Kepler orbit

Kepler orbit

Gravitational attraction is the force that makes the Solar system stick together with the planets orbiting the Sun and the Moon orbiting the Earth. Isaac Newton formulated the physical law for this gravitational attraction which explained Kepler's laws of planetary motion that had been discovered empirically by Johannes Kepler about 80 year earlier analysing data from astronomical observations .

This law of universal gravitation says:

Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses:

: F = G frac{m_1 m_2}{r^2},

where:

* "F" is the magnitude of the gravitational force between the two point masses,
* "G" is the gravitational constant,
* "m"1 is the mass of the first point mass,
* "m"2 is the mass of the second point mass,
* "r" is the distance between the two point masses.

The shapes of essentially all celestial bodies are close to spheres [Some minor asteroids have a shape strongly deviating from a sphere] . For symmetry reasons it is clear that the net gravitational force attracting a mass point towards a homogeneous sphere must be directed towards the centre of the sphere. The shell theorem (also proven by Isaac Newton) says that the magnitude of this force is the same as if all mass was concentrated in the middle of the sphere. [From this immediately follows that the attraction between two homogeneous spheres is as if both had its mass concentrated to centre.] Planets that rotate (like for example the Earth) take a slightly oblate shape because of the centrifugal force and with such an oblate shape the gravitational attraction will deviate somewhat from that of a homogeneous sphere. This phenomenon is quite noticeable for artificial Earth satellites, especially those in low orbits, but at a large distance the effect of this oblateness is very small and the planetary motions in the Solar system can be computed with full precision assuming the gravitational attraction between any two bodies of the Solar system to follow the law

: F = G frac{m_1 m_2}{r^2},

where "r" is the distance between the centres of the celestial bodies.

The two body problem is the case that there are only two point masses (or homogeneous spheres).

If the two mass points (or homogenous spheres) have the masses m_1 and m_2 and the position vectors ar r_1 and ar r_2 relative a point fixed with respect to inertial space (for example relative their common centre of mass) the equations of motion for the two mass points are

: m_1 cdot ddot {ar r_1}=-G frac{m_1 m_2}r^2 cdot hat r : m_2 cdot ddot {ar r_2}= G frac{m_1 m_2}r^2 cdot hat r

where

:r = |ar r_1 - ar r_2|

is the distance between the bodies

and : hat r = frac{ar r_1 - ar r_2}{r}

is the unit vector pointing from body 2 to body 1

Dividing with the factors m_1 and m_2and subtracting the resulting equations one gets the differential equation : ddot {ar r } = -mu cdot frac {hat r } {r^2} (1)

for the vector from body 2 to body 1

: ar r = ar r_1 - ar r_2

where

: mu=G cdot (m_1+m_2)

This differential equation for the two body case can be completely solved mathematically and the resulting orbit which follows Kepler's laws of planetary motion is called a "Kepler orbit". The orbits of all planets are to a very high accuracy Kepler orbits around the Sun (as observed by Johannes Kepler!), the small deviations being due to the much weaker gravitational attractions between the planets. Also the orbits around the Earth of the Moon and of the artificial satellites are with a fair approximation Kepler orbits. In fact, the gravitational acceleration towards the Sun is about the same for the Earth and the satellite and therefore in a first approximation "cancels out". Also in high accuracy applications for which the equation of motion must be integrated numerically with all gravitational and non-gravitational forces [The non-gravitational forces that affect artificial Earth satellites are solar radiation pressure and air drag] being taken into account the Kepler orbit concepts are of paramount importance and heavily used.

For example, the orbital elements

*Semi-major axis
*eccentricity
*inclination
*right ascension of the ascending node
*argument of perigee
*True anomaly

are only defined for a Kepler orbit. The use of orbital elements therefore always implies that the orbit is approximated with the Kepler orbit having these orbital elements.

_________________

Mathematical solution of the differential equation (1) above

Like for the movement under any central force, i.e. a force aligned with hat{r} , the angular

momentum ar{H} = ar{r} imes {dot{ar{r} stays constant:: dot {ar{H = dot {overbrace{ar{r} imes {dot{ar{r} = dot{ar{r imes {dot{ar{r} +

ar{r} imes {ddot{ar{r} =ar{0} + ar{0} = ar{0}

Introducing a coordinate system hat{x} , hat{y} in the plane orthogonal to ar{H}

and polar coordinates: ar{r} = r cdot ( cos heta cdot hat{x} + sin heta cdot hat{y})

the differential equation (1) takes the form (see "Polar coordinates#Vector calculus")

: r^2 cdot dot{ heta} = H (2) : ddot{r} - r cdot {dot{ heta^2 = - frac {mu} {r^2} (3)

Taking the time derivative of (2) one gets : ddot{ heta} = - frac {2 cdot H cdot dot{r {r^3} (4)

Using the chain rule for differentiation one gets : dot{r} = frac {dr} {d heta} cdot dot { heta} (5)

: ddot{r} = frac {d^2r} {d heta^2} cdot {dot { heta^2 + frac {dr} {d heta} cdot ddot { heta}

(6)

Using the expressions for dot{ heta}, ddot{ heta}, dot{r}, ddot{r} of equations (2), (4),

(5) and (6) all time derivatives in (3) can be replaced by derivatives of r as function of

heta. After some simplification one gets: ddot{r} - r cdot {dot{ heta^2 = frac {H^2} {r^4} cdot left ( frac{d^2 r} {d heta ^2} - 2 cdot frac{left (frac {dr} {d heta} ight ) ^2}

{r} - r ight )= - frac {mu} {r^2} (7)The differential equation (7) can be solved analytically by the variable substitution:r=frac{1} {s} (8)

Using the chain rule for differentiation one gets:

:frac {dr} {d heta} = -frac {1} {s^2} cdot frac {ds} {d heta} (9):frac {d^2r} {d heta^2} = frac {2} {s^3} cdot left (frac {ds} {d heta} ight )^2 - frac {1} {s^2} cdot frac {d^2s} {d heta^2} (10)

Using the expressions (10) and (9) for frac {d^2r} {d heta^2} and frac {dr} {d heta}one gets:H^2 cdot left ( frac {d^2s} {d heta^2} + s ight ) = mu (11)with the general solution:s = frac {mu} {H^2} cdot left ( 1 + e cdot cos ( heta- heta_0) ight ) (12)

where e and heta_0 are constants of integration depending on the initial values for

s and frac {ds} {d heta}.

Instead of using the constant of integration heta_0 explicitly one introduces the convention that the

unit vectors hat{x} , hat{y} defining the coordinate system in the orbital plane are selected

such that heta_0 takes the value zero and e is positive. This then means that

heta is zero at the point where s is maximal and therefore r= frac {1}{s} is minimal. Defining the parameter p as frac {H^2}{mu} one has that :r = frac {1}{s} = frac {p}{1 + e cdot cos heta} (13)

This is the equation in polar coordinates for a conic section with origin in a focal point. The argument heta is called "true anomaly".

For e = 0 this is a circle with radius p.

For 0 < e < 1 this is an ellipse with : a = frac {p}{1-e^2} (14): b = frac {p}{sqrt{1-e^2 = a cdot sqrt{1-e^2} (15).

For e = 1 this is a parabola with focal length frac {p}{2}

For e > 1 this is a hyperbola with : a = frac {p}{e^2-1} (16): b = frac {p}{sqrt{e^2-1 = a cdot sqrt{e^2-1} (17)

The following image illustrates an ellipse(red), a parabola (green) and a hyperbola (blue)

The lower branch of the hyperbola is irrelevant here (image from Wikimedia Commons).

The point below the focal point F is the point with heta = 0 for which the distance to the focus takes the minimal value frac {p}{1 + e}, the pericentre. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value frac {p}{1 - e}. For the hyperbola the range for heta is :left [ -cos^{-1}(-frac{1}{e}) < heta < cos^{-1}(-frac{1}{e}) ight ] and for a parobola the range is:left [ -pi < heta < pi ight ]

Using the chain rule for differentiation (5), the equation (2) and the definition of p as frac {H^2}{mu} one gets that the radial velocity component is: V_r = dot{r} = frac {H}{p} cdot e cdot sin heta = sqrt{frac {mu}{p cdot e cdot sin heta (18)

and that the tangential component is: V_t = r cdot dot{ heta} = frac {H}{r} = sqrt{frac {mu}{p cdot (1 + e cdot cos heta)

(19)

The connection between the polar argument heta and time t is slightly different for elliptic and hyperbolic orbits. For an elliptic orbit one switches to the "eccentric anomaly" E for which : x = a cdot (cos E -e) (20): y = b cdot sin E (21)and consequently: dot{x} = -a cdot sin E cdot dot{E} (22): dot{y} = b cdot cos E cdot dot{E} (23)

and the angular momentum H is: H = x cdot dot{y} - y cdot dot{x}=a cdot b cdot ( 1 - e cdot cos E) cdot dot{E} (24)

Integrating with respect to time t one gets: H cdot t = a cdot b cdot ( E - e cdot sin E) (25)

under the assumption that time t=0 is selected such that the integration constant is zero.

As by definition of p one has: H = sqrt{mu cdot p} (26)this can be written: t = a cdot sqrt{frac{a} {mu ( E - e cdot sin E) (27)

For a hyperbolic orbit one uses the hyperbolic functions for the parameterisation : x = a cdot (e - cosh E) (28): y = b cdot sinh E (29)for which one has: dot{x} = -a cdot sinh E cdot dot{E} (30): dot{y} = b cdot cosh E cdot dot{E} (31)and the angular momentum H is: H = x cdot dot{y} - y cdot dot{x}=a cdot b cdot ( e cdot cosh E-1) cdot dot{E} (32)Integrating with respect to time t one gets: H cdot t= a cdot b cdot ( e cdot sinh E-E) (33)i.e.: t = a cdot sqrt{frac{a} {mu (e cdot sinh E-E) (34)To find what time t that corresponds to a certain true anomaly heta one computes corresponding

parameter E connected to time with relation (27) for an elliptic and with relation (34) for an hyperbolic orbit.

Note that the relations (27) and (34) define a mapping between the ranges:left [ -infin < t < infin ight ] longleftrightarrow left [-infin < E < infin ight ]

ome additional formulas

For an elliptic orbit one gets from (20) and (21) that:r = a cdot (1-e cdot cos E) (35)and therefore that: cos heta = frac{x} {r} =frac{cos E-e}{1-e cdot cos E} (36)From (36) then follows that: an^2 frac{ heta}{2} = frac{1-cos heta}{1+cos heta}= frac{1-frac{cos E-e}{1-e cdot cos E{1+frac{cos E-e}{1-e cdot cos E= frac{1-e cdot cos E - cos E+e}{1-e cdot cos E + cos E-e}= frac{1+e}{1-e} cdot frac{1-cos E}{1+cos E}= frac{1+e}{1-e} cdot an^2 frac{E}{2}

From the the geometrical construction defining the eccentric anomaly it is clear that the vectors ( cos E , sin E ) and ( cos heta , sin heta ) are on the same side of the x-axis. From this then follows that the vectors ( cosfrac{E}{2} , sinfrac{E}{2} ) and ( cosfrac{ heta}{2} , sinfrac{ heta}{2} ) are in the same quadrant. One therefore has that : an frac{ heta}{2} = sqrt{frac{1+e}{1-e cdot an frac{E}{2} (37)

and that

: heta = 2 cdot operatorname{arg}(sqrt{1-e} cdot cos frac{E}{2} , sqrt{1+e} cdot sinfrac{E}{2})+ ncdot 2pi (38).:E = 2 cdot operatorname{arg}(sqrt{1+e} cdot cos frac{ heta}{2} , sqrt{1-e} cdot sinfrac{ heta}{2})+ ncdot 2pi (39)

where "operatorname{arg}(x , y)" is the polar argument of the vector ( x , y ) and n is selected such that left |E - heta ight| < pi

For the numerical computation of operatorname{arg}(x , y) the standard function ATAN2(y,x)(or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN can be used.

Note that this is a mapping between the ranges

:left [ -infin < heta < infin ight ] longleftrightarrow left [-infin < E < infin ight ]

For an hyperbolic orbit one gets from (28) and (29) that:r = a cdot (e cdot cosh E-1) (40)and therefore that: cos heta = frac{x} {r} =frac{e-cosh E}{e cdot cosh E-1} (41)

As: an^2 frac{ heta}{2} = frac{1-cos heta}{1+cos heta}= frac{1-frac{e-cosh E}{e cdot cosh E-1{1+frac{e-cosh E}{e cdot cosh E-1= frac{e cdot cosh E - e +cosh E}{e cdot cosh E + e -cosh E}= frac{e+1}{e-1} cdot frac{cosh E-1}{cosh E+1}= frac{e+1}{e-1} cdot anh^2 frac{E}{2} and as an frac{ heta}{2} and anh frac{E}{2} have the same sign it follows that: an frac{ heta}{2} = sqrt{frac{e+1}{e-1 cdot anh frac{E}{2} (42)This relation is convenient for passing between "true anomaly" and the parameter E, the latter being connected to time through relation (34). Note that this is a mapping between the ranges

:left [ -cos^{-1}(-frac{1}{e}) < heta < cos^{-1}(-frac{1}{e}) ight ] longleftrightarrow left [-infin < E < infin ight ]

and that frac{E}{2} can be computed using the relation

: anh ^{-1}x=frac{1}{2}ln left( frac{1+x}{1-x} ight)

From relation (27) follows that the orbital period P for an elliptic orbit is: P = 2pi cdot a cdot sqrt{frac{a} {mu (43)

As the potential energy corresponding to the force field of relation (1) is:-frac {mu} {r}it follows from (13) , (14), (18) and (19) that the sum of the kinetic and the

potential energy

:fracV_r}^2+{V_t}^2}{2}-frac {mu} {r}

for an elliptic orbit is

:-frac {mu} {2 cdot a} (44)and from (13) , (16), (18) and (19) that the sum of the kinetic and the

potential energy for a hyperbolic orbit is:frac {mu} {2 cdot a} (45)

Relative the inertial coordinate system

: hat{x} , hat{y}

in the orbital plane with hat{x} towards pericentre one gets from (18) and (19) that the velocity componets are

: V_x = cos heta cdot V_r - sin heta cdot V_t = -sqrt{frac {mu}{p cdot sin heta (44): V_y = sin heta cdot V_r + cos heta cdot V_t = sqrt{frac {mu}{p cdot (e +cos heta) (45)

Determination of the Kepler orbit that corresponds to a given initial state

This is the "initial value problem" for the differential equation (1) which is a first order equation for the 6-dimensional "state vector" ( ar{r} ,ar{v} ) when written as

: dot {ar{v = -mu cdot frac {hat{r {r^2} (46): dot {ar{r = ar{v} (47)

For any values for the initial "state vector" ( ar{r_0} ,ar{v_0} ) the Kepler orbit corresponding to the solution of this initial value problem can be found with the following algorithm:

Define the orthogonal unit vectors (hat{r} , hat{t}) through

: ar{r_0} = r cdot hat{r} (48): ar{v_0} = V_r cdot hat{r} + V_t cdot hat{t} (49)

with r > 0 and V_t > 0

From (13), (18) and (19) follows that by setting

: p = frac(r cdot V_t)}^2}{mu } (50)

and by defining e ge 0 and heta such that

: e cdot cos heta = frac{V_t} {V_0} - 1 (51): e cdot sin heta = frac{V_r} {V_0} (52)

where

: V_0 = sqrt{frac{mu}{p (53)

one gets a Kepler orbit that for true anomaly heta has the same r, V_r and V_t values as those defined by (48) and (49).

If this Kepler orbit then also has the same (hat{r} , hat{t}) vectors for this true anomaly heta as the ones defined by (48) and (49) the state vector (ar{r} , ar{v}) of the Kepler orbit takes the desired values ( ar{r_0} ,ar{v_0} ) for true anomaly heta.

The standard inertially fixed coordinate system (hat{x} , hat{y}) in the orbital plane with (hat{x} directed from the centre of the homogeneous sphere to the pericentre) defining the orientation of the conical section (ellipse, parabola or hyperbola) can then be determined with the relation

: hat{x} = cos heta cdot hat{r} - sin heta cdot hat{t} (54): hat{y} = sin heta cdot hat{r} + cos heta cdot hat{t} (55)

Note that the relations (51) and (52) has a singularity when V_r=0 and

:V_t=V_0=sqrt{frac{mu}{p=sqrt{frac{mu}{frac(r cdot V_t)}^2}{mu }

i.e.

:V_t=sqrt{frac{mu}{r (56)

which is the case that it is a circular orbit that is fitting the initial state ( ar{r_0} ,ar{v_0} )

The osculating Kepler orbit

For any state vector ( ar{r} , ar{v} ) the Kepler orbit corresponding to this state can be computed with the algorithm defined above.First the parameters p , e , heta are determined from r , V_r , V_t and then the orthogonal unit vectors in the orbital plane hat{x} , hat{y} using the relations (54) and (55).

If now the equation of motion is

: ddot {ar{r = operatorname{ar{F(ar{r},dot {ar{r,t) (57)

where

:operatorname{ar{F(ar{r},dot {ar{r,t)

is another function then

:-mu cdot frac {hat{r {r^2}

the resulting parameters

p , e , heta, hat{x} , hat{y}

defined by ar{r},dot {ar{r will all vary with time as opposed to the case of a Kepler orbit for which only the parameter heta will vary

The Kepler orbit computed in this way having the same "state vector" as the solution to the "equation of motion" (57) at time t is said to be "osculating" at this time.

This concept is for example useful in case :operatorname{ar{F(ar{r},dot {ar{r,t)=-mu cdot frac {hat{r {r^2}+operatorname{ar{f(ar{r},dot {ar{r,t)

where

:operatorname{ar{f(ar{r},dot {ar{r,t)

is a small "perturbing force" due to for example a faint gravitational pull from other celestial bodies. The parameters of the osculating Kepler orbit will then only slowly change and the osculating Kepler orbit is a good approximation to the real orbit for a considerable time period before and after the time of osculation.

This concept can also be useful for a rocket during powered flight as it then tells which Kepler orbitthe rocket would continue in in case the thrust is switched-off.

For a "close to circular" orbit the concept "eccentricity vector" defined as ar{e}=e cdot hat{x} is useful. From (51), (52) and (54) follows that

:ar{e}=frac{(V_t-V_0) cdot hat{r} - V_r cdot hat{t{V_0} (58)

i.e. ar{e} is a smooth differentiable function of the state vector ( ar{r} ,ar{v} ) also if this state corresponds to acircular orbit.

Orbital elements

A Kepler orbit is specified by six orbital element that normally area - semi-major axis

e - eccentricity

i - inclination

Omega - right ascension of ascending node

omega - argument of perigee

heta - the true anomaly that corresponds to a specified time

The angles Omega , i, omega are the euler angles (alpha , eta, gamma with the

notations of that article) characterising the orientation of the coordinate system

:hat{x},hat{y},hat{z}

with hat{x},hat{y} in the orbital plane and with hat{x} in the direction to the pericentre.

The transformation from the euler angles Omega , i, omega to hat{x},hat{y},hat{z} is:

:x_1= cos Omega cdot cos omega - sin Omega cdot cos i cdot sin omega:x_2= sin Omega cdot cos omega + cos Omega cdot cos i cdot sin omega:x_3= sin i cdot sin omega:y_1=-cos Omega cdot sin omega - sin Omega cdot cos i cdot cos omega :y_2=-sin Omega cdot sin omega + cos Omega cdot cos i cdot cos omega :y_3= sin i cdot cos omega :z_1= sin i cdot sin Omega :z_2=-sin i cdot cos Omega :z_3= cos i

The transformation from hat{x},hat{y},hat{z} to euler angles Omega , i, omega is:

:Omega= operatorname{arg}( -z_2 , z_1 ):i = operatorname{arg}( z_3 , sqrtz_1}^2 + {z_2}^2} ):omega= operatorname{arg}( y_3 , x_3 )

where operatorname{arg}(x , y) signifies the polar argument that can be computed withthe standard function ATAN2(y,x) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN.

[Category:Orbits]


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