The science behind a planetary sized humanoid

Table of contents

Introduction
Weight of a planetary sized human
- 2a. Limited weight
- 2b. De-Limited weight
Destroying the earth
Orbiting Sana
Escaping Sana
Living on Sana
A credit to Sana’s biology
Feeding Sana
- 8a. Surface area of limited Sana
- 8b. Radiance of objects
Conclusion
Questions and Answers

1. Introduction

This is Sana.

Member of Hololive EN 2nd Gen (Holo-Council) and to quote Sana.

Sana is big.

Amongst all Hololive English members, Sana is the tallest at 169 cm (5ft 6′). Indeed of all active Hololive members Sana is 2nd tallest. In Australia (where her accent comes from) only one in six would be taller than her but we are ignoring something important.

The little pink twirly thing on Sana’s head is a size limiter. Sana is actually bigger when she takes it off. Much bigger.

1,697,840km tall.

At a certain point this simply becomes a number that you are unable to picture. So here’s another way of showing this. Here is a scale image of the Earth, the Moon, the separation between the earth and the moon and Sana.

(Spoilers: I do the math, this is also as close as Sana could be to the earth and not immediately destroy it due to the gravitational forces involved. Sure it would be a bad time and likely result in a lot of death, but at least we would know what happened.)

2. Weight of a planetary sized human

Now a lot of people I’ve seen doing similar things and they just approximate Sana’s size limited weight to be that of an average human female. I draw issues with this.

Average weight is practically pointless. The average weight of a human female aged 20-69 is 61.5kg. With that said 95% of females aged 20-69 fall within the weight range of 39kg and 83kg.
Operate under the assumption that the person is of average height (proveably false for Sana)
Operate under the assumption that the person is within an age range (Sana’s age is that of the universe)
I don’t think we can call an entity with a life span this long and naturally this big can be equated as human
Anime
I’ve similar things before, so it isn’t too much work to do again.

Instead of all this we are going to work with two assumptions:

With her limiter on Sana has the same proportions as a human (Certainly possible)
Sana has a similar density (I.E. is made of the same materials) as a human (unlikely but I have to work with something).

2a. Limited weight

$FemaleWeight = -40.2 + 0.42 \times (AbdominalCircumference) + 1.3 \times (ThighCircumference)$

$(AbdominalCircumference) = \frac{169cm}{872} \times 124 \times \pi$

$(ThighCircumference) = \frac{169cm}{872} \times 52 \times \pi$

$FemaleWeight = -40.2 + 0.42 \times (\frac{169cm}{872} \times 124 \times \pi) + 1.3 \times (\frac{169cm}{872} \times 52 \times \p) = 32.668kg$

2b. De-Limited weight

Under the assumption that the limiter isn’t doing something off with Sana’s weight or density (which it likely is for reasons we get to). Under such assumptions seeming as her height by a factor of 1,000,000,000 when taking off her limiter her volume grows by 1,000,000,000,000,000,000,000,000,000 (10²⁷). Making Sana’s weight 3.21×10²⁸ kg. Approximately 1.6% of the mass of the sun or 17.2 times the mass of Jupiter.

3. Destroying the earth

Now of course the instantaneous growth from 169cm to 1,697,840km is going to destroy the earth. The impact alone would achieve this. That said Sana could leave the earth on a rocketship, travel far enough that she wouldn’t hit the planet, then remove her limiter and still spontaneously destory the planet without ever touching the planet.

In space any pair of objects have a distance associated with them called the Roche limit, if the two objects ever reach a separation less than the Roche limit the lighter object is torn apart. Either becoming part of the larger body or turning into a ring. The calculation for this is:

$Roche = r(\frac{2M}{m})^{\frac{1}{3}}$

Where M is the mass of the larger body, m is the mass of the smaller body and r is the radius of the smaller body. So some numbers later:

$Roche = 6378.137km\times (\frac{2\times3.2\times10^{28}}{6\times10^{24}})^{\frac{1}{3}}=137,412.8km$

Which actually isn’t that far. The moon is further away. Issue then however is of course that we’d orbit Sana, but at least the earth doesn’t immediately evaporate.

4. Orbiting Sana

So I’ve seen some “If Sana was at moon distance” pictures before. Which we now know wouldn’t immediately destroy the planet. Here is my own rendition.

But let’s talk about what would happen.

The earth would have to orbit around Sana’s mid section. The forces don’t really support an orbit of Sana head to tail and if we orbit around anything else the forces would eventually lead it to orbit around Sana’s centre of mass.
The earth would tidally lock to Sana. Meaning only one side of the planet would get Sana but would get to look at Sana all the time. The other side gets the universe.
Sana would either have to move or interrupt the earth on a frequent basis to ensure everyone gets sunlight. The other options being
- Sana Stands relative to the sun, Earth’s Sana side might get some light during sunrise/sunset but the “day” would be eclipsed by Sana
- Sana lies down head/feet towards the sun. Earth orbits like Neptunes rings. Leaving one side of the earth in perpetual light and the other in darkness.
Completing a full orbit of Sana takes less time than a month.

$T = \sqrt{\frac{4 \times \pi^2 \times r^3}{G \times M}}$

$T = \sqrt{\frac{4 \times \pi^2 \times 384400000^3}{G \times 3.2\times10^{28}}}$

T = 32069.5s = 8 hours 54 minutes 29.5 seconds.

5. Escaping Sana

Escape velocity equation:

$V_c = \sqrt{\frac{2GM}{r}}$

Now big thing with this equation is quite simply it assumes the body you are escaping from is spherical. This is quite obviously not true. There are other assumptions made but this is one of the most obviously wrong ones. Now on one hand I could run through the deviation of a similar equation only break down a human into a number of shapes that are summed together. But integration on this sort of scale is a lot of work and ultimately wouldn’t matter that much.

This equation is derived by calculating the amount of energy it would take to move a point mass from a surface of a sphere to a distance infinitely far away. By summing up all gravitational force at all points between the objects surface and infinity. Now as you move further away the overall shape becomes less important. The gravitational force imparted by a cube and a sphere are different due to the differing shape but if you take a sphere and a cube of the same weight and same dimensions by the time you are 4 cubes/spheres away the difference in the force imparted is only 0.06%. In much the same way as you get far enough away from any shape it ends up like looking like a circular speck. So rather go through a lot of work, I’m just going to use the figures as if Sana was a sphere and use them as ballpark figures

Question then is, where do we try and escape Sana from? Well the closer we are to Sana’s centre of mass the harder it is to escape. Knowing this someone takes the (long) trek to the place furthest from her centre of mass (bottom of feet or top of head).

The centre of mass of a person is located roughly central to their height (just behind the belly button). So the seperation from the center of mass would be half Sana’s height. 848,920km

Some math later we end up with an escape velocity of 68,680 m/s, 153,632 mph, 247,248 kmph or mach 200

(For reference of how much easier the trip makes it standing on her midsection and attempting to take off, the distance is approximately 120,161 km therefore escape velocity is 182,560m/s, 408,375 mph, 657,216kmph or mach 523).

As a comparison the escape velocity for earth (the speed rockets need to reach to leave earth) is only 11,186 m/s, 25,022mph, 40,269kmph or mach 32 and they take advantage of the rotation of the earths surface to reduce the work needed to get this speed. Pretty much short of getting Sana to help you by throwing you or something you aren’t escaping. This however leads into a much bigger issue.

6. Living on Sana

As evidenced by the much higher escape velocities the gravity on Sana is much larger than that on earth. Living on Sana’s head the local gravity would be 262,000 times that of earth (on her midsection it would be 1,850,000 times that of earth). While humans can survive hundreds of times the force of gravity they can only do so for exceptionally short stints. As a more realistic comparison humans struggle to survive a force of 16 times the earths gravity when it is sustained for 1 minute. This is probably the biggest argument as to why Sana isn’t human. She quite simply wouldn’t survive. She would die under the forces that her own body pushes on her.

7. A credit to Sana’s biology.

Assuming Sana’s head weighs as much as a humans head weighs in comparison to the rest of a human body Sana’s head is 8.2% of her body mass or

$3.2 \times 10^{28} \times 8.2 \% = 2.6 \times 10^{27} kg$

This means that the force between her head and the rest of her body can be found by:

$F = \frac{G \times M_1 \times M_ 2}{r^2}$

$F = \frac{G \times (3.2 \times 10^{28} \times 8.2 \%) \times (3.2 \times 10^{28} \times 91.8 \%)}{1697840km^2}$

$F = 1.86 \times 10 ^{27}N$

Her neck would be experiencing a pressure of

$\frac{1.86 \times 10 ^{27}N}{\pi \times (\frac{1697840km}{872} \times 32)^2}$

$1.525 \ times 10^{11} Pa = 22,118,255 psi$

Which is enough to crush diamonds, graphene and all other materials I can think of.

So the bad news is that if Sana were a normal human she would immediately collapse into a very large ball of flesh (good news, as the mass required to become a star is over 4 times Sana’s de-limited weight she wouldn’t turn into a flaming ball of gas).

So Sana is likely made of some special space material we don’t quite fully know about. There is tonnes of the stuff in the high tempratures and pressures of stars and planets we don’t have the means to fully analyse.

8. Feeding Sana.

Unfortunately the issues for our overly big humanoid do not end there. There are other, planetary sized humanoids in fiction which I probably could have focused this on but Sana is the one that got me interested in it. Off the top of my head another planetary sized humanoid is Galactus. Who in the marvel universe goes around eating planets. Which is a very good point because the energy an being of this size would need to consume is not insignificant. Granted the biggest animal on earth, the blue whale, may feed off one of the smallest but in total it eats 240,000 times more than a human in any given day. Working out exactly how much Sana would need to eat is actually quite hard, but we can work out a bare minimum number. Namely how much would Sana need to consume in order to keep her body temperature within normal human body temperature limits.

8a. Surface area of limited Sana.

Simply a case of putting Sana’s weight and height into online calculators.

$1.2475m^2.$

8b. Radiance of objects.

The radiance of an object (how much energy is lost due to the radiation of heat alone is calculated by

$\frac{4 \times \sigma \times T^4 \times A}{1-a}$

Where σ is a constant, T is the temperature (in kelvin), A is the area and a is called albedo, which for humans is 0.97

$\frac{4 \times \sigma \times 310.15^4 \times (1.2475 \times 1000000000^2)}{1-0.97}$

$8.7 \times 10^{22}$

Now this would mean that in order to sustain a liveable human temperature Sana’s biology would have to consume (assuming she has the means to digest it all):

All of the known fossil fuels on the planet earth
All of the known uranium supplies on planet earth
AND
All of the sunlight that strikes the earth in any given 24 hour period

every second. So either she’s eating a LOT or they don’t sustain a human body temperature.

9. Conclusion

How does the size limiter work?

Now to go backwards a bit explain Hololive. Hololive is a part of a Japanese company that creates Idols that operate behind virtual avatars. While they are an Idol company, their members are probably better known as YouTube entertainers. Making songs and performing concerts yes but its grown to encapsulate other things. They stream playing video games, some stream them drawing art, some stream ASMR, some just like to chat, so on and so forth. It’s existed since 2017 and has various branches (Holostars is hololive but male, there is also an Indonesian section and an English section). Overtime each new group that joined started to get themed. Hololives 2nd English group (of which Sana is a member) is referred to as the holo-council and each member is meant to represent an abstract concept. Space, Nature, Time, Civilisation and Chaos. Sana is the “speaker of space”. To represent space. Another member Kronii is the “speaker of time” and meant to represent time.

Kronii and Sana’s first collaboration (video live stream with just the two of them) ended up with Sana taking about Kronii as a multi-dimensional entity. Kronii frequently “messes around with time”. Seeming as she is able to freely move in the three dimensions of space (Forward & Back, Left & Right, Up & Down) and freely in the fourth dimension of time she must be a multi-dimensional entity. (Aside: Humans can technically move freely in the three dimensions of space, but confined in their movement through time. Forward, one second every second). Sana then proceeded to explain other dimensions than the 4 spatial dimension and 1 temporal dimension but I’ve digressed.

No one really knows how to explain all the higher dimensions simply because our brains aren’t built that way. I can at least attempt to describe 1 additional dimension. Note: I did not create this description but always liked it and am adapting it from a memory a few years old now.

Imagine you are a two-dimensional ant living on a flat piece of paper. You have left, right forwards and back but no concept of up and down. One day in your flat ant world there exists a circle that wasn’t there before. As you watch the circle it appears from nowhere, starts small, grows bigger until it reaches a maximum size, shrinks back down and vanishes. Now all the ants in this flat ant world are confused as to what happened. A three dimensional being watching from above accidently dropped a sphere through their flat ant world and for the ants that had no concept of up or down they only saw the circular cross sections of the sphere as it was dropped through their world. Something that doesn’t exist growing, becoming big reaching a maximum size, then shrinking back down until it doesn’t exist anymore. In the same way if our same three dimensional being had dropped a cone on our flat ant world point first it would look like a circle that came from nowhere, grew up to a maximum size then immediately vanished.

Imagining a fourth spatial dimension is ni on impossible. Imagining an object suddenly appearing, growing and/or shrinking then disappearing however is not. Some may be seeing where I am going with this.

In physics space and time are intrinsically linked. Hence spacetime. The speaker of time has exhibited her abilities as a multi-dimensional being. Would it be too much to say that she wasn’t the only one.

10. Questions and Answers.

Sure this isn’t perfect. this but it allows us to hand wave many questions away.

How can Kronii freely manipulate time but Sana not freely manipulate her size?
- Well Kronii has literally all the time to practice with.
Why doesn’t Sana immediately collapse into a ball of flesh?
- She is being supported by herself in the 4th dimension.
Where does Sana get the energy to exist in her un-limited size?
- Somewhere in amongst the dimension we are unable to see.
Why does small Sana need a size limiter on her head but big sana simply exist?
- When was the last time you tried to balance a cone on it’s point instead of it’s larger side.
Where does all the energy for Sana size change come from?
- Technically she isn’t changing size. A 1 meter 3 dimensional cube measures 1 meter in each of the three dimensions. A 1 meter four dimensional cube measures 1 meter in each of the four dimensions, it just so happens we can’t see the fourth one so it appears to grow/shrink despite not actually changing size.
How does the size limiter work?
- Balancing a cone on it’s point is far easier with superglue.
Is the size limiter four dimensional?
- Probably only slightly. Like you can call a flat piece of superglue a 2 dimensional object supporting a three dimensional object provided you don’t look too hard. (In reality though it is actually 3 dimensional just very small in that third dimension.)
Could sneezing dislodge a technically four dimensional size limiter from a three dimensional Sana?
- Outside forces. It depends on how the four dimensional object interacts with the four dimensional Sana. But if an entire four dimensional entity sneezes, imagine shaking your cone balanced on its point that you stuck down with superglue.
I thought higher dimensions were folded into smaller sections of space. How would a higher dimensional being grow?
- I don’t pretend to understand higher dimensional physics, I’ve studied it for maybe 10 hours. 2 of which was listening to the same talk twice.
Could we make Sana survivable?
- One of (but by far not the only) issue with living on Sana would be the gravitational force. Now technically there is nothing saying that big Sana has to have all the internal organs/muscles etc a human needs (think of a hollowed out cone shape). Her internal bodily functions could be managed on a smaller scale and smaller Sana then effectively puppets Big Sana from an unseen dimension. This technically solves far more than just this issue as well (I.E. Sana no longer crushes herself due to the weight of her own body/Solve the energy issues as big Sana doesn’t technically the energy consumption scaled up. Not that this necessarily means that big Sana would feel as if she were a puppet as from an extradimensional point of view she would still be a single entity.