Somigliana Normal Gravity Calculator

Description

Calculate Somigliana normal gravity from an explicit ellipsoid and compatible equatorial/polar constants.

Geodesy distinguishes a mathematical reference ellipsoid, a realized coordinate datum and the physical gravity field. Geodetic latitude follows the ellipsoid normal; geocentric latitude follows a radius from the center. Ellipsoidal height, orthometric height and geoid undulation are separate signed quantities. Keep the declared ellipsoid, coordinate axes, angular units and time scale with a result.

Calculate Somigliana normal gravity from an explicit ellipsoid and compatible equatorial/polar constants.

GRS80 provides a reference ellipsoid and a normal gravity model. Geometric flattening and model normal gravity are reference quantities; neither identifies a measured local gravity or a datum realization by itself.^[1]

When to use this tool

Use this tool when you need to calculate Somigliana normal gravity from an explicit ellipsoid and compatible equatorial/polar constants.

How to use the tool and how it works

Enter semiMajorMeters, flattening, latitudeDegrees, equatorialGravity, polarGravity using the definitions below.
Run the operation. Resolve input-format errors before interpreting an output.
Read result.gravityMetersPerSecondSquared, using the stated boundary, ordering, sign and absence conventions.

Let r=b/a=1−f and e²=f(2−f). Calculate [gammaEquator cos²(phi)+r gammaPole sin²(phi)]/sqrt(1−e²sin²(phi)), returning m/s².^[2]

Input field	Meaning and units	Accepted input
semiMajorMeters	Equatorial semiaxis a in meters, 1–10⁸. It is a radius, not a diameter.	Finite numeric value satisfying the stated comparisons.
flattening	Dimensionless ordinary flattening f=(a-b)/a, from zero through 0.02; supply f rather than inverse flattening.	Finite numeric value satisfying the stated comparisons.
latitudeDegrees	Geodetic latitude in degrees, north positive; the operation’s pole restrictions apply.	Finite numeric value satisfying the stated comparisons.
equatorialGravity	Positive normal gravity at the ellipsoid equator in m/s², at most 100.	Finite numeric value satisfying the stated comparisons.
polarGravity	Positive normal gravity at the ellipsoid pole in m/s², at most 100.	Finite numeric value satisfying the stated comparisons.

Output field	Interpretation
result	Object containing the fields below.
result.gravityMetersPerSecondSquared	Modeled normal gravity in m/s², not an observed or height-corrected gravity.

json

{
  "input": {
    "semiMajorMeters": 10,
    "flattening": 0,
    "latitudeDegrees": 0,
    "equatorialGravity": 9.8,
    "polarGravity": 9.9
  },
  "output": {
    "result": {
      "gravityMetersPerSecondSquared": 9.8
    }
  }
}

Limitations and assumptions

Both caller-supplied normal gravities are in (0,100] m/s². This is normal gravity on the reference ellipsoid, not observed gravity at arbitrary elevation or a terrain/anomaly correction.
All quantities are finite and retain the units stated for this operation. Ellipsoid-parameter tools use semi-major axis 1–10⁸ m and dimensionless flattening 0–0.02; explicit-axis tools use 0.9a≤b≤a unless a tighter bound is stated. These tools do not fetch datums, geoid grids, satellite observations or Earth-orientation updates.
Every declared outer field is required and unknown outer fields are rejected. Strings and JSON text have size limits, lists are bounded to 128 entries, and JSON nesting/node counts are bounded. Tighter operation-specific bounds stated above take precedence.
Numeric results must be finite. Large physical magnitudes are allowed by the output evidence checks, while integer fields and source schemas retain their exactness rules. Source indexes are zero-based. Floating-point residues can remain even where an ideal formula gives an exact zero.

Alternative or Complementary methods

Use a coordinate transform for a change of representation and explicit datum parameters for a change of reference frame. Preserve model and convergence metadata, and use observation and datum information appropriate to a real survey before interpreting a computed tolerance as accuracy.

References

Geodetic Reference System 1980 (corrected reference) — H. Moritz; International Association of Geodesy, hosted by MIT
Tool behavior and example: Somigliana Normal Gravity Theoretical Formula Calculator — tool.io

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