True, Eccentric, and Mean Anomaly¶

This section deals with the conversion between true, eccentric, and mean anomaly.

True anomaly, frequently denoted $\nu$, is the angular parameter that defines the position of an object moving along a Keplerian orbit. It is the angle between the eccentricity vector (vector pointing from the main pericenter to the periapsis) and the current position of the body in the orbital plane itself.

The eccentric anomaly, $E$, is another angular parameter that defines the position of an object moving along a Keplerian orbit if viewed from the center of the ellipse.

Finally, the mean anomaly, $M$, defines the fraction of an orbital period that has elapsed since the orbiting object has passed its periapsis. It is the angle from the pericenter an object moving on a fictitious circular orbit with the same semi-major axis would have progressed through in the same time as the body on the true elliptical orbit.

Conversion between all types of angular anomaly is possible. However, there is no known direct conversion between true and mean anomaly. Conversion between the two is accomplished by transformation through eccentric anomaly.

True and Eccentric Anomaly Conversions¶

To convert from true anomaly to eccentric anomaly, you can use the function anomaly_eccentric_to_true. To perform the reverse conversion use anomaly_true_to_eccentric.

Eccentric anomaly can be converted to true anomaly by using equations derived using equations from Vallado¹. Starting from Equation (2-12) $$ \sin{\nu} = \frac{\sin{E}\sqrt{1-e^2}}{1 - e\cos{E}} $$ can be divided by $$ \cos{\nu} = \frac{\cos{E}-e}{1 - e\cos{E}} $$ and rearrange to get $$ \nu = \arctan{\frac{\sin{E}\sqrt{1-e^2}}{\cos{E}-e}} $$

This conversion is what is implemented by anomaly_eccentric_to_true. Similarly, we can derive $$ E = \arctan{\frac{\sin{\nu}\sqrt{1-e^2}}{\cos{\nu}+e}} $$ which allows for conversion from true anomaly to eccentric anomaly and is implemented in anomaly_true_to_eccentric.

PythonRust

import brahe as bh

bh.initialize_eop()

nu = 45.0  # Starting true anomaly (degrees)
e = 0.01  # Eccentricity

# Convert to eccentric anomaly
ecc_anomaly = bh.anomaly_true_to_eccentric(nu, e, angle_format=bh.AngleFormat.DEGREES)
print(f"True anomaly:      {nu:.3f} deg")
print(f"Eccentric anomaly: {ecc_anomaly:.3f} deg")

# Convert back from eccentric to true anomaly
nu_2 = bh.anomaly_eccentric_to_true(ecc_anomaly, e, angle_format=bh.AngleFormat.DEGREES)
print(f"Round-trip result: {nu_2:.3f} deg")

# Verify round-trip accuracy
print(f"Difference:        {abs(nu - nu_2):.2e} deg")

# Expected output:
# True anomaly:      45.000 deg
# Eccentric anomaly: 44.596 deg
# Round-trip result: 45.000 deg
# Difference:        0.00e0 deg

use brahe as bh;

fn main() {
    bh::initialize_eop().unwrap();

    let nu = 45.0; // Starting true anomaly (degrees)
    let e = 0.01;  // Eccentricity

    // Convert to eccentric anomaly
    let ecc_anomaly = bh::orbits::anomaly_true_to_eccentric(nu, e, bh::constants::AngleFormat::Degrees);
    println!("True anomaly:      {:.3} deg", nu);
    println!("Eccentric anomaly: {:.3} deg", ecc_anomaly);

    // Convert back from eccentric to true anomaly
    let nu_2 = bh::orbits::anomaly_eccentric_to_true(ecc_anomaly, e, bh::constants::AngleFormat::Degrees);
    println!("Round-trip result: {:.3} deg", nu_2);

    // Verify round-trip accuracy
    println!("Difference:        {:.2e} deg", (nu - nu_2).abs());

    // Expected output:
    // True anomaly:      45.000 deg
    // Eccentric anomaly: 44.596 deg
    // Round-trip result: 45.000 deg
    // Difference:        0.00e0 deg
}

Plot Source

fig_anomaly_true_eccentric.py
import os
import pathlib
import sys
import plotly.graph_objects as go
import brahe as bh

# Add plots directory to path for importing brahe_theme
sys.path.insert(0, str(pathlib.Path(__file__).parent))
from brahe_theme import get_theme_colors, save_themed_html

# Configuration
SCRIPT_NAME = pathlib.Path(__file__).stem
OUTDIR = pathlib.Path(os.getenv("BRAHE_FIGURE_OUTPUT_DIR", "./docs/figures/"))

# Ensure output directory exists
os.makedirs(OUTDIR, exist_ok=True)

# Generate data

# Generate range of true anomalies (degrees)
nu = [x for x in range(0, 360)]

# Compute eccentric anomaly for range of eccentricities
eccentricities = [0.0, 0.1, 0.3, 0.5, 0.7, 0.9]
ecc_data = {}
for e in eccentricities:
    # Take output mod 360 to wrap from 0 to 360 degrees
    ecc_data[e] = [
        bh.anomaly_true_to_eccentric(x, e, angle_format=bh.AngleFormat.DEGREES) % 360
        for x in nu
    ]

# Create figure with theme support


def create_figure(theme):
    """Create figure with theme-specific colors."""
    colors = get_theme_colors(theme)

    fig = go.Figure()

    # Color palette for different eccentricities
    color_palette = [
        colors["primary"],
        colors["secondary"],
        colors["accent"],
        colors["error"],
        colors["primary"],
        colors["secondary"],
    ]

    # Add traces for each eccentricity
    for i, e in enumerate(eccentricities):
        fig.add_trace(
            go.Scatter(
                x=nu,
                y=ecc_data[e],
                mode="lines",
                line=dict(color=color_palette[i % len(color_palette)], width=2),
                name=f"e = {e:.1f}",
                showlegend=True,
            )
        )

    # Configure axes
    fig.update_xaxes(
        tickmode="linear",
        tick0=0,
        dtick=30,
        title_text="True Anomaly (deg)",
        range=[0, 360],
        showgrid=False,
    )

    fig.update_yaxes(
        tickmode="linear",
        tick0=0,
        dtick=30,
        title_text="Eccentric Anomaly (deg)",
        range=[0, 360],
        showgrid=False,
    )

    return fig


# Generate and save both themed versions
light_path, dark_path = save_themed_html(create_figure, OUTDIR / SCRIPT_NAME)
print(f"✓ Generated {light_path}")
print(f"✓ Generated {dark_path}")

Eccentric and Mean Anomaly Conversions¶

To convert from true anomaly to eccentric anomaly, you can use the function anomaly_eccentric_to_mean. To perform the reverse conversion use anomaly_mean_to_eccentric.

Conversion from eccentric anomaly to mean anomaly is accomplished by application of Kepler's equation $$ M = E - e\sin{E} $$ which is implemented in anomaly_eccentric_to_mean.

Converting back from mean anomaly to eccentric anomaly is more challenging. There is no known closed-form solution to convert from mean anomaly to eccentric anomaly. Instead, we introduce the auxiliary equation $$ f(E) = E - e\sin(E) - M $$ And treat the problem as numerically solving for the root of $f$ for a given $M$. This iteration can be accomplished using Newton's method. Starting from an initial guess $E_0$ the value of $E_*$ can be iteratively updated using $$ E_{i+1} = \frac{f(E_i)}{f^\prime(E_i)}= E_i - \frac{E_i - e\sin{E_i} - M}{1 - e\cos{E_i}} $$ This update is performed until a coverage value of $$ |E_{i+1} - E_i| \leq \Delta_{\text{tol}} $$ is reached. The value set as 100 times floating-point machine precision 100 * f64::epsilon. This conversion is provided by anomaly_mean_to_eccentric.

Warning

Because this is a numerical method, convergence is not guaranteed. There is an upper limit of 10 iterations to reach convergence. Since convergence may not occur the output of the function is a Result, forcing the user to explicitly handle the case where the algorithm does not converage.

Since Python lacks Rust's same error handling mechanisms, non-convergence will result in a runtime error.

PythonRust

import brahe as bh

bh.initialize_eop()

ecc = 45.0  # Starting eccentric anomaly (degrees)
e = 0.01  # Eccentricity

# Convert to mean anomaly
mean_anomaly = bh.anomaly_eccentric_to_mean(ecc, e, angle_format=bh.AngleFormat.DEGREES)
print(f"Eccentric anomaly: {ecc:.3f} deg")
print(f"Mean anomaly:      {mean_anomaly:.3f} deg")

# Convert back from mean to eccentric anomaly
ecc_2 = bh.anomaly_mean_to_eccentric(
    mean_anomaly, e, angle_format=bh.AngleFormat.DEGREES
)
print(f"Round-trip result: {ecc_2:.3f} deg")

# Verify round-trip accuracy
print(f"Difference:        {abs(ecc - ecc_2):.2e} deg")

# Expected output:
# Eccentric anomaly: 45.000 deg
# Mean anomaly:      44.595 deg
# Round-trip result: 45.000 deg
# Difference:        0.00e0 deg

use brahe as bh;

fn main() {
    bh::initialize_eop().unwrap();

    let ecc = 45.0; // Starting eccentric anomaly (degrees)
    let e = 0.01;   // Eccentricity

    // Convert to mean anomaly
    let mean_anomaly = bh::orbits::anomaly_eccentric_to_mean(ecc, e, bh::constants::AngleFormat::Degrees);
    println!("Eccentric anomaly: {:.3} deg", ecc);
    println!("Mean anomaly:      {:.3} deg", mean_anomaly);

    // Convert back from mean to eccentric anomaly
    let ecc_2 = bh::orbits::anomaly_mean_to_eccentric(mean_anomaly, e, bh::constants::AngleFormat::Degrees).unwrap();
    println!("Round-trip result: {:.3} deg", ecc_2);

    // Verify round-trip accuracy
    println!("Difference:        {:.2e} deg", (ecc - ecc_2).abs());

    // Expected output:
    // Eccentric anomaly: 45.000 deg
    // Mean anomaly:      44.595 deg
    // Round-trip result: 45.000 deg
    // Difference:        0.00e0 deg
}

Plot Source

fig_anomaly_eccentric_mean.py
import os
import pathlib
import sys
import plotly.graph_objects as go
import brahe as bh

# Add plots directory to path for importing brahe_theme
sys.path.insert(0, str(pathlib.Path(__file__).parent))
from brahe_theme import get_theme_colors, save_themed_html

# Configuration
SCRIPT_NAME = pathlib.Path(__file__).stem
OUTDIR = pathlib.Path(os.getenv("BRAHE_FIGURE_OUTPUT_DIR", "./docs/figures/"))

# Ensure output directory exists
os.makedirs(OUTDIR, exist_ok=True)

# Generate data

# Generate range of eccentric anomalies (degrees)
ecc = [x for x in range(0, 360)]

# Compute mean anomaly for range of eccentricities
eccentricities = [0.0, 0.1, 0.3, 0.5, 0.7, 0.9]
mean_data = {}
for e in eccentricities:
    # Take output mod 360 to wrap from 0 to 360 degrees
    mean_data[e] = [
        bh.anomaly_eccentric_to_mean(x, e, angle_format=bh.AngleFormat.DEGREES) % 360
        for x in ecc
    ]

# Create figure with theme support


def create_figure(theme):
    """Create figure with theme-specific colors."""
    colors = get_theme_colors(theme)

    fig = go.Figure()

    # Color palette for different eccentricities
    color_palette = [
        colors["primary"],
        colors["secondary"],
        colors["accent"],
        colors["error"],
        colors["primary"],
        colors["secondary"],
    ]

    # Add traces for each eccentricity
    for i, e in enumerate(eccentricities):
        fig.add_trace(
            go.Scatter(
                x=ecc,
                y=mean_data[e],
                mode="lines",
                line=dict(color=color_palette[i % len(color_palette)], width=2),
                name=f"e = {e:.1f}",
                showlegend=True,
            )
        )

    # Configure axes
    fig.update_xaxes(
        tickmode="linear",
        tick0=0,
        dtick=30,
        title_text="Eccentric Anomaly (deg)",
        range=[0, 360],
        showgrid=False,
    )

    fig.update_yaxes(
        tickmode="linear",
        tick0=0,
        dtick=30,
        title_text="Mean Anomaly (deg)",
        range=[0, 360],
        showgrid=False,
    )

    return fig


# Generate and save both themed versions
light_path, dark_path = save_themed_html(create_figure, OUTDIR / SCRIPT_NAME)
print(f"✓ Generated {light_path}")
print(f"✓ Generated {dark_path}")

True and Mean Anomaly Conversions¶

Methods to convert from true anomaly to mean anomaly are provided for convenience. These methods simply wrap successive calls to two anomaly_true_to_mean. To perform the reverse conversion use anomaly_mean_to_true.

PythonRust

import brahe as bh

bh.initialize_eop()

nu = 45.0  # Starting true anomaly (degrees)
e = 0.01  # Eccentricity

# Convert to mean anomaly
mean_anomaly = bh.anomaly_true_to_mean(nu, e, angle_format=bh.AngleFormat.DEGREES)
print(f"True anomaly:      {nu:.3f} deg")
print(f"Mean anomaly:      {mean_anomaly:.3f} deg")

# Convert back from mean to true anomaly
nu_2 = bh.anomaly_mean_to_true(mean_anomaly, e, angle_format=bh.AngleFormat.DEGREES)
print(f"Round-trip result: {nu_2:.3f} deg")

# Verify round-trip accuracy
print(f"Difference:        {abs(nu - nu_2):.2e} deg")

# Expected output:
# True anomaly:      45.000 deg
# Mean anomaly:      44.194 deg
# Round-trip result: 45.000 deg
# Difference:        0.00e0 deg

use brahe as bh;

fn main() {
    bh::initialize_eop().unwrap();

    let nu = 45.0; // Starting true anomaly (degrees)
    let e = 0.01;  // Eccentricity

    // Convert to mean anomaly
    let mean_anomaly = bh::orbits::anomaly_true_to_mean(nu, e, bh::constants::AngleFormat::Degrees);
    println!("True anomaly:      {:.3} deg", nu);
    println!("Mean anomaly:      {:.3} deg", mean_anomaly);

    // Convert back from mean to true anomaly
    let nu_2 = bh::orbits::anomaly_mean_to_true(mean_anomaly, e, bh::constants::AngleFormat::Degrees).unwrap();
    println!("Round-trip result: {:.3} deg", nu_2);

    // Verify round-trip accuracy
    println!("Difference:        {:.2e} deg", (nu - nu_2).abs());

    // Expected output:
    // True anomaly:      45.000 deg
    // Mean anomaly:      44.194 deg
    // Round-trip result: 45.000 deg
    // Difference:        0.00e0 deg
}

Plot Source

fig_anomaly_true_mean.py
import os
import pathlib
import sys
import plotly.graph_objects as go
import brahe as bh

# Add plots directory to path for importing brahe_theme
sys.path.insert(0, str(pathlib.Path(__file__).parent))
from brahe_theme import get_theme_colors, save_themed_html

# Configuration
SCRIPT_NAME = pathlib.Path(__file__).stem
OUTDIR = pathlib.Path(os.getenv("BRAHE_FIGURE_OUTPUT_DIR", "./docs/figures/"))

# Ensure output directory exists
os.makedirs(OUTDIR, exist_ok=True)

# Generate data

# Generate range of true anomalies (degrees)
nu = [x for x in range(0, 360)]

# Compute mean anomaly for range of eccentricities
eccentricities = [0.0, 0.1, 0.3, 0.5, 0.7, 0.9]
mean_data = {}
for e in eccentricities:
    # Take output mod 360 to wrap from 0 to 360 degrees
    mean_data[e] = [
        bh.anomaly_true_to_mean(x, e, angle_format=bh.AngleFormat.DEGREES) % 360
        for x in nu
    ]

# Create figure with theme support


def create_figure(theme):
    """Create figure with theme-specific colors."""
    colors = get_theme_colors(theme)

    fig = go.Figure()

    # Color palette for different eccentricities
    color_palette = [
        colors["primary"],
        colors["secondary"],
        colors["accent"],
        colors["error"],
        colors["primary"],
        colors["secondary"],
    ]

    # Add traces for each eccentricity
    for i, e in enumerate(eccentricities):
        fig.add_trace(
            go.Scatter(
                x=nu,
                y=mean_data[e],
                mode="lines",
                line=dict(color=color_palette[i % len(color_palette)], width=2),
                name=f"e = {e:.1f}",
                showlegend=True,
            )
        )

    # Configure axes
    fig.update_xaxes(
        tickmode="linear",
        tick0=0,
        dtick=30,
        title_text="True Anomaly (deg)",
        range=[0, 360],
        showgrid=False,
    )

    fig.update_yaxes(
        tickmode="linear",
        tick0=0,
        dtick=30,
        title_text="Mean Anomaly (deg)",
        range=[0, 360],
        showgrid=False,
    )

    return fig


# Generate and save both themed versions
light_path, dark_path = save_themed_html(create_figure, OUTDIR / SCRIPT_NAME)
print(f"✓ Generated {light_path}")
print(f"✓ Generated {dark_path}")

D. Vallado, Fundamentals of Astrodynamics and Applications (4th Ed.), 2010
https://celestrak.com/software/vallado-sw.php ↩