3 ms·

Measuring the Shape of the Earth

18 points by Hooke 3y ago

Pinus 3y ago
"Map showing the triangulation across the Andes, 1744. Bibliothèque nationale de France." Fancy, I didn’t know that Sangis, Torneå and Kemi were in the Andes! :-)
AlbertCory 3y ago
One of my favorite questions is: "What point on Earth is the farthest from the Earth's center?" If it's being asked at all, you know it's probably not Everest, and indeed it's not: https://en.wikipedia.org/wiki/Summits_farthest_from_the_Earth's_center https://en.wikipedia.org/wiki/Summits_farthest_from_the_Eart...
0_____0 3y ago
And on top of all this, there are tidal forces to consider. I asked an astrophysicist relative how much a point on the Earth at its equator might rise and fall with every rotation, and he reckoned a few meters at least IIRC.
- raattgift 3y ago
  The article is mainly about an interesting chapter in the history of the metre <https://en.wikipedia.org/wiki/History_of_the_metre#Meridional_definition https://en.wikipedia.org/wiki/History_of_the_metre#Meridiona...> which effectively won versus the second/metre pendulum approach (which was fiddly to calibrate and keep true for any given environment at a given point on Earth). The author is a well-known historian of science in the English enlightenment, and notably Newton, https://en.wikipedia.org/wiki/Patricia_Fara https://en.wikipedia.org/wiki/Patricia_Fara . I would not expect the article to even touch on 21st century geodesy. A good theory for the solid earth tide only arrived in first dozen years of the 20th century and was only practically measured with the arrival decades later of very long baseline interferometry (up to extragalactic radio sources), satellite (and lunar) laser ranging, geodetic GNSS, and other tools in modern geodesy. The solid earth tide has a maximum of a few tens of centimetres, not metres. So even in the 21st century, solid Earth tides, being small and difficult to measure without specialist equipment, are for most practical purposes completely negligible. Your astrophysicist relative may be interested in some details. At <https://geodesyworld.github.io/SOFTS/solid.htm https://geodesyworld.github.io/SOFTS/solid.htm> there is a calculator (and notes) giving more information than a lay person is ever likely to want to know ("Fine print: This section is not for the faint of heart."); there is a reimplementation and modification of solid in R at <https://rdrr.io/cran/solidearthtide/ https://rdrr.io/cran/solidearthtide/>; there is the ESA page with useful simplified formulae and graphs at <https://gssc.esa.int/navipedia/index.php/Solid_Tides https://gssc.esa.int/navipedia/index.php/Solid_Tides> since solid earth tides can be relevant for astronomy; or alternatively the wikipedia page for https://en.wikipedia.org/wiki/Earth_tide https://en.wikipedia.org/wiki/Earth_tide is decent and more accessible, and has a link to tidal Love numbers at the bottom. Your relative can look up the fluid and tidal Love numbers for main sequence stars (and how dynamics and dissipation works with fast rotation), and there is literature for relativistic Love numbers for white dwarfs and neutron stars and so forth (although nonspinning black holes have none: Binnington & Poisson "Relativistic theory of tidal Love numbers", <https://arxiv.org/abs/0906.1366 https://arxiv.org/abs/0906.1366>), and should have little trouble following methods for arriving at tidal Love numbers given different equations of state for visecoelastically differentiated bodies. Earth has https://en.wikipedia.org/wiki/Planetary_differentiation https://en.wikipedia.org/wiki/Planetary_differentiation and when you dig into solid Earth tides in detail you get to confront all sorts of unknowns. In practice, it is easier to start with the precise measurement of rotation of a real body e.g. Earth, Saturn, Io, super-Earth exoplanets (a nice example is at <https://doi.org/10.1051/0004-6361/201731775 https://doi.org/10.1051/0004-6361/201731775>) and work backwards to the interior structure. This is perhaps less true for the Earth since we can use time-of-travel seismology and gravimetry and have ready access to rock samples. ETA: (in part so that I don't lose the reference), this astronomy stackexchange summary of the tidal Love numbers is extremely well written: https://astronomy.stackexchange.com/posts/46889/revisions https://astronomy.stackexchange.com/posts/46889/revisions (and I'm sure your astrophysicist relative would appreciate it too).
  - 0_____0 3y ago
    Oh wow thank you for this!!
    - raattgift 3y ago
      You're welcome. Tides are fun, and that's mostly why I jumped in. Tides are also hard and have occupied a lot of thinking time for gravitational scientists, and naval people with respect to Earth's ocean: https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2 https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2 (be sure to see the photos). Your comment about the solid earth tide was pretty neat because few people who aren't professionally into geodesy or the study of Earth's gravitation think about that. As I mentioned, there are tides in all sorts of bodies elsewhere in our solar system (and beyond). The ones people think about are atmospheric tides and ocean tides (within e.g. Europa), or how it couples to volcanism (e.g. Io's magma and Enceladus's ice). However, scattered about our solar system there are "asteroids" in pairs, where one is what amounts to a blob of kitty litter to large gravel, and the other is a more solid body. The rubble-pile is especially interesting in a tidal context. I just submitted https://news.ycombinator.com/item?id=39561656 https://news.ycombinator.com/item?id=39561656 which the Royal Astronomical Society announced yesterday in various places including <https://twitter.com/RAS_Journals/status/1763219322732786054 https://twitter.com/RAS_Journals/status/1763219322732786054> (the image is worth a click, but also appears as Fig. 1 in the full article at the HN submission; if you hate X and/or extra clicks, this is a direct link: <https://academic.oup.com/view-large/figure/441211342/stae325fig1.jpg https://academic.oup.com/view-large/figure/441211342/stae325...>). When general relativity students learn about decompositions of the Riemann curvature tensor they are sometimes taken to the idea of a sort of cloud of coffee-grounds (with the special provision that they pass through each other, always have gaps between them, or are otherwise non-collisional and therefore non-clumping; and they have no internal pressure) to see a variety of ways in which one can find "tidal forces" encoded in e.g. the Weyl curvature tensor (which is extracted from the Riemann curvature tensor). Baez is an example of this <https://math.ucr.edu/home/baez/gr/outline2.html https://math.ucr.edu/home/baez/gr/outline2.html> and <https://math.ucr.edu/home/baez/gr/ricci.weyl.html https://math.ucr.edu/home/baez/gr/ricci.weyl.html>, although textbooks generally stick to "test particles". Rubble pile asteroids, while essentially not needing any post-Newtonian gravitational corrections at all, are a neat step up in difficulty because they do clump. But unlike larger bodies, they aren't fused by e.g. melting; the rubble can shift around unlike a fused rock or icy concretion; and internal pressures and shear stresses are low. So the rubble (how it shifts around, what shape it tends to adopt) will be affected by tides induced by the sun and possibly other solar system bodies closer to them. Personally I like the idea of depositing a crash test dummy loaded with sensors onto the surface of (probable) rubble-pile 101955 Bennu. Is it like a ball pit, or like a bean-bag chair? Does the dummy sink or get buried in landslides or fountains of material? (We already know we can send things there (and return them): <https://www.nasa.gov/news-release/nasas-bennu-asteroid-sam https://www.nasa.gov/news-release/nasas-bennu-asteroid-sam>).
deleted 3y ago
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18 points by Hooke 3y ago

"Map showing the triangulation across the Andes, 1744. Bibliothèque nationale de France." Fancy, I didn’t know that Sangis, Torneå and Kemi were in the Andes! :-)

AlbertCory 3y ago

One of my favorite questions is: "What point on Earth is the farthest from the Earth's center?" If it's being asked at all, you know it's probably not Everest, and indeed it's not: https://en.wikipedia.org/wiki/Summits_farthest_from_the_Earth's_center https://en.wikipedia.org/wiki/Summits_farthest_from_the_Eart...

0_____0 3y ago

And on top of all this, there are tidal forces to consider. I asked an astrophysicist relative how much a point on the Earth at its equator might rise and fall with every rotation, and he reckoned a few meters at least IIRC.

raattgift 3y ago
The article is mainly about an interesting chapter in the history of the metre <https://en.wikipedia.org/wiki/History_of_the_metre#Meridional_definition https://en.wikipedia.org/wiki/History_of_the_metre#Meridiona...> which effectively won versus the second/metre pendulum approach (which was fiddly to calibrate and keep true for any given environment at a given point on Earth). The author is a well-known historian of science in the English enlightenment, and notably Newton, https://en.wikipedia.org/wiki/Patricia_Fara https://en.wikipedia.org/wiki/Patricia_Fara . I would not expect the article to even touch on 21st century geodesy. A good theory for the solid earth tide only arrived in first dozen years of the 20th century and was only practically measured with the arrival decades later of very long baseline interferometry (up to extragalactic radio sources), satellite (and lunar) laser ranging, geodetic GNSS, and other tools in modern geodesy. The solid earth tide has a maximum of a few tens of centimetres, not metres. So even in the 21st century, solid Earth tides, being small and difficult to measure without specialist equipment, are for most practical purposes completely negligible. Your astrophysicist relative may be interested in some details. At <https://geodesyworld.github.io/SOFTS/solid.htm https://geodesyworld.github.io/SOFTS/solid.htm> there is a calculator (and notes) giving more information than a lay person is ever likely to want to know ("Fine print: This section is not for the faint of heart."); there is a reimplementation and modification of solid in R at <https://rdrr.io/cran/solidearthtide/ https://rdrr.io/cran/solidearthtide/>; there is the ESA page with useful simplified formulae and graphs at <https://gssc.esa.int/navipedia/index.php/Solid_Tides https://gssc.esa.int/navipedia/index.php/Solid_Tides> since solid earth tides can be relevant for astronomy; or alternatively the wikipedia page for https://en.wikipedia.org/wiki/Earth_tide https://en.wikipedia.org/wiki/Earth_tide is decent and more accessible, and has a link to tidal Love numbers at the bottom. Your relative can look up the fluid and tidal Love numbers for main sequence stars (and how dynamics and dissipation works with fast rotation), and there is literature for relativistic Love numbers for white dwarfs and neutron stars and so forth (although nonspinning black holes have none: Binnington & Poisson "Relativistic theory of tidal Love numbers", <https://arxiv.org/abs/0906.1366 https://arxiv.org/abs/0906.1366>), and should have little trouble following methods for arriving at tidal Love numbers given different equations of state for visecoelastically differentiated bodies. Earth has https://en.wikipedia.org/wiki/Planetary_differentiation https://en.wikipedia.org/wiki/Planetary_differentiation and when you dig into solid Earth tides in detail you get to confront all sorts of unknowns. In practice, it is easier to start with the precise measurement of rotation of a real body e.g. Earth, Saturn, Io, super-Earth exoplanets (a nice example is at <https://doi.org/10.1051/0004-6361/201731775 https://doi.org/10.1051/0004-6361/201731775>) and work backwards to the interior structure. This is perhaps less true for the Earth since we can use time-of-travel seismology and gravimetry and have ready access to rock samples. ETA: (in part so that I don't lose the reference), this astronomy stackexchange summary of the tidal Love numbers is extremely well written: https://astronomy.stackexchange.com/posts/46889/revisions https://astronomy.stackexchange.com/posts/46889/revisions (and I'm sure your astrophysicist relative would appreciate it too).
- 0_____0 3y ago
  Oh wow thank you for this!!
  - raattgift 3y ago
    You're welcome. Tides are fun, and that's mostly why I jumped in. Tides are also hard and have occupied a lot of thinking time for gravitational scientists, and naval people with respect to Earth's ocean: https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2 https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2 (be sure to see the photos). Your comment about the solid earth tide was pretty neat because few people who aren't professionally into geodesy or the study of Earth's gravitation think about that. As I mentioned, there are tides in all sorts of bodies elsewhere in our solar system (and beyond). The ones people think about are atmospheric tides and ocean tides (within e.g. Europa), or how it couples to volcanism (e.g. Io's magma and Enceladus's ice). However, scattered about our solar system there are "asteroids" in pairs, where one is what amounts to a blob of kitty litter to large gravel, and the other is a more solid body. The rubble-pile is especially interesting in a tidal context. I just submitted https://news.ycombinator.com/item?id=39561656 https://news.ycombinator.com/item?id=39561656 which the Royal Astronomical Society announced yesterday in various places including <https://twitter.com/RAS_Journals/status/1763219322732786054 https://twitter.com/RAS_Journals/status/1763219322732786054> (the image is worth a click, but also appears as Fig. 1 in the full article at the HN submission; if you hate X and/or extra clicks, this is a direct link: <https://academic.oup.com/view-large/figure/441211342/stae325fig1.jpg https://academic.oup.com/view-large/figure/441211342/stae325...>). When general relativity students learn about decompositions of the Riemann curvature tensor they are sometimes taken to the idea of a sort of cloud of coffee-grounds (with the special provision that they pass through each other, always have gaps between them, or are otherwise non-collisional and therefore non-clumping; and they have no internal pressure) to see a variety of ways in which one can find "tidal forces" encoded in e.g. the Weyl curvature tensor (which is extracted from the Riemann curvature tensor). Baez is an example of this <https://math.ucr.edu/home/baez/gr/outline2.html https://math.ucr.edu/home/baez/gr/outline2.html> and <https://math.ucr.edu/home/baez/gr/ricci.weyl.html https://math.ucr.edu/home/baez/gr/ricci.weyl.html>, although textbooks generally stick to "test particles". Rubble pile asteroids, while essentially not needing any post-Newtonian gravitational corrections at all, are a neat step up in difficulty because they do clump. But unlike larger bodies, they aren't fused by e.g. melting; the rubble can shift around unlike a fused rock or icy concretion; and internal pressures and shear stresses are low. So the rubble (how it shifts around, what shape it tends to adopt) will be affected by tides induced by the sun and possibly other solar system bodies closer to them. Personally I like the idea of depositing a crash test dummy loaded with sensors onto the surface of (probable) rubble-pile 101955 Bennu. Is it like a ball pit, or like a bean-bag chair? Does the dummy sink or get buried in landslides or fountains of material? (We already know we can send things there (and return them): <https://www.nasa.gov/news-release/nasas-bennu-asteroid-sam https://www.nasa.gov/news-release/nasas-bennu-asteroid-sam>).

deleted 3y ago

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