## Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly Solution

STEP 0: Pre-Calculation Summary
Formula Used
Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2))
F = 2*atanh(sqrt((eh-1)/(eh+1))*tan(θ/2))
This formula uses 4 Functions, 3 Variables
Functions Used
tan - Trigonometric tangent function, tan(Angle)
sqrt - Square root function, sqrt(Number)
tanh - Hyperbolic tangent function, tanh(Number)
atanh - Inverse hyperbolic tangent function, atanh(Number)
Variables Used
Eccentric Anomaly in Hyperbolic Orbit - (Measured in Radian) - Eccentric Anomaly in Hyperbolic Orbit is an angular parameter that characterizes the position of an object within its hyperbolic trajectory.
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.
True Anomaly - (Measured in Radian) - 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.
STEP 1: Convert Input(s) to Base Unit
Eccentricity of Hyperbolic Orbit: 1.339 --> No Conversion Required
True Anomaly: 109 Degree --> 1.90240888467346 Radian (Check conversion here)
STEP 2: Evaluate Formula
Substituting Input Values in Formula
F = 2*atanh(sqrt((eh-1)/(eh+1))*tan(θ/2)) --> 2*atanh(sqrt((1.339-1)/(1.339+1))*tan(1.90240888467346/2))
Evaluating ... ...
F = 1.19067631954554
STEP 3: Convert Result to Output's Unit
1.19067631954554 Radian --> No Conversion Required
1.19067631954554 1.190676 Radian <-- Eccentric Anomaly in Hyperbolic Orbit
(Calculation completed in 00.004 seconds)
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## Credits

Created by Harsh Raj
Indian Institute of Technology, Kharagpur (IIT KGP), Kharagpur, West Bengal
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## < 5 Orbital Position as Function of Time Calculators

Time since Periapsis in Hyperbolic Orbit given Hyperbolic Eccentric Anomaly
Time since Periapsis = Angular Momentum of Orbit^3/([GM.Earth]^2*(Eccentricity of Hyperbolic Orbit^2-1)^(3/2))*(Eccentricity of Hyperbolic Orbit*sinh(Eccentric Anomaly in Hyperbolic Orbit)-Eccentric Anomaly in Hyperbolic Orbit)
True Anomaly in Hyperbolic Orbit given Hyperbolic Eccentric Anomaly and Eccentricity
True Anomaly = 2*atan(sqrt((Eccentricity of Hyperbolic Orbit+1)/(Eccentricity of Hyperbolic Orbit-1))*tanh(Eccentric Anomaly in Hyperbolic Orbit/2))
Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly
Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2))
Mean Anomaly in Hyperbolic Orbit given Hyperbolic Eccentric Anomaly
Mean Anomaly in Hyperbolic Orbit = Eccentricity of Hyperbolic Orbit*sinh(Eccentric Anomaly in Hyperbolic Orbit)-Eccentric Anomaly in Hyperbolic Orbit
Time since Periapsis in Hyperbolic Orbit given Mean Anomaly
Time since Periapsis = Angular Momentum of Orbit^3/([GM.Earth]^2*(Eccentricity of Hyperbolic Orbit^2-1)^(3/2))*Mean Anomaly in Hyperbolic Orbit

## Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly Formula

Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2))
F = 2*atanh(sqrt((eh-1)/(eh+1))*tan(θ/2))

## Why are parabolic trajectories also called escape trajectories?

If the body of some mass m is launched on a parabolic trajectory, it will coast to infinity, arriving there with zero velocity relative to central body. It will not return. Parabolic paths are therefore called escape trajectories.

## How to Calculate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?

Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly calculator uses Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2)) to calculate the Eccentric Anomaly in Hyperbolic Orbit, The Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly formula is defined as angle that defines the object's position within the hyperbolic orbit, given the true anomaly. Eccentric Anomaly in Hyperbolic Orbit is denoted by F symbol.

How to calculate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly using this online calculator? To use this online calculator for Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly, enter Eccentricity of Hyperbolic Orbit (eh) & True Anomaly (θ) and hit the calculate button. Here is how the Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly calculation can be explained with given input values -> 1.190676 = 2*atanh(sqrt((1.339-1)/(1.339+1))*tan(1.90240888467346/2)).

### FAQ

What is Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?
The Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly formula is defined as angle that defines the object's position within the hyperbolic orbit, given the true anomaly and is represented as F = 2*atanh(sqrt((eh-1)/(eh+1))*tan(θ/2)) or Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2)). Eccentricity of Hyperbolic Orbit describes how much the orbit differs from a perfect circle, and this value typically falls between 1 and infinity & 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.
How to calculate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly?
The Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly formula is defined as angle that defines the object's position within the hyperbolic orbit, given the true anomaly is calculated using Eccentric Anomaly in Hyperbolic Orbit = 2*atanh(sqrt((Eccentricity of Hyperbolic Orbit-1)/(Eccentricity of Hyperbolic Orbit+1))*tan(True Anomaly/2)). To calculate Hyperbolic Eccentric Anomaly given Eccentricity and True Anomaly, you need Eccentricity of Hyperbolic Orbit (eh) & True Anomaly (θ). With our tool, you need to enter the respective value for Eccentricity of Hyperbolic Orbit & True Anomaly and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
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