How do we measure height on celestial bodies with no significant atmospheric pressure, no seas, or no surface to speak of?

Question

On the Earth, measuring elevation above sea level is relatively easy. Throughout the years of evolving these measurements to not rely on different reference points of the Mean Sea Level (MSL) of various seas and oceans, which would be the average height of some ocean or the halfway point between its mean high tide and its mean low tide, and since moving to Standard Sea Level (SSL) on an international scale, we have agreed on a standard to measure elevation of geographic features and other objects that is acceptable to all and not too difficult to measure using all kinds of different equipment, from barometers to triangulating position by the use of satellites that all work with the same reference point (SSL). Having agreed on that, we can then also agree on how high Mount Everest is, for example.

But how do we measure elevation on other celestial bodies, that might not have oceans and seas, or even any atmosphere to speak of? Measuring elevation relative to some surface is easy enough, we can use radars, lidars,... anything that will bounce off, so to speak, and then measure the time it took for it to do the round trip. But where is the sea level on such bodies? Is there any international standard, an agreement that defines methods for zero sea level on other celestial bodies, or does simply each nation, agency, even observatory define their own reference points arbitrarily?

For example, here is an example of an elevation (topographical) map from the Mars Global Surveyor's Mars Orbital Laser Altimeter (MOLA):

   

Or the topographical map of Venus:

   

These elevation maps both use surface elevation relative to some reference zero elevation. How are these defined on celestial bodies with no atmosphere (e.g. the Moon), bodies with no large bodies of water or oceans of other liquids (e.g. Mars), or both?

Additionally, how do we define depth for gas giants like Jupiter and Saturn? Or even the Sun? We would also have to define some reference point first, no? Who decides where that reference point is and what methods are employed to determine it?

Answer 1

Introduction to selecting a reference surface

The surface of any celestial body can be anything but uniform. The oceans, where existing, can be treated as reasonably uniform, but the surface or topography of the land masses can exhibit large vertical variations between mountains and valleys. These variations make it impossible to approximate the shape of the celestial body with any reasonably simple mathematical model.

In case of the Earth, two main reference surfaces have been established to approximate the shape of it. One reference surface is called the geoid, the other reference surface is the ellipsoid. These are illustrated in the figure below:

    

    ^{The body's surface, and two reference surfaces used to approximate it: the geoid, and a reference ellipsoid.}

Leaving further clarification to a few short excerpts from Wikipedia pages on geoid and reference ellipsoid, we get these descriptions:

• The geoid is the shape that the surface of the oceans would take under the influence of Earth's gravity and rotation alone, in the absence of other influences such as winds and tides. All points on that surface have the same scalar potential—there is no difference in potential energy between any two.
• In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.

So we have defined two possible candidates for how to establish solid body reference surface, or as it's sometimes called, particularly when in reference to the mean sea levels, the vertical datum. For establishing the reference surface by using the ellipsoid method, this can be further differentiated to those measured over a local, or global area of the celestial body's surface:

      

      ^{The Geoid, a globally best fitting ellipsoid, and a regionally or locally best fitting ellipsoid, for a chosen region.}

---

OK, so we have established possible methods, possible areas over which they apply, and we've also learned a bit on the terminology used. But how does this translate to actual uses on celestial bodies of our Solar system? This is what I was able to find out;

Zero elevation on Mars

Since Mars has no oceans and hence no 'sea level', it is convenient to define an arbitrary zero-elevation level or "datum" for mapping the surface. The datum for Mars is arbitrarily defined in terms of a constant atmospheric pressure. During the Mariner 9 mission, this was chosen as 610.5 Pa (6.105 mbar), on the basis that below this pressure liquid water can never be stable (i.e., the triple point of water is at this pressure). This value is only 0.6% of the pressure at sea level on Earth, which forms the zero elevation datum for our planet. Note that the choice of this value does not mean that liquid water does exist below this elevation, just that it could were the temperature to exceed 273.16 K.

Quoted excerpt source and Mars elevation map above: Wikipedia on Geography of Mars

Wikipedia is however a bit outdated and more recently, the data from the Mars Orbiting Laser Altimeter (MOLA) of the NASA's Mars Global Surveyor and a spherical-harmonic representation of an equipotential surface is recommended to be used as the reference for elevations by the Mars geodesy / cartography working group. MOLA transmits infrared laser pulses towards the Martian surface at a rate of 10 times per second, to produce a highly detailed map of the surface. This is an excerpt from Mars geodesy / cartography working group recommendations on Mars cartographic constants and coordinate systems (PDF):

MOLA topographic model will be used for accurate projection of images and other remote-sensing data onto the planet, and a spherical-harmonic representation of an equipotential surface will be used as the reference for elevations.

      

      ^{Mars Orbiting Laser Altimeter (MOLA) Tracks}

Zero elevation on the Moon

According to dr. Simon O'Toole from the Australian Astronomical Observatory, a similar instrument to the one employed to map elevation of Mars was used by NASA's Lunar Reconnaissance Orbiter spacecraft to provide a precise topographic model of the Moon.

Because it has no ocean or significant atmosphere, the zero elevation point of the Moon is the average diameter.

   

   ^{Colored global elevation map based on terrain data from the SELENE / Kaguya orbiter, Japan Aerospace Exploration Agency (Source: USGS)}

According to The Unified Lunar Control Network 2005 article (PDF) published by USGS (U.S. Department of the Interior U.S. Geological Survey):

This report documents a new general unified lunar control network and lunar topographic model based on a combination of Clementine images and a previous network derived from Earth based and Apollo photographs, and Mariner 10 and Galileo images. This photogrammetric network solution is the largest planetary control network ever completed. It includes the determination of the 3-D positions of 272,931 points on the lunar surface and the correction of the camera angles for 43,866 Clementine images, using 546,126 tie point measurements.

Reference surface area of gas giants

With a fair bit of digging around, I have finally stumbled upon one explanation to where the reference surface of gas giants is measured from, according to which we can actually measure depth and heights of clouds and other weather phenomena forming on them, how deep our atmospheric probes have penetrated these planets before they finally gave in and imploded, and similar. In a IAU (International Astronomical Union) / IAG (International Association of Geodesy) working group report on cartographic coordinates and rotational elements from 2006 (PDF), this quote can be found:

The radii and axes of the large gaseous planets, Jupiter, Saturn, Uranus, and Neptune in Table 4 refer to a one-bar-pressure surface. The radii given in the tables are not necessarily the appropriate values to be used in dynamical studies; the radius actually used to derive a value of J₂ (for example) should always be used in conjunction with it.

The table it refers to doesn't really matter for us, but the definition of the reference surface area does. This means that, by international standards these two bodies (IAU/IAG) represent, the reference surface area is defined by the point at one-bar-pressure of their atmosphere. This same paper gives us another clue:

Reference surface area of minor planets and comets

For irregularly shaped bodies the ellipsoid is obviously useless, except perhaps for dynamical studies. For very irregular bodies, the concept of a reference ellipsoid ceases to be useful for most purposes. For these bodies, topographic shapes are usually represented by a grid of radii to the surface as a function of planetocentric latitude and longitude (when possible, or also by a set of vertices and polygons).

Another problem with small bodies is that two coordinates (i.e. spherical angular measures) may not uniquely identify a point on the surface of the body. In other words it is possible to have a line from the center of the object intersect the surface more than once.

It goes on to describe examples and cartographers' ad hoc tricks for specific maps, and later gives a recommendation that longitudes on minor planets and comets should be measured positively from 0 to 360 degrees using a righthand system from a designated prime meridian, where the origin is the center of mass, to the extent known.

---

As for the international authoritative bodies defining standards of measure for the reference surface areas of celestial bodies, it appears IAU (International Astronomical Union) and IAG (International Association of Geodesy) are as good as it gets, and who's to argue that?

Answer 2

For Mars, the current definition of 0 km is derived from data from the Mars Orbiting Laser Altimeter (MOLA) data from Mars Global Surveyor. In fact the altitude reference is referred to as "MOLA altitude". You would say for example: "minus 1.4 km MOLA".

From the paper:

Zero elevation on Mars from MOLA is defined as the equipotential surface (gravitational plus rotational) whose average value at the equator is equal to the mean radius as determined by MOLA (cf. Table4).

Also recognizing the established practice and officially setting the standard in this IAU paper:

The topographic reference surface of Mars is that specified in the final MOLA Mission Experiment Gridded Data Record (MEGDR) Products (Smith et al. 2003). In particular, the 128 pixels/◦ resolution, radius and topographic surfaces are recommended, although the lower resolution versions may be used where appropriate and documented, and for the areas poleward of ±88◦ latitude.

There was an older definition used for Mariner 9 and Viking data using the triple point of water as the definition for 0 km. There was a short period of annoyance in converting from the old to the new system, since they were about 1.6 km apart from each other.

Note that even on Earth, we don't really use sea level anymore. We use a reference datum that does not change with time, which was based on some average sea level measurement. So if sea level rises, which in fact it's doing, the height of Mount Everest will not change. (Except of course that Everest is still rising very slowly due to tectonic forces.)

For Venus, the planet is sufficiently spherical and has such a low rotation rate, and since the topographic data is of relatively coarse resolution, a sphere whose radius is the mean radius of the planet, 6051.881 km, is used. See this paper.

At Jupiter and the other gas giants, 0 km is defined as the radius of 1 bar pressure.

The "surface" of the Sun is defined to be the uppermost visible layer, the top of the photosphere. Layers above that, referred to as the Sun's "atmosphere", are too thin or cool to be seen by the naked eye. (Which shouldn't be looking at the Sun by the way.)

Answer 3

The reference datum is usually chosen to be about the average altitude. For example, on Mars, the datum, known as the Mars areoid, is very close to the average radius of Mars, as measured round the equator.

(It was defined by the height at which the pressure corresponds to the triple point of water.)

In reality it wouldn't matter very much what the reference was, as long as all interested parties agree on it.

(Still trying to find out who made the decisions- for Mars it appears to have been defined by NASA, but I can't find definite evidence)