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If the Earth's radius were increased from 6371.0 km to 9010.0 km while maintaining the same surface gravity, how much mass do I need to add?

Besides the main question, I also have a few other additional questions but relevant to the topic:

  1. Would the added mass have any effect if it's located in the crust?
  2. How realistic would it be to hollow out the lower crust and upper mantle with densely interconnected caves (like ridiculously gigantic ant nests)?
  3. With the increased diameter and maintaining the speed of rotation, would this affect the centripetal force and overall shape of the planet?
  4. Related to question #3, how would the tidal forces be affected?
  5. Since the Earth is now "bigger" in both size and mass, how much closer would the Moon end up being?
  6. Same as question #5, would the Earth end up becoming closer to the Sun? By how much?
  7. Would the atmosphere still remain the same with such changes?
  8. Would existing structures, flora and fauna remain stable with such changes?

Certain key points:

  • Radius increased from 6371.0 km to 9010.0 km.
  • Surface gravity should remain 9.80665 m/s^2.
  • Day time should remain the same.
  • No significant changes in weather.
  • Flora and fauna should be largely unaffected.
  • Structures, be it man-made or natural, should remain stable.

Notes:

This is largely fantasy, I also plan on adding exotic elements, magic and whatever comes to mind.

Clarifications:

The question is mostly about making the changes not affect the things already on Earth, the title itself says "making it practical", so just baselessly making the Earth's radius bigger wouldn't work out as too many other parameters would get affected.

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  • $\begingroup$ I don't know if that's possible, but if it doesn't change anything ... why? $\endgroup$
    – DonQuiKong
    Commented Feb 4, 2018 at 12:22
  • $\begingroup$ larger landmass and i could conveniently introduce things "below" the crust, to be precise, the landmass would be double at the very least, but i think the ocean level would drop with the increased surface area thus the total landmass would be much bigger than just double. $\endgroup$
    – Kita
    Commented Feb 4, 2018 at 12:43
  • $\begingroup$ You did not ask about oceans, but yes, you're right on that. Water, just like air, gets redistributed. $\endgroup$
    – LSerni
    Commented Feb 4, 2018 at 13:01
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    $\begingroup$ There are way too many surface gravity questions on the site already. Please take a look at any of them using the search feature. Here is an answer with equations that you an plug in to. This site is for providing information, but we aren't supposed to just help you do math... $\endgroup$
    – kingledion
    Commented Feb 4, 2018 at 13:37
  • $\begingroup$ thanks, that would be useful for future reference. also, the other questions are also relevant, so i still had to ask none the less. $\endgroup$
    – Kita
    Commented Feb 4, 2018 at 13:46

1 Answer 1

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The relationship between mass, gravity and radius is

$g = \frac{m}{r^2}$

so if you want to increase radius by 37%, while keeping g the same, you need to increase the mass by 88% (1.37*1.37 = 1.88).

But by increasing radius by 37%, you increase volume by 158%. This means that average density (mass over volume) has to go down by 37%.

In other words, the 0.88 Earth masses you add must have a volume of 1.58 Earths, which makes the newly added mass need to have a density of only 56% of Earth average density, which is 5.51 g/cm3, thus giving 3.08g/cm3.

Most silicates are in the 2-3 g/cm3 range, so you're golden.

You also need to add a significant quantity of air since the increased radius will increase the surface by that same 88%. Which means that the height of the atmosphere, once distributed over a surface 88% larger, will be proportionately lower (down to 53% of normal). Unless you add as much air as surface, which means adding 88% volume to the atmosphere.

Your key points are all satisfied provided you add the material in an orderly and careful way, and add the required air.

For the rest:

would the added mass have any effect if its located in the crust?

No, provided it's distributed uniformly.

how realistic would it be to hollow out the lower crust and upper mantle with densely interconnected caves (like ridiculously gigantic ant nests)?

Not very, I'm afraid. Some cave systems are feasible though. Not too deep, given the pressure they'd have to withstand.

with the increased diameter and maintaining the speed of rotation, would this affect the centripetal force and overall shape of the planet?

Yes, but not all that much. The centripetal acceleration is $a_c=\frac {v^2}r=\omega^2r$, so it would go up that same 37%. The equatorial bulge would bulge a little more, that's all.

related to question #3, how would the tidal forces be affected?

UPDATE I had started answering "not very". But that was assuming no changes to the Moon. The Moon will orbit nearer, so its attraction will be felt four times stronger. But tidal forces are forces of the third order, and tidal acceleration is approximately

$a_r = 2 \frac{RGM}{d^3}$

Here we increase R by 37%, and decrease d by about 50%. The overall effect is to increase tidal acceleration by a factor of eleven. You can expect proportionally stronger tides, and a settling period with a good number of earthquakes since tidal forces affect the mantle too (in their case, the force increase is only eight-fold, since the mantle's distance R from the center of the Earth has not changed).

since earth is now "bigger" in both size and mass, how much closer would the moon end up being?

The Earth is now 88% heavier, so the Moon is pulled all that much stronger. If its kinetic energy stays the same, since its acceleration is proportional to $\frac{v^2}{r}$, and needs to go 88% higher, this means that the orbital radius needs to decrease to about 53% of its former value. The orbit being now half as long, it is completed in half the time, or 14 days; since the Moon keeps rotating on its axis in 28 days, it ends up appearing to rotate as seen from the Earth. Every other full Moon will show what before was its far side (actually not quite - it's 53%, not 50% exactly - so there will appear to be a slow rotation).

same as question #5, would earth end up becoming closer to the sun? by how much?

No, the radius of the orbit depends almost exclusively on the mass of the primary. Here, the Sun. The Sun's mass is unchanged, so the orbit doesn't change; and we posited no change in the Earth's rotation. Therefore, the duration of the day and year remains the same. The year will now have 26 shorter months, though.

would the atmosphere still remain the same with such changes?

Yes, if more atmosphere is added (see above).

would existing structures, flora and fauna remain stable with such changes?

Yes, if the extra material is added slowly enough and does not break onto the surface. Otherwise, earthquakes and dead zones of newborn silicates are to be expected.

In the first case, however, the topsoil will need to be "stretched", so the panorama will change appreciably and several small-scale quakes will occur. In some places, the topsoil structure might be disrupted enough to cause ecological changes.

Another change

We have added a silicate layer, okay. But the magnetic field is still generated by the same core, which is now three thousand kilometers farther underground. The magnetic intensity on the surface is proportionally lower, and the Van Allen belts all that nearer.

You'll get some hell of aurorae, possibly all the way to the Equator.

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  • $\begingroup$ the underground caves could be made out of magical exotic elements, i plan to add adamantium, mithril, orichalcum and such to begin with so it fits the scenario, the caves would become one of the source to reduce overall density. silicates and exotic elements would be "designed" to form natural interconnected caves between the crust and mantle to push up the crust, it would later on connect to the surface in certain spots. i take it opening up fault-lines would cause massive damages right? as for adding more gas to the atmosphere, this shouldn't be impractical. $\endgroup$
    – Kita
    Commented Feb 4, 2018 at 13:13
  • $\begingroup$ Well, 'exotic elements' may have whatever characteristics you desire. You might even posit that it was the pressure and gravity gradient at a certain depth that 'squeezed' the material from a parallel magic universe; once enough material had squeezed through to alter conditions on our side, the leak stopped. Something like urethanic foam being injected underground beneath a building to prevent it from subsiding and pull it back up. $\endgroup$
    – LSerni
    Commented Feb 4, 2018 at 14:48
  • $\begingroup$ help me confirm if i got everything right: radius = 6371.0 KM to 9010.0 KM (41.2% increase) density = 5.51 g/cm³ to 3.01 g/cm³ (45.3% decrease) volume = 1.083 × 10^24 KM³ to 3.064 × 10^24 KM³ (183% increase) total mass = 5.972 × 10^24 Kg to 1.1944 × 10^25 Kg (100% increase) atmosphere mass = 5.1480 × 10^18 Kg to 1.0296 × 10^19 Kg (100% increase) landmass = 29% land + 71% water to 64.5% land + 35.5% water (122.4% increase) moon orbital radius 385,000 km to 192,500 km (50% decrease) moon orbital period 27.3days to 13.65days (50% decrease) $\endgroup$
    – Kita
    Commented Feb 4, 2018 at 14:57
  • $\begingroup$ Wait, the density of 3.01 refers only to the new mass. Average density needs to go down by 37%, from 5.51 to 4.02; this is done by keeping the existing material with its density of 5.51, and adding silicates with density of around 3.0. Mass, surface and atmospheric volume all will have to increase by 88%. $\endgroup$
    – LSerni
    Commented Feb 4, 2018 at 18:18
  • $\begingroup$ i see, and i just saw the "update", would speeding up the moon's orbit to match the previous 27.3days fix the massive increase in tidal forces? as for the magnetic field, would speeding up the core's spin compensate for the increased radius? if that negatively affects the planet then i suppose i'd just dump a couple of million tons of superconductive exotic elements to boost it's efficiency. $\endgroup$
    – Kita
    Commented Feb 4, 2018 at 23:28

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