and |zsx| are equal,
• ${\displaystyle M=\mathrm{\angle }zcy,}$ the mean anomaly.

The sector areas are related by ${\displaystyle |zsp|=\frac{b}{a}\cdot |zsx|.}$

The circular sector area ${\displaystyle |zcy|=\frac{a^{2}M}{2}.}$

The area swept since perihelion,

$${\displaystyle |zsp|=\frac{b}{a}\cdot |zsx|=\frac{b}{a}\cdot |zcy|=\frac{b}{a}\cdot \frac{a^{2}M}{2}=\frac{abM}{2},}$$

is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.

$${\displaystyle M=nt,}$$

where n is the mean motion.

Eccentric anomaly, E

When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ = f(M) is, however, not elementary. Kepler's solution is to use

$${\displaystyle E=\mathrm{\angle }zcx,}$$

x as seen from the centre, the eccentric anomaly as an intermediate variable, and first compute E as a function of M by solving Kepler's equation below, and then compute the true anomaly θ from the eccentric anomaly E. Here are the details.

$${\displaystyle \begin{array}{rl}|zcy|& =|zsx|=|zcx|-|scx|\\ with|scx|& =\frac{|cs|.|dx|}{2}\\ \frac{a^{2}M}{2}& =\frac{a^{2}E}{2}-\frac{a\epsilon \cdot a\sin E}{2}\end{array}}$$

Division by a²/2 gives Kepler's equation

$${\displaystyle M=E-\epsilon \sin E.}$$

This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used.

Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.

But note: Cartesian position coordinates with reference to the center of ellipse are (a cos E, b sin E)

With reference to the Sun (with coordinates (c,0) = (ae,0) ), r = (a cos E – ae, b sin E)

True anomaly would be arctan(r_y/rx), magnitude of r would be √r · r.

True anomaly, θ

Note from the figure that

$${\displaystyle |cd|=|cs|+|sd|}$$

so that

$${\displaystyle a\cos E=a\epsilon +r\cos \theta .}$$

Dividing by ${\displaystyle a}$ and inserting from Kepler's first law

$${\displaystyle \frac{r}{a}=\frac{1-{\epsilon }^2}{1+\epsilon \cos \theta }}$$

to get

$${\displaystyle \cos E=\epsilon +\frac{1-{\epsilon }^2}{1+\epsilon \cos \theta }\cos \theta =\frac{\epsilon (1+\epsilon \cos \theta )+\left(1-{\epsilon }^2\right)\cos \theta }{1+\epsilon \cos \theta }=\frac{\epsilon +\cos \theta }{1+\epsilon \cos \theta }.}$$

The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.

A computationally more convenient form follows by substituting into the trigonometric identity:

$${\displaystyle {\tan }^2\frac{x}{2}=\frac{1-\cos x}{1+\cos x}.}$$

Get

$${\displaystyle \begin{array}{rl}{\tan }^2\frac{E}{2}& =\frac{1-\cos E}{1+\cos E}=\frac{1-\frac{\epsilon +\cos \theta }{1+\epsilon \cos \theta }}{1+\frac{\epsilon +\cos \theta }{1+\epsilon \cos \theta }}\\ & =\frac{(1+\epsilon \cos \theta )-(\epsilon +\cos \theta )}{(1+\epsilon \cos \theta )+(\epsilon +\cos \theta )}=\frac{1-\epsilon }{1+\epsilon }\cdot \frac{1-\cos \theta }{1+\cos \theta }=\frac{1-\epsilon }{1+\epsilon }{\tan }^2\frac{\theta }{2}.\end{array}}$$

Multiplying by 1 + ε gives the result

$${\displaystyle (1-\epsilon ){\tan }^2\frac{\theta }{2}=(1+\epsilon ){\tan }^2\frac{E}{2}}$$

This is the third step in the connection between time and position in the orbit.

Distance, r

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:

$${\displaystyle r(1+\epsilon \cos \theta )=a\left(1-{\epsilon }^2\right)}$$

Using the relation above between θ and E the final equation for the distance r is:

$${\displaystyle r=a(1-\epsilon \cos E).}$$

Egyptian pyramids

 From Wikipedia, the free encyclopedia

A view of the Giza pyramid complex from the plateau to the south of the complex. From left to right, the three largest are: the Pyramid of Menkaure, the Pyramid of Khafre and the Great Pyramid of Giza. The three smaller pyramids in the foreground are subsidiary structures associated with Menkaure's pyramid.

Famous pyramids (cut-through with internal labyrinth layout).

The Egyptian pyramids are ancient masonry structures located in Egypt. Sources cite at least 118 identified "Egyptian" pyramids. Approximately 80 pyramids were built within the Kingdom of Kush, now located in the modern country of Sudan. Of those located in modern Egypt, most were built as tombs for the country's pharaohs and their consorts during the Old and Middle Kingdom periods.

The earliest known Egyptian pyramids are found at Saqqara, northwest of Memphis, although at least one step-pyramid-like structure has been found at Saqqara, dating to the First Dynasty: Mastaba 3808, which has been attributed to the reign of Pharaoh Anedjib, with inscriptions, and other archaeological remains of the period, suggesting there may have been others. The otherwise earliest among these is the Pyramid of Djoser built c. 2630–2610 BCE during the Third Dynasty. This pyramid and its surrounding complex are generally considered to be the world's oldest monumental structures constructed of dressed masonry.

The most famous Egyptian pyramids are those found at Giza, on the outskirts of Cairo. Several of the Giza pyramids are counted among the largest structures ever built. The Pyramid of Khufu is the largest Egyptian pyramid and the last of the Seven Wonders of the Ancient World still in existence, despite being the oldest by about 2,000 years.

Name

Unicode: 𓍋𓅓𓂋𓉴
Pyramid in hieroglyphs

The name for a pyramid in Egyptian is myr, written with the symbol 𓉴 (O24 in the Gardner Sign List). Myr is preceded by three other signs used as phonetics. The meaning of myr is unclear, as it only self-references the built object itself. By comparison, some similar architectural terms become compound words, such as the word for 'temple' (per-ka) becoming a compound of the words for 'house' and 'soul'. It has been speculated myr belongs to a class of words like djed and ankh, which refer to objects already in existence when the Egyptian language split off from afroasiatic.  A typical translation of myr is given as 'high place'. By graphical analysis, myr uses the same sign, O24, as benben. The benben is the mound of existence that arose out of the abyss, known as nun in the Egyptian creation myth. The relationship between myr and benben is further linked by the capstone architectural element of pyramids and obelisks, which was named benbenet, the feminine form of benben.

Sign O24 related terms

Hieroglyph	Sign	Egyptian	English
	O24	myr	Pyramid
	O24	benben	Primeval Mound
	O24	benbent	Pyramidon
	O24	Aaa	Pyramid tomb

Historical development

The Mastabat al-Fir’aun at Saqqara

Preceded by assumed earlier sites in the Eastern Sahara, tumuli with megalithic monuments developed as early as 4700 BCE in the Saharan region of Niger. Fekri Hassan (2002) indicates that the megalithic monuments in the Saharan region of Niger and the Eastern Sahara may have served as antecedents for the mastabas and pyramids of ancient Egypt. During Predynastic Egypt, tumuli were present at various locations (e.g., Naqada, Helwan).

From the time of the Early Dynastic Period (c. 3150–2686 BCE), Egyptians with sufficient means were buried in bench-like structures known as mastabas. At Saqqara, Mastaba 3808, dating from the latter part of the 1st Dynasty, was discovered to contain a large, independently built step-pyramid-like structure enclosed within the outer palace facade mastaba. Archaeological remains and inscriptions suggest there may have been other similar structures dating to this period.

The first historically documented Egyptian pyramid is attributed by Egyptologists to the 3rd Dynasty pharaoh Djoser. Although Egyptologists often credit his vizier Imhotep as its architect, the dynastic Egyptians themselves, contemporaneously or in numerous later dynastic writings about the character, did not credit him with either designing Djoser's pyramid or the invention of stone architecture. The Pyramid of Djoser was first built as a square mastaba-like structure, which as a rule were known to otherwise be rectangular, and was expanded several times by way of a series of accretion layers, to produce the stepped pyramid structure we see today. Egyptologists believe this design served as a gigantic stairway by which the soul of the deceased pharaoh could ascend to the heavens.

Though other pyramids were attempted in the 3rd Dynasty after Djoser, it was the 4th Dynasty, transitioning from the step pyramid to true pyramid shape, which gave rise to the great pyramids of Meidum, Dahshur, and Giza. The last pharaoh of the 4th Dynasty, Shepseskaf, did not build a pyramid and beginning in the 5th Dynasty; for various reasons, the massive scale and precision of construction decreased significantly leaving these later pyramids smaller, less well-built, and often hastily constructed. By the end of the 6th Dynasty, pyramid building had largely ended and it was not until the Middle Kingdom that large pyramids were built again, though instead of stone, mudbrick was the main construction material.

Long after the end of Egypt's own pyramid-building period, a burst of pyramid-building occurred in what is present-day Sudan, after much of Egypt came under the rule of the Kingdom of Kush, which was then based at Napata. Napatan rule, known as the 25th Dynasty, lasted from 750 BCE to 664 BCE. The Meroitic period of Kushite history, when the kingdom was centered on Meroë, (approximately in the period between 300 BCE and 300 CE), experienced a full-blown pyramid-building revival, which saw about 180 Egyptian-inspired indigenous royal pyramid-tombs constructed in the vicinity of the kingdom's capital cities.

Al-Aziz Uthman (1171–1198), the second Ayyubid Sultan of Egypt, tried to destroy the Giza pyramid complex. He gave up after only damaging the Pyramid of Menkaure because the task proved too large.

Pyramid symbolism

Diagram of the interior structures of the Great Pyramid. The inner line indicates the pyramid's present profile, the outer line indicates the original profile.

The shape of Egyptian pyramids is thought to represent the primordial mound from which the Egyptians believed the earth was created. The shape of a pyramid is also thought to be representative of the descending rays of the sun, and most pyramids were faced with polished, highly reflective white limestone, in order to give them a brilliant appearance when viewed from a distance. Pyramids were often also named in ways that referred to solar luminescence. For example, the formal name of the Bent Pyramid at Dahshur was The Southern Shining Pyramid, and that of Senusret II at El Lahun was Senusret Shines.

While it is generally agreed that pyramids were burial monuments, there is continued disagreement on the particular theological principles that might have given rise to them. One suggestion is that they were designed as a type of "resurrection machine."

The Egyptians believed the dark area of the night sky around which the stars appear to revolve was the physical gateway into the heavens. One of the narrow shafts that extend from the main burial chamber through the entire body of the Great Pyramid points directly towards the center of this part of the sky. This suggests the pyramid may have been designed to serve as a means to magically launch the deceased pharaoh's soul directly into the abode of the gods.

All Egyptian pyramids were built on the west bank of the Nile, which, as the site of the setting sun, was associated with the realm of the dead in Egyptian mythology.

Number and location of pyramids

For a more comprehensive list, see List of Egyptian pyramids.

In 1842, Karl Richard Lepsius produced the first modern list of pyramids—now known as the Lepsius list of pyramids—in which he counted 67. A great many more have since been discovered. At least 118 Egyptian pyramids have been identified. The location of Pyramid 29 which Lepsius called the "Headless Pyramid", was lost for a second time when the structure was buried by desert sands after Lepsius's survey. It was found again only during an archaeological dig conducted in 2008.

Many pyramids are in a poor state of preservation or buried by desert sands. If visible at all, they may appear as little more than mounds of rubble. As a consequence, archaeologists are continuing to identify and study previously unknown pyramid structures.

The most recent pyramid to be discovered was that of Neith, a wife of Teti.

All of Egypt's pyramids, except the small Third Dynasty pyramid at Zawyet el-Maiyitin, are sited on the west bank of the Nile, and most are grouped together in a number of pyramid fields. The most important of these are listed geographically, from north to south, below.

Abu Rawash

Main article: Abu Rawash

The largely destroyed Pyramid of Djedefre

Abu Rawash is the site of Egypt's most northerly pyramid (other than the ruins of Lepsius pyramid number one), the mostly ruined Pyramid of Djedefre, son and successor of Khufu. Originally it was thought that this pyramid had never been completed, but the current archaeological consensus is that not only was it completed, but that it was originally about the same size as the Pyramid of Menkaure, which would have placed it among the half-dozen or so largest pyramids in Egypt.

Its location adjacent to a major crossroads made it an easy source of stone. Quarrying, which began in Roman times, has left little apart from about fifteen courses of stone superimposed upon the natural hillock that formed part of the pyramid's core. A small adjacent satellite pyramid is in a better state of preservation.

Giza

Main article: Giza pyramid complex

Map of the Giza pyramid complex

Aerial view of the Giza pyramid complex

The Giza Plateau is the location of the Pyramid of Khufu (also known as the "Great Pyramid" and the "Pyramid of Cheops"), the somewhat smaller Pyramid of Khafre (or Chephren), the relatively modest-sized Pyramid of Menkaure (or Mykerinus), along with a number of smaller satellite edifices known as "Queen's pyramids", and the Great Sphinx of Giza. Of the three, only Khafre's pyramid retains part of its original polished limestone casing, near its apex. This pyramid appears larger than the adjacent Khufu pyramid by virtue of its more elevated location, and the steeper angle of inclination of its construction—it is, in fact, smaller in both height and volume.

The Giza pyramid complex has been a popular tourist destination since antiquity and was popularized in Hellenistic times when the Great Pyramid was listed by Antipater of Sidon as one of the Seven Wonders of the Ancient World. Today it is the only one of those wonders still in existence.

Zawyet el-Aryan

See also: Zawyet el'Aryan

This site, halfway between Giza and Abusir, is the location for two unfinished Old Kingdom pyramids. The northern structure's owner is believed to be pharaoh Nebka, while the southern structure, known as the Layer Pyramid, may be attributable to the Third Dynasty pharaoh Khaba, a close successor of Sekhemkhet. If this attribution is correct, Khaba's short reign could explain the seemingly unfinished state of this step pyramid. Today it stands around 17 m (56 ft) high; had it been completed, it is likely to have exceeded 40 m (130 ft).

Abusir

Main article: Abusir

The Pyramid of Sahure at Abusir, viewed from the pyramid's causeway

There are a total of fourteen pyramids at this site, which served as the main royal necropolis during the Fifth Dynasty. The quality of construction of the Abusir pyramids is inferior to those of the Fourth Dynasty—perhaps signaling a decrease in royal power or a less vibrant economy. They are smaller than their predecessors and are built of low-quality local limestone.

The three major pyramids are those of Niuserre, which is also the best-preserved, Neferirkare Kakai and Sahure. The site is also home to the incomplete Pyramid of Neferefre. Most of the major pyramids at Abusir were built using similar construction techniques, comprising a rubble core surrounded by steps of mudbricks with a limestone outer casing. The largest of these Fifth Dynasty pyramids, the Pyramid of Neferirkare Kakai, is believed to have been built originally as a step pyramid some 70 m (230 ft) high and then later transformed into a "true" pyramid by having its steps filled in with loose masonry.

Saqqara

Main article: Saqqara

The Pyramid of Djoser

Major pyramids located here include the Pyramid of Djoser—generally identified as the world's oldest substantial monumental structure to be built of dressed stone—the Pyramid of Userkaf, the Pyramid of Teti and the Pyramid of Merikare, dating to the First Intermediate Period of Egypt. Also at Saqqara is the Pyramid of Unas, which retains a pyramid causeway that is one of the best-preserved in Egypt. Together with the pyramid of Userkaf, this pyramid was the subject of one of the earliest known restoration attempts, conducted by Khaemweset, a son of Ramesses II. Saqqara is also the location of the incomplete step pyramid of Djoser's successor Sekhemkhet, known as the Buried Pyramid. Archaeologists believe that had this pyramid been completed, it would have been larger than Djoser's.

South of the main pyramid field at Saqqara is a second collection of later, smaller pyramids, including those of Pepi I, Djedkare Isesi, Merenre, Pepi II and Ibi. Most of these are in a poor state of preservation.

The Fourth Dynasty pharaoh Shepseskaf either did not share an interest in or have the capacity to undertake pyramid construction like his predecessors. His tomb, which is also sited at south Saqqara, was instead built as an unusually large mastaba and offering temple complex. It is commonly known as the Mastabat al-Fir’aun.

A previously unknown pyramid was discovered in north Saqqara in late 2008. Believed to be the tomb of Teti's mother, it currently stands approximately 5 m (16 ft) high, although the original height was closer to 14 m (46 ft).

Dahshur

Main article: Dahshur

Sneferu's Red Pyramid

This area is arguably the most important pyramid field in Egypt outside Giza and Saqqara, although until 1996 the site was inaccessible due to its location within a military base and was relatively unknown outside archaeological circles.

The southern Pyramid of Sneferu, commonly known as the Bent Pyramid, is believed to be the first Egyptian pyramid intended by its builders to be a "true" smooth-sided pyramid from the outset; the earlier pyramid at Meidum had smooth sides in its finished state, but it was conceived and built as a step pyramid, before having its steps filled in and concealed beneath a smooth outer casing of dressed stone. As a true smooth-sided structure, the Bent Pyramid was only a partial success—albeit a unique, visually imposing one; it is also the only major Egyptian pyramid to retain a significant proportion of its original smooth outer limestone casing intact. As such it serves as the best contemporary example of how the ancient Egyptians intended their pyramids to look. Several kilometres to the north of the Bent Pyramid is the last—and most successful—of the three pyramids constructed during the reign of Sneferu; the Red Pyramid is the world's first successfully completed smooth-sided pyramid. The structure is also the third-largest pyramid in Egypt, after the pyramids of Khufu and Khafra at Giza.

Also at Dahshur is one of two pyramids built by Amenemhat III, known as the Black Pyramid, as well as a number of small, mostly ruined subsidiary pyramids.

Mazghuna

Main article: Mazghuna

Located to the south of Dahshur, several mudbrick pyramids were built in this area in the late Middle Kingdom, perhaps for Amenemhat IV and Sobekneferu.

The Pyramid of Amenemhet I at Lisht

Lisht

Main article: Lisht

Two major pyramids are known to have been built at Lisht: those of Amenemhat I and his son, Senusret I. The latter is surrounded by the ruins of ten smaller subsidiary pyramids. One of these subsidiary pyramids is known to be that of Amenemhat's cousin, Khaba II. The site which is in the vicinity of the oasis of the Faiyum, midway between Dahshur and Meidum, and about 100 kilometres south of Cairo, is believed to be in the vicinity of the ancient city of Itjtawy (the precise location of which remains unknown), which served as the capital of Egypt during the Twelfth Dynasty.

Meidum

Main article: Meidum

The pyramid at Meidum

The pyramid at Meidum is one of three constructed during the reign of Sneferu, and is believed by some to have been started by that pharaoh's father and predecessor, Huni. However, that attribution is uncertain, as no record of Huni's name has been found at the site. It was constructed as a step pyramid and then later converted into the first "true" smooth-sided pyramid, when the steps were filled in and an outer casing added. The pyramid suffered several catastrophic collapses in ancient and medieval times. Medieval Arab writers described it as having seven steps, although today only the three uppermost of these remain, giving the structure its odd, tower-like appearance. The hill on which the pyramid is situated is not a natural landscape feature, it is the small mountain of debris created when the lower courses and outer casing of the pyramid gave way.

Hawara

Main article: Hawara

The Pyramid of Amenemhet III at Hawara

Amenemhat III was the last powerful ruler of the Twelfth Dynasty, and the pyramid he built at Hawara, near the Faiyum, is believed to post-date the so-called "Black Pyramid" built by the same ruler at Dahshur. It is the Hawara pyramid that is believed to have been Amenemhet's final resting place.

El Lahun

Main article: El Lahun

The Pyramid of Senusret II. The pyramid's natural limestone core is clearly visible as the yellow stratum at its base.

The Pyramid of Senusret II at El Lahun is the southernmost royal-tomb pyramid structure in Egypt. Its builders reduced the amount of work necessary to construct it by using as its foundation and core a 12-meter-high natural limestone hill.

El-Kurru

Main article: El-Kurru

Piye's pyramid at El-Kurru

Piye, the king of Kush who became the first ruler of the Twenty-fifth Dynasty, built a pyramid at El-Kurru. He was the first Egyptian pharaoh to be buried in a pyramid in centuries.

Nuri

Main article: Nuri

Taharqa's pyramid at Nuri

Taharqa, a Kushite ruler of the Twenty-fifth Dynasty, built his pyramid at Nuri. It was the largest in the area (North Sudan).

Construction dates and heights

The following table lays out the chronology of the construction of most of the major pyramids mentioned here. Each pyramid is identified through the pharaoh who ordered it built, his approximate reign, and its location.

Pyramid (Pharaoh)	Reign	Field	Height
Pyramid of Djoser (Djoser)	c. 2670 BCE	Saqqara	62 meters (203 feet)
Red Pyramid (Sneferu)	c. 2612–2589 BCE	Dahshur	104 meters (341 feet)
Meidum Pyramid (Sneferu)	c. 2612–2589 BCE	Meidum	65 meters (213 feet) (ruined) Would have been 91.65 meters (301 feet) or 175 Egyptian Royal cubits.
Great Pyramid of Giza (Khufu)	c. 2589–2566 BCE	Giza	146.7 meters (481 feet) or 280 Egyptian Royal cubits
Pyramid of Djedefre (Djedefre)	c. 2566–2558 BCE	Abu Rawash	60 meters (197 feet)
Pyramid of Khafre (Khafre)	c. 2558–2532 BCE	Giza	136.4 meters (448 feet) Originally: 143.5 m (471 ft) or 274 Egyptian Royal cubits
Pyramid of Menkaure (Menkaure)	c. 2532–2504 BCE	Giza	65 meters (213 feet) or 125 Egyptian Royal cubits
Pyramid of Userkaf (Userkaf)	c. 2494–2487 BCE	Saqqara	48 meters (161 feet)
Pyramid of Sahure (Sahure)	c. 2487–2477 BCE	Abusir	47 meters (155 feet)
Pyramid of Neferirkare (Neferirkare Kakai)	c. 2477–2467 BCE	Abusir	72.8 meters (239 feet)
Pyramid of Nyuserre (Nyuserre Ini)	c. 2416–2392 BCE	Abusir	51.68 m (169.6 feet) or 99 Egyptian Royal cubits
Pyramid of Amenemhat I (Amenemhat I)	c. 1991–1962 BCE	Lisht	55 meters (181 feet)
Pyramid of Senusret I (Senusret I)	c. 1971–1926 BCE	Lisht	61.25 meters (201 feet)
Pyramid of Senusret II (Senusret II)	c. 1897–1878 BCE	el-Lahun	48.65 m (159.6 ft; 93 Egyptian Royal cubits) or 47.6 m (156 ft; 91 Egyptian Royal cubits)
Black Pyramid (Amenemhat III)	c. 1860–1814 BCE	Dahshur	75 meters (246 feet)
Pyramid of Khendjer (Khendjer)	c. 1764–1759 BCE	Saqqara	about 37 metres (121 ft), now completely ruined
Pyramid of Piye (Piye)	c. 721 BCE	El-Kurru	20 meters (66 feet) or 30 meters (99 feet)
Pyramid of Taharqa (Taharqa)	c. 664 BCE	Nuri	40 meters (132 feet) or 50 meters (164 feet)

Construction techniques

Drawing showing transportation of a colossus. The water poured in the path of the sledge, long dismissed by Egyptologists as ritual, but now confirmed as feasible, served to increase the stiffness of the sand, and likely reduced by 50% the force needed to move the statue.

Main article: Egyptian pyramid construction techniques

Further information: Diary of Merer

Constructing the pyramids involved moving huge quantities of stone. While most blocks came from nearby quarries, special stones were transported on great barges from distant locations, for instance white limestone from Tura and granite from Aswan.

In 2013, papyri, named Diary of Merer, were discovered at an ancient Egyptian harbor at the Red Sea coast. They are logbooks written over 4,500 years ago by an official with the title inspector, who documented the transport of white limestone from the Tura quarries, along the Nile River, to the Great Pyramid of Giza, the tomb of the Pharaoh Khufu.

It is possible that quarried blocks were then transported to the construction site by wooden sleds, with sand in front of the sled wetted to reduce friction. Droplets of water created bridges between the grains of sand, helping them stick together. Workers cut the stones close to the construction site, as indicated by the numerous finds of cutting tools. The finished blocks were placed on the pre-prepared foundations. The foundations were levelled using a rough square level, water trenches and experienced surveyors.