# Questions about the Dynamic Solid Tide

I found the following plot in the book Satellite Orbits; Models, Methods, Applications by Oliver Montenbruck and Eberhard Gill, Springer, 2000. The figure and description can also be found in google books. It's a low quality snapshot but it's hard to capture a dozen different dependencies over 20 orders of magnitude without showing the whole thing.

This answer points out that the orbit-perturbing effects of the Moon on a satellite in LEO are stronger than the orbit-perturbing effects of the changing Earth gravity field due to the Moon (and Sun, which I'd forgotten about).

After seeing the plot below, I realized that I'd also forgotten about the solid tide. And then I also noticed that there is no line for the ocean tide. I think these are different - incompressible water flows in order to make a tidal bulge, but can we think of the solid tide similarly - as flowing magma? Does the crustal shell outside just flex and bend to accommodate it? Does the crust significantly restrict it?

Any idea why ocean tide is not shown, is it's effect on Earth's gravity field so much lower than the solid tide that it wouldn't show up on this twenty orders of magnitude plot?

I also notices that the plot for the Dynamic Solid Tide parallels the two $$J_2$$ lines about four orders of magnitude lower than $$J_{2,0}$$ (looks like a $$1/r^4$$ slope). Does that reflect the relative amplitudes of the Earth's dynamic and static bulges?

• Could not figure out how to include "magma flowing underground" within the question; youtu.be/I1wg1DNHbNU?t=49 (Tom Hanks version; youtu.be/iPjCUlh-JMo?t=6) – uhoh Jun 14 '17 at 6:02
• Great question! Have you looked into research by Steven Nerem or Matt Talpe? Nerem leads the geodesic lab of CCAR and Talpe researched specifically the effects of water displacement on gravity models using data from JASON-2 and other spacecraft. – ChrisR Jun 14 '17 at 7:48

What Montenbruck and Gill call the "dynamic solid tide" is exactly what I was referring to in my answer to your related question.

Can we think of the solid tide similarly - as flowing magma?

Absolutely not. There are isolated pockets of partial melt near the crust/mantle boundary, and a layer of weak material just below (the asthenosphere). However, it's best to look at the Earth's crust and mantle as solid rock rather than flowing magma. On the timescale of half of a day (the timescale of the key tidal components, even the asthenosphere is rock solid. There is no tidal flow in the solid body tides. Just because it's "solid" doesn't mean that it doesn't deform under pressure. It does. As an extreme, look to the Earth's solid inner core. It's most iron and nickel, compressed by 40%.

Any idea why ocean tide is not shown, is it's effect on Earth's gravity field so much lower than the solid tide that it wouldn't show up on this twenty orders of magnitude plot?

Modeling the ocean tides is ridiculously hard. If you're working on model the orbits of the LAGEOS satellites, you'll need to worry about ocean tides. Otherwise, you can pretty much forget about them. They're a couple of orders of magnitude smaller than are the time-varying effects of the solid body tides.

I also notices that the plot for the Dynamic Solid Tide parallels the two $J_2$ lines about four orders of magnitude lower than $J_{2,0}$ (looks like a $1/r^4$ slope). Does that reflect the relative amplitudes of the Earth's dynamic and static bulges?

The primary influence of the solid body tides is to make the supposedly constant $J_2$ terms vary over time. So the effect is roughly $1/r^4$, the same as for the $J_2$ terms. The solid tides also affect the $J_3$ and $J_4$ harmonics, but since those fall as $1/r^5$ and $1/r^6$, it's the changes to the $J_2$ terms that dominates. And yes, that four-plus order of magnitude reduction compared to $J_{2,0}$ is what I meant in my previous answer when I wrote that these effects are small.

• What's actually happening within the Earth has always been a bit fuzzy for me, thanks for nailing things down. – uhoh Jun 15 '17 at 3:04
• btw I just ran into some more quaternions :) – uhoh Jun 15 '17 at 3:06

Tides are the effect, a deformation, caused by a force, and any real solid is more or less deformed by a force. Tides may be more noticeable in fluids, but they are measurable in solid bodies too...

• Thanks, but this is not really up to the level of a good stackexchange answer. It's better left as a comment, or expanded into a proper answer. – uhoh Jun 14 '17 at 7:58