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Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly Formula

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Eccentric Anomaly in Hyperbolic Orbit is an angular parameter that characterizes the position of an object within its hyperbolic trajectory. Check FAQs

F = 2 \cdot a \tanh (\sqrt{\frac{e h - 1}{e h + 1}} \cdot \tan (\frac{θ}{2}))

F - Eccentric Anomaly in Hyperbolic Orbit?e_h - Eccentricity of Hyperbolic Orbit?θ - True Anomaly?

Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly Example

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Here is how the Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly equation looks like with Values.

Here is how the Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly equation looks like with Units.

Here is how the Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly equation looks like.

68.2207 = 2 \cdot a \tanh (\sqrt{\frac{1.339 - 1}{1.339 + 1}} \cdot \tan (\frac{109}{2}))

68.2207° = 2 \cdot a \tanh (\sqrt{\frac{1.339 - 1}{1.339 + 1}} \cdot \tan (\frac{109°}{2}))

68.2207 = 2 \cdot a \tanh (\sqrt{\frac{1.339 - 1}{1.339 + 1}} \cdot \tan (\frac{109}{2}))

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Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly Solution

Follow our step by step solution on how to calculate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?

FIRST Step Consider the formula

F = 2 \cdot a \tanh (\sqrt{\frac{e h - 1}{e h + 1}} \cdot \tan (\frac{θ}{2}))

Next Step Substitute values of Variables

∴

F = 2 \cdot a \tanh (\sqrt{\frac{1.339 - 1}{1.339 + 1}} \cdot \tan (\frac{109°}{2}))

Next Step Convert Units

∴

F = 2 \cdot a \tanh (\sqrt{\frac{1.339 - 1}{1.339 + 1}} \cdot \tan (\frac{1.9024rad}{2}))

Next Step Prepare to Evaluate

∴

F = 2 \cdot a \tanh (\sqrt{\frac{1.339 - 1}{1.339 + 1}} \cdot \tan (\frac{1.9024}{2}))

Next Step Evaluate

∴

F = 1.19067631954554rad

Next Step Convert to Output's Unit

∴

F = 68.2207278761425°

LAST Step Rounding Answer

∴

F = 68.2207°

Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly Formula Elements

Variables

Functions

Eccentric Anomaly in Hyperbolic Orbit

Eccentric Anomaly in Hyperbolic Orbit is an angular parameter that characterizes the position of an object within its hyperbolic trajectory.

Symbol: F

Measurement: AngleUnit: °

Note: Value should be greater than 0.

Formulas to find Eccentric Anomaly in Hyperbolic Orbit

Eccentricity of Hyperbolic Orbit

Eccentricity of Hyperbolic Orbit describes how much the orbit differs from a perfect circle, and this value typically falls between 1 and infinity.

Symbol: e_h

Measurement: NAUnit: Unitless

Note: Value should be greater than 1.

Formulas to find Eccentricity of Hyperbolic Orbit

True Anomaly

True Anomaly measures the angle between the object's current position and the perigee (the point of closest approach to the central body) when viewed from the focus of the orbit.

Symbol: θ

Measurement: AngleUnit: °

Note: Value can be positive or negative.

Formulas to find True Anomaly

tan

The tangent of an angle is a trigonometric ratio of the length of the side opposite an angle to the length of the side adjacent to an angle in a right triangle.

Syntax: tan(Angle)

sqrt

A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number.

Syntax: sqrt(Number)

tanh

The hyperbolic tangent function (tanh) is a function that is defined as the ratio of the hyperbolic sine function (sinh) to the hyperbolic cosine function (cosh).

Syntax: tanh(Number)

atanh

The inverse hyperbolic tangent function returns the value whose hyperbolic tangent is a number.

Syntax: atanh(Number)

Other formulas in Orbital Position as Function of Time category

Go Mean Anomaly in Hyperbolic Orbit given Hyperbolic Eccentric Anomaly

M h = e h \cdot \sinh (F) - F

Go Time since Periapsis in Hyperbolic Orbit given Mean Anomaly

t = \frac{{h h}^{3}}{{[GM.Earth]}^{2} \cdot {({e h}^{2} - 1)}^{\frac{3}{2}}} \cdot M h

Go Time since Periapsis in Hyperbolic Orbit given Hyperbolic Eccentric Anomaly

t = \frac{{h h}^{3}}{{[GM.Earth]}^{2} \cdot {({e h}^{2} - 1)}^{\frac{3}{2}}} \cdot (e h \cdot \sinh (F) - F)

Go True Anomaly in Hyperbolic Orbit given Hyperbolic Eccentric Anomaly and Eccentricity

θ = 2 \cdot a \tan (\sqrt{\frac{e h + 1}{e h - 1}} \cdot \tanh (\frac{F}{2}))

How to Evaluate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?

Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly evaluator uses Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2)) to evaluate the Eccentric Anomaly in Hyperbolic Orbit, The Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly formula is a parameter used to describe the position of an object in a hyperbolic orbit relative to its central body, it's similar to the concept of eccentric anomaly used in elliptical orbits, but adapted for hyperbolic trajectories, given the eccentricity and the true anomaly of the hyperbolic orbit, the hyperbolic eccentric anomaly can be calculated using a hyperbolic analog of Kepler's equation. Eccentric Anomaly in Hyperbolic Orbit is denoted by F symbol.

How to evaluate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly using this online evaluator? To use this online evaluator for Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly, enter Eccentricity of Hyperbolic Orbit (e_h) & True Anomaly (θ) and hit the calculate button.

FAQs on Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly

What is the formula to find Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?

The formula of Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly is expressed as Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2)). Here is an example- 3908.76 = 2*atanh(sqrt((1.339-1)/(1.339+1))*tan(1.90240888467346/2)).

How to calculate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?

With Eccentricity of Hyperbolic Orbit (e_h) & True Anomaly (θ) we can find Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly using the formula - Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2)). This formula also uses Tangent (tan)Square Root (sqrt)Hyperbolic Tangent (tanh), Inverse Hyperbolic Tangent (atanh) function(s).

Can the Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly be negative?

No, the Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly, measured in Angle cannot be negative.

Which unit is used to measure Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?

Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly is usually measured using the Degree[°] for Angle. Radian[°], Minute[°], Second[°] are the few other units in which Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly can be measured.

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