Orbital elements
Orbital elements, also known as Keplerian elements, are a set of six parameters that fully define the trajectory of a celestial body orbiting another in a two-body system, specifying the orbit's size, shape, orientation, and the body's position along it.[1] These elements derive from Johannes Kepler's laws of planetary motion and are essential for predicting and analyzing orbits in astronomy, astrodynamics, and space mission planning.[1] The classical six orbital elements consist of:- Semi-major axis (a): The average distance from the orbiting body to the central body, determining the orbit's scale.[1]
- Eccentricity (e): A measure of the orbit's deviation from a perfect circle, where e = 0 indicates a circular orbit and values between 0 and 1 describe ellipses.[1]
- Inclination (i): The angle between the orbital plane and a reference plane (such as the ecliptic or equatorial plane), defining the orbit's tilt.[1]
- Longitude of the ascending node (Ω): The angle from a reference direction to the point where the orbit crosses the reference plane moving northward, setting the orbit's rotational orientation.[1]
- Argument of periapsis (ω): The angle from the ascending node to the point of closest approach (periapsis), locating the orbit's closest point relative to the node.[1]
- Time of periapsis passage (τ) or mean anomaly (M) at epoch: The time or angular position indicating the body's location in the orbit at a specific reference time.[1]
Fundamentals of Orbital Elements
Definition and Purpose
Orbital elements are a set of six parameters that uniquely define the trajectory of a body in a two-body gravitational system, specifying the orbit's size, shape, orientation relative to a reference frame, and the body's position within that orbit at a given time.[1] The classical Keplerian elements consist of the semi-major axis (a), which determines the orbit's size; the eccentricity (e), which describes its shape; the inclination (i), which measures the tilt of the orbital plane; the longitude of the ascending node (Ω), which locates the point where the orbit crosses a reference plane; the argument of periapsis (ω), which indicates the orientation of the orbit's closest approach to the central body; and the mean anomaly (M) at epoch, which indicates the body's mean angular position in the orbit at a reference time.[1] These parameters provide a compact, geometrically intuitive representation of the orbit compared to time-varying position and velocity vectors in Cartesian coordinates.[3] The primary purpose of orbital elements is to reduce the complex, continuous description of orbital motion—encompassing infinite possible paths in three-dimensional space—to a finite set of scalar values that facilitate analytical solutions, long-term predictions, cataloging of celestial objects, and mission planning.[4] Unlike Cartesian coordinates, which require six time-dependent components to fully specify position and velocity at every instant, orbital elements leverage the symmetries of gravitational motion to enable efficient propagation of trajectories using closed-form equations, particularly in the unperturbed two-body problem.[4] This parameterization is especially valuable for astrodynamics, where it supports the design and analysis of satellite orbits and interplanetary transfers by allowing quick assessment of orbital stability and energy requirements.[4] In the context of Keplerian motion, orbits take the form of conic sections—ellipses for bound trajectories, parabolas for marginally escaping paths, or hyperbolas for unbound flybys—governed by the inverse-square law of gravitation, providing the foundational geometry for these elements. Their importance lies in enabling precise, long-term predictions of unperturbed motion, which serve as a baseline for accounting for real-world perturbations in both natural systems, such as planetary and asteroidal orbits, and artificial ones, like Earth-orbiting satellites.[4] This framework, originating from the work of Johannes Kepler and Isaac Newton on planetary motion, remains central to modern orbital mechanics despite extensions for multi-body effects.[1]Historical Development
The origins of orbital elements lie in ancient astronomical models that sought to describe planetary motions using geometric parameters. In the 2nd century CE, Claudius Ptolemy's geocentric system employed deferents and epicycles to account for observed irregularities in planetary paths, with parameters such as the epicycle's radius and angular speed serving as early analogs to modern eccentricity and orbital period concepts.[5] Similarly, Nicolaus Copernicus's heliocentric model in the 16th century retained epicycle-deferent structures but shifted the frame to the Sun, introducing parameters for orbital radii and velocities that prefigured later elements like semi-major axis.[6] Johannes Kepler's groundbreaking work in the early 17th century marked a pivotal shift toward elliptical orbits, supplanting circular assumptions. In his 1609 publication Astronomia Nova, Kepler formulated his first two laws, describing planetary orbits as ellipses with the Sun at one focus, thereby introducing the semi-major axis as a measure of orbital size and eccentricity as a quantifyer of orbital shape.[7] His third law, published in 1619 in Harmonices Mundi, related the square of the orbital period to the cube of the semi-major axis, providing a foundational relation for orbit determination. These innovations laid the groundwork for the classical Keplerian elements. Isaac Newton's Philosophiæ Naturalis Principia Mathematica in 1687 provided the theoretical framework by establishing the universal law of gravitation, enabling the analytical prediction of orbital paths from first principles and allowing elements to be derived mathematically rather than empirically fitted.[8] This gravitational inverse-square law unified Kepler's descriptive laws with causal mechanics, facilitating the computation of stable elliptical orbits under central forces. In the 19th century, Carl Friedrich Gauss advanced orbit determination techniques, particularly through his 1801 treatise Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium, where he applied least-squares methods to compute orbital elements from sparse observational data, successfully predicting the rediscovery of the asteroid Ceres.[9] Gauss's approach incorporated refinements to elements describing three-dimensional geometry, including orbital inclination—the angle between the orbital plane and a reference plane—and the longitude of the ascending node, which defines the orientation of the orbital plane relative to a fixed direction. These parameters, building on earlier work by Leonhard Euler and Joseph-Louis Lagrange, became essential for precise astronomical predictions.[10] The 20th century saw further evolution to address perturbations in real orbits, with Dirk Brouwer and Gerald M. Clemence's 1961 Methods of Celestial Mechanics providing a comprehensive treatment of variational equations for how gravitational influences alter classical elements over time.[11] The launch of Sputnik 1 in 1957, the first artificial satellite, spurred the standardization of orbital data formats, leading to the adoption of Two-Line Element (TLE) sets by the U.S. Department of Defense's NORAD for tracking Earth-orbiting objects and disseminating mean Keplerian elements globally.[12] This milestone integrated historical elements into practical satellite operations, culminating in the classical Keplerian set as a cornerstone of astrodynamics.Classification of Orbital Elements
Core Parameters for Orbit Determination
The fundamental inputs for determining orbital elements are the position vector \vec{r} and the velocity vector \vec{v} of the orbiting body at a specific instant, providing six scalar components in three-dimensional space (three coordinates for each vector). These state vectors represent the instantaneous dynamical state of the body relative to the central attracting body, such as Earth or the Sun, and serve as the basis for computing the classical orbital elements in the two-body problem of classical mechanics.[13] The position and velocity vectors determine all classical orbital elements through conserved quantities known as integrals of motion, including the total mechanical energy, which relates to the orbit's size, and the angular momentum vector, which informs its plane and shape. For instance, the magnitude of the angular momentum vector \vec{h} = \vec{r} \times \vec{v} establishes the specific angular momentum, a key invariant used to derive subsequent elements. These parameters encapsulate the full dynamical state without prior assumptions about the orbit's geometry, making them essential for initial orbit determination from observational data.[14][15] To fully specify the orbit, an epoch—a precise time reference—must accompany the vectors, such as the Julian Date or Coordinated Universal Time (UTC) at the moment of measurement, ensuring the state is tied to a unique instant. Without this temporal anchor, the vectors alone cannot distinguish between evolving orbits, as the body's motion is time-dependent under gravitational influence. The Julian Date system, for example, provides a continuous count of days since January 1, 4713 BCE, facilitating high-precision calculations in astrodynamics.[16] This combination of \vec{r}, \vec{v}, and epoch defines the osculating orbit, which is the instantaneous Keplerian (conic-section) trajectory that exactly matches the body's position and velocity at that epoch, effectively "kissing" the true trajectory at that point. In perturbed environments, such as near-Earth space with atmospheric drag or non-spherical gravity, the osculating orbit approximates the local motion but requires updates over time to account for deviations. These core parameters thus feed into the computation of shape descriptors like the semi-major axis, providing a foundational bridge to the classical orbital elements.[17]Shape and Size Parameters
The shape and size of an orbit in the two-body problem are primarily characterized by two parameters: the semi-major axis a, which determines the scale or overall size of the orbit, and the eccentricity e, which governs its deviation from a circular path. These parameters are derived from the conserved quantities of specific energy E and specific angular momentum h, under the influence of the gravitational parameter \mu = G(M + m), where G is the gravitational constant and M, m are the masses of the two bodies. For bound orbits, such as those of planets or satellites, a represents half the length of the major axis of the elliptical path, effectively averaging the distances from periapsis to apoapsis.[18] The semi-major axis a is related to the specific orbital energy E by the equation a = -\frac{\mu}{2E}, where E < 0 for closed orbits, ensuring a > 0. This relation stems from the conservation of total mechanical energy in the central force field, where the negative sign reflects the bound nature of the system. For elliptical orbits, a quantifies the average radial distance, influencing key dynamical properties like the orbital period via Kepler's third law.[18][19] The eccentricity e defines the orbit's shape, with e = 0 corresponding to a perfect circle, $0 < e < 1 to an ellipse, e = 1 to a parabola, and e > 1 to a hyperbola. It is given by e = \sqrt{1 + \frac{2 E h^2}{\mu^2}}, where h is the magnitude of the specific angular momentum vector, perpendicular to the orbital plane. This formula arises from the geometry of the conic section solution to the equations of motion, linking energy and angular momentum to the focal displacement of the orbit. As e increases from zero, the orbit elongates, concentrating motion near the central body at periapsis and slowing it at apoapsis.[19] Key distances along the major axis are the periapsis distance r_p = a(1 - e), the closest approach to the focus (central body), and the apoapsis distance r_a = a(1 + e), the farthest point. These expressions follow directly from the polar form of the conic section equation r = \frac{a(1 - e^2)}{1 + e \cos \theta}, evaluated at \theta = 0 (periapsis) and \theta = \pi (apoapsis). For circular orbits (e = 0), r_p = r_a = a, simplifying the geometry.[20] The vis-viva equation provides the speed v at any radial distance r: v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right). This relation, derived from energy conservation, holds for all conic sections and allows computation of velocity without specifying position angles, highlighting how speed varies inversely with a for a given r. At periapsis and apoapsis, it yields maximum and minimum speeds, respectively, underscoring the orbit's energy-driven dynamics.[18][19] The signs of a and e distinguish bounded from unbound orbits. For bound (closed) paths like ellipses, E < 0 implies a > 0, enabling periodic motion confined within r_a. Unbound orbits, such as parabolas (a \to \infty) and hyperbolas (E > 0, a < 0), represent escape trajectories where the object approaches from or recedes to infinity, with the negative a conventionally maintaining consistency in the conic equations despite the open geometry. This classification is fundamental for assessing orbital stability in celestial mechanics.[21][19]Orientation Parameters
The orientation parameters in orbital elements define the spatial attitude of an orbit relative to a chosen reference frame, specifying its tilt and rotational position without regard to the body's location within the plane. These parameters are essential for distinguishing orbits that share the same shape and size but differ in their alignment with respect to the central body's coordinate system. Together with shape and size parameters, they provide a complete geometric description of the orbit's path in three-dimensional space.[1] Inclination, denoted as i, measures the angle between the orbital plane and the reference plane, typically ranging from 0° to 180°. An inclination of 0° corresponds to a prograde equatorial orbit lying within the reference plane, while 90° indicates a polar orbit perpendicular to it, and 180° denotes a retrograde equatorial orbit. For Earth-centered orbits, the reference plane is usually the equatorial plane, whereas for solar system bodies, the ecliptic plane—defined by Earth's orbit around the Sun—serves as the standard.[4][1] The longitude of the ascending node, denoted as \Omega, quantifies the rotational position of the orbit's line of nodes within the reference plane, measured as the angle from a fixed reference direction—such as the vernal equinox—to the ascending node, where the orbit crosses the reference plane from south to north. This parameter ranges from 0° to 360° and helps describe phenomena like nodal precession due to gravitational influences. In equatorial reference frames, \Omega is often expressed as the right ascension of the ascending node, equivalent to its longitude but aligned with celestial coordinates. However, for orbits with i = 0^\circ or i = 180^\circ, the line of nodes is undefined, rendering \Omega indeterminate or conventionally set to zero in computational models.[4][22] The argument of periapsis, denoted as \omega, specifies the orientation of the orbit's closest approach point (periapsis) relative to the ascending node, measured along the orbital plane from the node to the periapsis direction. It ranges from 0° to 360° and completes the rotational description within the inclined plane. For circular orbits where eccentricity is zero, \omega becomes undefined due to the absence of a distinct periapsis, though it is often assigned a default value of 0° in practice. In collinear configurations, such as equatorial orbits, \omega aligns directly with the reference direction when \Omega is undefined.[4][1]Time and Epoch Parameters
Time and epoch parameters in orbital elements specify the position of a body along its orbit at a particular instant, providing a temporal reference within the otherwise geometrically defined path. These parameters are essential for determining the body's location relative to the fixed orbital frame established by other elements, such as orientation and shape. The epoch serves as the foundational reference time, while the sixth orbital element—typically the mean anomaly at epoch or the time of periapsis passage—defines the initial condition, with other anomalies derived for position calculations. The epoch denotes the specific reference time at which the set of orbital elements is defined and valid, ensuring consistency in predictions over short durations. For instance, in many astronomical applications, the standard epoch is J2000.0, corresponding to noon on January 1, 2000, Terrestrial Time, which aligns the elements with a common temporal baseline for ephemeris calculations. This choice maintains short-term accuracy, as orbital elements evolve due to perturbations, necessitating updates for longer predictions.[1] The true anomaly, denoted \nu, measures the angle at the focus (typically the central body) from the periapsis to the orbiting body's current position, ranging from 0° to 360°. It is a time-dependent quantity that indicates the body's location in the orbital plane relative to the closest approach point and is computed from the mean anomaly and eccentricity.[1][23] The mean anomaly, M, provides a linear measure of time progression, assuming uniform motion in a circular orbit with the same period as the actual ellipse. It is given by M = n(t - T), where n is the mean motion, t is the current time, and T is the time of periapsis passage (often coinciding with the epoch). The mean motion links to the semi-major axis a via n = \sqrt{\mu / a^3}, where \mu is the gravitational parameter, connecting temporal aspects to the orbit's size. This formulation simplifies time-of-flight computations by treating angular progress as proportional to elapsed time. As the sixth orbital element, the mean anomaly at epoch specifies the body's initial position.[23][24] For elliptical orbits, the eccentric anomaly E serves as an auxiliary angle measured at the ellipse's center from the periapsis to the projection of the body's position onto a circumscribing circle of radius a. It relates the mean anomaly to the true geometry through Kepler's equation: M = E - e \sin E, where e is the eccentricity; solving this transcendental equation yields E from M, facilitating position determination. The true anomaly is then derived from E.[23][24] Relations between these anomalies enable time-of-flight calculations, converting uniform time measures to actual positions. The eccentric anomaly bridges the mean and true anomalies, with the true anomaly derived from E using geometric projections that account for eccentricity. These transformations ensure precise orbital positioning without direct time dependencies in the anomaly definitions themselves.[23]Standard Orbital Element Sets
Classical Keplerian Elements
The classical Keplerian elements comprise six parameters that completely specify the trajectory of a body in a two-body orbit under a central inverse-square gravitational force. These elements are the semi-major axis a, eccentricity e, inclination i, right ascension of the ascending node \Omega (often denoted as longitude of the ascending node), argument of periapsis \omega, and true anomaly \nu (or mean anomaly M specified at an epoch for time-dependent propagation).[25] The semi-major axis a quantifies the orbit's size as half the length of the major axis of the ellipse. The eccentricity e characterizes the orbit's shape, with $0 \leq e < 1 yielding bound elliptical paths ( e = 0 for circles). The inclination i measures the tilt of the orbital plane relative to a reference plane, such as the ecliptic. The longitude of the ascending node \Omega defines the orbital plane's orientation by the angle from a reference direction to the point where the orbit crosses the reference plane ascendingly. The argument of periapsis \omega locates the closest approach (periapsis) within the orbital plane, measured from the ascending node. The true anomaly \nu gives the angular position of the orbiting body relative to the periapsis in the orbital plane; alternatively, the mean anomaly M at epoch serves as an initial condition for temporal evolution.[25] The position in the orbit is first obtained in polar form within the orbital plane, where the radial distance r from the primary body is r = \frac{a(1 - e^2)}{1 + e \cos \nu}. This yields the position vector in the orbital plane as \mathbf{r}' = r (\cos \nu, \sin \nu, 0). To obtain Cartesian coordinates in an inertial reference frame, this vector is transformed via a composite rotation matrix accounting for the orbit's orientation: \mathbf{R} = R_3(-\Omega) R_1(-i) R_3(-\omega), where R_3(\theta) rotates about the z-axis by angle \theta: R_3(\theta) = \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}, and R_1(\theta) rotates about the x-axis by \theta: R_1(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{pmatrix}. The final position is then \mathbf{r} = \mathbf{R} \mathbf{r}'.[25] These elements facilitate analytical solutions for unperturbed orbital motion by solving Kepler's equation, M = E - e \sin E (with E the eccentric anomaly), which relates time to position without requiring numerical integration.[26] They are particularly suited to ideal two-body scenarios involving point masses under central attraction.[25] Limitations arise from the underlying assumptions of point-mass primaries and a purely central force, rendering the elements inadequate for perturbed environments like those influenced by additional gravitational bodies or oblateness. Singularities also occur: \omega becomes undefined for circular orbits (e = 0), and \Omega for equatorial orbits (i = 0).[25] As an illustrative case, Earth's orbit around the Sun features a semi-major axis a \approx 1 AU and eccentricity e \approx 0.0167, producing a nearly circular trajectory.[27]Elements Tailored to Body Types
Orbital elements are modified to suit the dynamical environments of specific celestial or artificial bodies, incorporating reference frames and parameters that address non-Keplerian influences such as multi-body effects or observational constraints. These adaptations extend the classical Keplerian set by selecting appropriate central bodies and adding descriptors for unbound or perturbed trajectories, ensuring accurate representation of orbits under real conditions.[28] For planetary orbits, elements are often expressed in a barycentric reference frame, with the origin at the solar system's center of mass, to account for the collective gravitational influence of all major bodies. This approach is particularly useful for outer planets, where the Sun's motion around the barycenter introduces noticeable offsets; for instance, JPL's Development Ephemerides use barycentric positions for the nine planets and the Sun relative to this center. In contrast, heliocentric elements, centered on the Sun, are standard for inner planets like Mercury and Venus, where the barycenter's displacement is minimal and solar dominance prevails.[29][30][28] Asteroids and comets on unbound paths require hyperbolic elements, where the eccentricity e > 1 indicates trajectories that do not close, originating from interstellar space or distant perturbations. These elements include a negative semimajor axis a (conventionally defined for hyperbolas) alongside inclination, longitude of the ascending node, and argument of perihelion, derived from position and velocity vectors. To characterize the approach, incoming asymptote angles—such as θ (inclination relative to the ecliptic) and φ (azimuthal orientation)—define the direction of the velocity at infinity, facilitating orbit determination from sparse observations like those spaced 7 days apart using Gauss's method.[31][32] Artificial satellites orbiting Earth employ geocentric elements, referenced to the planet's center, with adjustments for oblateness via the J₂ perturbation term, which represents the equatorial bulge's gravitational asymmetry (J₂ ≈ 1.083 × 10⁻³). This non-spherical potential causes secular precession in the right ascension of the ascending node (retrograde at rate proportional to cos I) and argument of perigee (prograde or retrograde depending on inclination I, vanishing at I ≈ 63.4°), while leaving semimajor axis, eccentricity, and inclination unaffected to first order. These modifications enable short-term predictions by incorporating J₂ into the mean motion and element evolution equations.[33][34] Exoplanets detected via radial velocity or transit methods yield specialized elements that emphasize measurable projections rather than full 3D geometry. Radial velocity provides the orbital period, eccentricity (often near zero for close-in planets), and minimum mass (m sin i from velocity semi-amplitude K), while transits deliver the impact parameter b = \frac{a \cos i}{R_\star}, where a is the semimajor axis, i the inclination (near 90° for edge-on views), and R_\star the stellar radius, quantifying the transit chord's centrality. Additionally, the projected obliquity λ, measured through the Rossiter-McLaughlin effect during transits, assesses spin-orbit alignment, with values near 0° indicating co-alignment in many hot Jupiter systems; true obliquity ψ incorporates stellar inclination for a 3D view. These parameters address detection biases, such as favoring low-inclination orbits, and are refined using hierarchical Bayesian models across ensembles of over 200 systems.[35][36][37]Two-Line Element Sets
Two-Line Element Sets (TLEs) represent a standardized, concise data format for disseminating orbital elements of Earth-orbiting satellites, enabling short-term position predictions through simplified propagation models. Developed by the North American Aerospace Defense Command (NORAD), TLEs encode key orbital parameters in a fixed-width text structure, facilitating widespread use in satellite tracking applications.[38][39] A TLE typically comprises three lines: an optional Line 0 for the satellite name, followed by mandatory Lines 1 and 2 containing numerical data. Line 0 consists of a 24-character name or identifier for the satellite, such as "NOAA 19", providing human-readable context without affecting computation.[38] Line 1 begins with the line number "1" in column 1, followed by the five-digit satellite catalog number (e.g., 25544) in columns 3–7, a classification indicator (e.g., "U" for unclassified) in column 8, and the international designator (e.g., "08064A") in columns 10–17, which includes launch year, launch sequence, and piece identifier. The epoch is specified in columns 19–32 as the year (last two digits) and day of year with fractional part (e.g., "25286.12345678"), marking the reference time for the elements. Subsequent fields include the first time derivative of mean motion (columns 34–43), second time derivative (columns 45–52), BSTAR drag term (columns 54–61, a scaled atmospheric drag coefficient), ephemeris type (column 63, typically 0), element set number (columns 65–68, incrementing with updates), and a checksum digit (column 69). Line 2 starts with "2" in column 1 and repeats the satellite number in columns 3–7, then lists the inclination (degrees, columns 9–16), right ascension of the ascending node (Ω, degrees, columns 18–25), eccentricity (as a seven-digit number implying a leading decimal point, e.g., 0001234 for 0.0001234, columns 27–33), argument of perigee (ω, degrees, columns 35–42), mean anomaly (M, degrees, columns 44–51), mean motion (n, revolutions per day, columns 53–63), revolution number at epoch (columns 64–68), and a final checksum (column 69). These parameters derive briefly from classical Keplerian elements but are adjusted for propagation needs.[38]| Line | Columns | Field | Description | Example |
|---|---|---|---|---|
| 1 | 01 | Line Number | Fixed as "1" | 1 |
| 1 | 03-07 | Satellite Number | NORAD catalog ID | 25544 |
| 1 | 08 | Classification | Security level (U=Unclassified) | U |
| 1 | 10-17 | International Designator | Launch details (YYNNNPP) | 08064A |
| 1 | 19-32 | Epoch | Year (YY) + day.fraction | 25286.12345678 |
| 1 | 34-43 | 1st Mean Motion Derivative | d n/dt (revs/day², decimal implied) | 0.00001234 |
| 1 | 45-52 | 2nd Mean Motion Derivative | d² n/dt² (revs/day³, decimal implied) | 00000-0 |
| 1 | 54-61 | BSTAR Drag Term | Scaled drag coefficient (decimal implied) | 12345-5 |
| 1 | 63 | Ephemeris Type | Model indicator (0=SGP) | 0 |
| 1 | 65-68 | Element Number | Update sequence | 999 |
| 1 | 69 | Checksum | Modulo 10 sum of digits | 7 |
| 2 | 01 | Line Number | Fixed as "2" | 2 |
| 2 | 03-07 | Satellite Number | NORAD catalog ID | 25544 |
| 2 | 09-16 | Inclination (i) | Degrees | 98.1234 |
| 2 | 18-25 | RAAN (Ω) | Degrees | 45.6789 |
| 2 | 27-33 | Eccentricity (e) | Dimensionless (decimal implied) | 0001234 |
| 2 | 35-42 | Argument of Perigee (ω) | Degrees | 90.5678 |
| 2 | 44-51 | Mean Anomaly (M) | Degrees | 180.9012 |
| 2 | 53-63 | Mean Motion (n) | Revolutions per day | 15.12345678 |
| 2 | 64-68 | Revolution Number | Orbits at epoch | 12345 |
| 2 | 69 | Checksum | Modulo 10 sum of digits | 8 |