In my last couple of posts I've been exploring the physics thought experiment called The Gravity Train. For simplicity I'm only interested in a path that goes directly through the centre of the Earth. Initially I used a simple constant density model of Earth to arrive at an analytic solution, although easy to solve, it's essentially wrong. To see what the results would be using some more realistic models I've done some extra simulations. Ideally I should have fired up Octave or another similar numerical package to model the scenarios, but for a quick ballpark answer I decided to just use a spreadsheet. It's less accurate but it's quick and easy.
1. Constant Density
Mass 5.9736 E+10^24 kg
Radius 6371 km
2. Linear Density
The density of the planet varies linearly in a radial direction.
Mass 5.9736 E+10^24 kg
Radius 6371 km
Inner Density 13000 kg/m^3
3. Preliminary Reference Earth Model
Uses the Preliminary Reference Earth Model (PREM) (Dziewonski & Anderson, 1981)
to model the Earth as a sphere of radius 6371 km.
|Earth's density at different depths|
The graph above shows the density of the Earth as you travel from one side to the other, with the middle of the graph being the centre of the Earth. The uniform density model is just a straight line representing a density of about 5500 kg/m^3. The linear density model starts at about 3000 kg/m^3 at the surface and rises to around 13000 kg/m^3 at the Earth's core. The Preliminary Reference Earth Model (PREM), the most realistic model I could find, shows distinct areas of different density that correspond to layers such as the core, inner core, and mantle.
speadsheet - EarthGravityModels.zip
The spreadsheet models the situation as a finite difference time domain problem, where an object is accelerated by the Earth with different accelerations at different depths. I won't go though the equations used in the spreadsheet but they're based on the formula in my last post, Mass Of a Spherical Shell With a Linearly Varying Density. The time taken for the object to reach the other side of the planet is noted and in theory the object should reach 0 velocity just at it reaches the other side. However, due to the inherent errors of numerical modelling, the velocity wasn't zero, but it was within a reasonable margin of error. As a point of comparison, using the data provided for the constant density model, the equation derived in my first post on this topic yields a travel time of 42 minutes and 10 seconds.
|Displacement vs Time for Different Models|
The times to travel from one side to the other side of Earth using different models are as follows.
Uniform 41 minutes 48.0 seconds
Linear 38 minutes 13.2 seconds
PREM 37 minutes 43.6 seconds
These results show a time decrease for the more realistic models which is what you'd expect. With more of the mass concentrated at the centre of the Earth, it can accelerate an object for a longer period of time. I'm fairly confident with those numbers. The time difference between the uniform numerical and analytic models is only 22 seconds, which is a reasonable amount of error for the simulation method.
An extra piece of information that can be extracted from the data is the gravitational acceleration of the Earth vs depth.
|Gravitational Acceleration Inside Earth|
Unlike the uniform density model I used in my first post, the acceleration inside Earth actually goes up before falling to zero in the other models. For the realistic PREM model there is a point approximately halfway to the core of the Earth where gravitational acceleration peaks at around 10.6 m/s^2 that about 1.1g's. Didn't know that.