Semi-Empirical Stellar Equations

I’ve spent a lot of time trying to answer certain questions in astronomy, where I just want a rough approximation for the purpose of a simulation, and don’t need an exact answer. These are some of the equations I’ve come up with.

Yerkes Classes’ T/Mv Relations

These equations are shown as functions where x is temperature in Kelvin, and f(x) is absolute magnitude (visual). Most are valid from 2400 K to a little bit past 30000 K, the exceptions are noted. For the hypergiants and white dwarfs, the range is within an elliptical region, of which these functions define the major axis; for all others, they are a trend line along a Yerkes classification.

Hypergiants (0):
f(x) = -8.9
Supergiants (Ia):
f(x) = -0.00135x3 + 0.0233x2 + 0.0187x – 7.349
Supergiants (Ib):
f(x) = 0.00329x3 – 0.0962x2 + 0.829x – 7.209
Bright Giants (II):
f(x) = 0.00557x3 – 0.166x2 + 1.505x – 6.816
Giants (III):
f(x) = 0.0135x3 – 0.373x2 + 3.019x – 7.233
Subgiants (IV):
f(x) = -0.151x2 + 2.216x – 5.128 (4450-100000 K only)
Dwarfs (V):
f(x) = 0.00193x5 – 0.0615x4 + 0.742x3 – 4.257x2 + 12.439x – 12.996 (note: this one is the least accurate)
Subdwarfs (VI):
f(x) = 0.131x3 – 3.275x2 + 27.576x – 71.7 (3050-6000 K only)
White Dwarfs (VII):
f(x) = 0.489x + 10.01 (4450-30000 K only)

Simple Conversions

Absolute Magnitude (visual) → Luminosity (solar luminosities):
Lsun = 100 * exp(-0.944 * Mv)
Accurate between 0-10 Mv. Probably continues accuracy relatively well.

Main Sequence Luminosity / Mass Relation:
Lstar / Lsun = (Mstar / Msun)3.5
Lstar = ( 3.68 * ln(Mstar) – 0.244 )e
(The second equation applies to all stars.)

Mass / Lifetime Relation:
Tyears = ( 23.4 – 2.68 * ln(Mstar) )e
(This applies to all stars.)

Spectral (UBVRI) Filters:
Ultraviolet, Blue, Visual, Red, Infrared.
Objects are listed in different indexes:
UV for the hottest objects (stellar remnants, galaxies), then BV (the majority of stars), and RI for the coolest (LTY “stars” and below).

Equations I no longer recommend

Provided for completeness, in case they are found to be useful.

Temperature (K) → Absolute Magnitude (visual):
Mv = 35.463 * exp(-0.000353 * T)
Roughly accurate between 2000-50000 Kelvin. Probably doesn’t continue accuracy at hotter temperatures. Previously this was at the top of this post, now I am not sure why (perhaps the simplicity). The Yerkes’ classifications are a better system.

B-V index (x) → Temperature (K):
T = -772.2x3 + 3152x2 – 6893x + 9500
Sorta accurate between 0-2. (This equation I am least comfortable with, and don’t plan to use.)

cc-pkg: A ComputerCraft Package Manager

ComputerCraft is a Minecraft mod that adds Lua-based computers. Over time, many programs have been created, and several package managers have come and gone. As I write this, all that I have seen are gone – their original authors have moved on, and shut down the servers hosting packages.

Now it’s my turn to sell you a package manager: cc-pkg. Unlike the others, I expect this to remain viable – even if I’m gone from the picture. If you want to skip to trying it, here’s how it’s installed (and how to ask for help):

pastebin get 9Li3u4Rc /bin/pkg
/bin/pkg help

(I’d also recommend installing the unix-like package, which adds /bin to your path, along with a few other small tweaks.)

Why is cc-pkg different?

  1. It is built on ComputerCraft’s pastebin integration.
  2. It does not require a maintainer.
  3. It is extremely simple – and flexible.

cc-pkg has the same three sub-commands of the default pastebin program: get, run, and put*. They each do exactly what you’d expect them to do, except they can use package names as well as pastebin IDs. Package names are alphanumeric characters and dashes, and there are two types of packages.

Both are plaintext lists of the form key=value. The main type of package is a list of file paths (all starting with a forward slash) as keys set to package names or pastebin IDs as values. This is how cc-pkg knows which files to download and where to put them. The second type is simply called a list; and contains package names as keys, set to package names or pastebin IDs. Lists are saved to a local file cc-pkg uses to resolve package names – overwriting any existing entries with the same package name, which is how updating is done.

With just those core features, I think the system is viable.

*The put command is not implemented as of version 1.4.2 1.5.9, the latest at the time of writing finishing updating this introduction.

Command Extensions

On top of this, if a file is saved to /lib/pkg-commands/<name> and a user runs pkg <name>, that file will be run with the other arguments. This allows adding new functions, and overwriting the core functions to add additional features, if desired.

Under the Hood

cc-pkg keeps metadata through the following files:

  • /etc/pkg/names.list: The master list of package names cc-pkg knows.
  • /etc/pkg/ids.list: A raw ordered list of every pastebin ID cc-pkg has successfully downloaded.
  • /etc/pkg/<package-name>: The file describing each installed package is itself stored by name.

API Loadable

It uses global functions so that it can be loaded as an API to make a more advanced package management system on top of it – or just to make programs automatically download requirements using cc-pkg.

  • get(name_or_id, path): The core function that installs a package (path is optional).
  • down(id): Downloads from a pastebin ID, returning its content or throwing an error.
  • save(path, data): Write to file. (Note: This is not a binary write.)
  • append(path, data): Append to file. (Note: This is not a binary write.)
  • id(name_or_id): Recursively checks the package names list until a pastebin ID is returned.
  • type(data): Recognizes data as a package, list, or unknown type.
  • src(data): Combines this data with existing package names (overwriting if duplicates exist).

An example of this is the pkg-search package, which adds a search command to look for specific package names within the master list. I am also considering adding an extension which downloads short descriptions of packages, allowing you to view what is available.

Backup Solutions

The following is current as of May 2019:

I’ve spent a few days researching software and services for backing up data. My requirements: encrypted backups, deduplication, low cost, compatible with Windows 10 & Linux, and preferably using off-site storage.


I have only considered two competitors: tarsnap and restic. Tarsnap can create keys with different permissions – a server can run backups with no danger of a compromise leading to the destruction of backups, but it is only compatible with for data storage.

Restic allows you to plug it into any system for storage. I prefer tarsnap’s extra layer of paranoia, but the service costs are where the battle ends for me.

Update: These tools both deduplicate at a block size rather than by file, and both utilize a cache for speeding up backups. Both chunk at a dynamic level, with data blobs/chunks/blocks usually being around 1MB. Tarsnap’s cache is local, but can be restored by scanning the backup server (at a network usage cost of approximately 0.1% the size of the data stored), while restic uses both a local cache and a cache on the destination. Restic also creates checkpoints while uploading backups to reduce duplication caused by interrupted uploads.


  • Tarsnap: $0.25/GB/month (transfer: $0.25/GB)
  • $0.04/GB/month (min: 200 GB)
  • Amazon S3: f***ing complicated pricing
  • Wasabi: $0.0059/GB/month (no other charges)
  • Backblaze B2: $0.005/GB/month (download: $0.01/GB)
  • Local: Hardware costs + electricity.

Obviously, price is not everything. offers daily snapshots, cheaper per-GB pricing with mass amounts of data storage needed, and additional features. Amazon S3 and Wasabi are designed for application services rather than storage. Backblaze’s B2 is probably the only cloud service (of those I examined) designed for this usage.

Ultimately, cost is my limiting factor. My backups are using restic and local hardware for now, but I plan to move to using Backblaze B2 as I can afford to.


Since publication, a few have reached out to me recommending alternative services or sharing their choices. I have not compared these as thoroughly as I did my shortlist, but I feel they deserve their own note for anyone pursuing this decision themselves:

  • SpiderOak One Backup: Starts at $0.04/GB for their 150GB plan, goes down to $0.0058/GB with a 5TB plan. I’d probably choose it if I had a bit more money to spend.
  • CrashPlan for Small Business: $10/computer, “unlimited” storage. Haven’t looked at the caveats included.

A Journey into λCalculus

I’m playing around in Minetest, and I have an idea. In order to execute this idea, I’m going to need a simple programming language. Asking Google… implementing a simple programming language

7 lines of code, 3 minutes: Implement a programming language from scratch
Sounds good! Ridiculously simple, fast, and gives us a fully usable language. (I would’ve understood brainfuck better..) Let’s see what this language looks like:

(λ v . e)   anonymous function with argument v and returning e
(f v)       call function f with argument v
(λ a . a)   example: identity function (returns its argument)

And the scary one:
(((λ f . (λ x . (f x))) (λ a . a)) (λ b . b))

Seems okay until I get to understanding that last example. If you’re curious about the steps I went through before I finally figured it out, here are my notes. I’m gonna skip to the good part:

(((λ f . (λ x . (f x))) (λ a . a)) (λ b . b))
two identity functions, let's name them I, & remove excess parenthesis
(λ f . (λ x . (f x))) I I
the syntax is still confusing me, let's make an "F" function
F(x) -> return (λ x . (f x))
F I I                          equivalent to ((F I) I)
substituting the function (identity) into our definition gives us
(λ x . (I x))         actually the same as the identity function
I(I)                  all comes to returning the identity function

λ x . x

Now, at this point I’ve learned a few small tricks for my understanding, as well as how lambda calculus works in general.


Solving a lambda calculus program is made of three (or 2.5) steps called reductions:

  • η-conversion (eta): Replace equivalent functions with simpler forms (λ x . f x) -> f
  • β-reduction (beta): Substitution (λ a . a) x -> x (essentially, THIS is solving it)
  • α-conversion (alpha): Rename conflicting names (λ a . a b) (λ a . a) -> (λ a . a b) (λ c . c)


This is where my journey ends for now. I started studying lambda calculus because of a desire to implement a simple programming language, but this will likely not satisfy my least not in this form. Here are additional resources:

Pocket Dragon (NSFW Fiction)

A short work involving anthropomorphic canid characters and a dragon. Inspired by these images (but not directly based on them).

It was just three of us, but we partied like a frat house. At some point I’d been dared to make myself 5 inches tall instead of 5 feet tall. Yeah, I was a little short for my girls, yet alone a dragon, and that was before the dare.

Well, I say my girls, but really I was theirs. A curse had bound me to Leslie, she was the black canine of indeterminate pedigree. Big floppy ears, a wolfish muzzle, but her voice was all fox. Anise was her best friend, a fennec who had to be gagged when her mate came to visit for a night or two.

So there I was, 5 inches tall thanks to Anise, and very very slowly realizing that the alcohol content in my blood did not shrink with the rest of me. If I were any other species, I’d have been dead already, fortunately dragons are a little harder to poison.

After recovering from a fit of giggles, Leslie looked a little concerned. “You okay there-” a snort, “lil.. little guy?” I’d already passed out, still standing.

The next thing I knew, I was waking up with my snout in a small warm divot on a large silky rug of some- I looked up, seeing the hills of my Leslie’s breasts and her face looking back down at me. The divot was her belly button. I blushed, there was a faint scent of musk in there from when she made me paint her with cum, and she hadn’t washed quite enough to remove it all. I got up onto my knees.

“Hey.. C’mere draggie.” Her hand, slightly larger than me, brushed against my back. The force of it knocked me right back over, dizzying me further.

Before I could react, her hand picked me up and pushed me under the hem of her panties. I immediately noticed that she was in only her panties, and the strong scent of alcohol-fueled arousal.

“C’mon dra.. draggie. Come in.. inside.” She gasped as her clumsy movements brushed my rough exterior against her bowling ball-sized clit. An involuntary movement pushed me harder against it, and my legs and tail were enveloped by sticky, puffy labia.

It was far more pleasurable at this scale than I expected, like being caressed by velvet.. that also happened to be a little too humid. The smell got a lot stronger as I felt her open up slightly to accept more. She finished shoving me inside with a moan and I was nearly folded in half in the process.

I took a moment to rest, breathing heavily in a way that made the all-encompassing pink walls around me quiver and further moisten. Thank goodness I didn’t need air for a while, especially at this smaller size, or I could have been in serious danger.

Grasping around – more than a little unsure of where I was with my level of intoxication – I must’ve found a sensitive spot as the walls of my cave trembled and squeezed me tighter. A muffled squeal came from deep within, and the squirt of lubrication that followed tried to drown me.

I don’t remember much after that, the lack of oxygen and booze enough to make me pass out again.