# Accurate way to programmatically generate a main sequence star?

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I am working on a (personal) programming project, part of which involves generating random stars and their parameters.

In order to create a procedurally generated star, I want to randomly select as few parameter values as possible, and then extrapolate the rest using formulas.

For instance by interpreting the Stefan-Boltzmann law I can derive the star's luminosity based on the surface temperature and the star's radius:

$$L = 4 pi R^2 sigma T_{e}^4$$

There are however other parameters that I would like to generate, such as the mass of the star. What seems to be able to help me is the mass-luminosity relation:

$$frac{L}{L_{odot}} = left(frac{M}{M_{odot}} ight)^a$$

By knowing $a$ (based on my randomly selected mass) it seems that I can derive the star's luminosity. By glancing at the Hertzsprung-Russell diagram (e.g. this one) I can see that I can perhaps somehow calculate the radius of the star, but I'm not entirely sure how.

There is an existing star "generator" which can be found here, however this one interpolates existing data using logarithms.

My full question would be, is there a way to "generate" a "regular", main sequence star, starting from one or two parameters and extrapolating the rest using formulas? Is there a sensible relation between the mass of the star and its radius that can be used to generate a "realistic" star? Or would most parameters need to be interpolated from existing data?

This is a tricky question, not immediate.

If we talk about Main Sequence stars, yes, you can constrain your fictional star by two parameters. The crucial parameters in the diagram, as you said, are the mass and the radius. This is why the HR diagram well represents the stellar population with two axis. Of course, for your scope, we need to do some assumptions. For example, the metallicity, that for your purposes can be fixed as the Solar one. Or, that the initial mass is already in a condition to contract.

You can see it this way: the luminosity is connected with the radius (because it depends on the irradiating surface), while the temperature is connected with the mass (the more is the mass, the more is the pressure at the center of the star).

If you have $L sim RT$ and $L sim M$, then you also have $RT sim M$. The degeneracy from one parameter can be overcame if you have your reference star (the Sun), which allows you to trace isoradii lines on the HR diagram. See here for example: Or, that is equivalent, you can express your first equation (Stefan-Boltzmann), in Solar units as well (as shown here).

Try it out just by your self: choose a mass, or observationally speaking, choose a luminosity. Then trace your line on the HR diagram, and find your star. That star has a defined temperature (spectral class), Mass and Radius as well.

Nice aim, anyway. Have fun!

Cool aim. I've been intermittently working on a similar program, feel free to use any pieces you like.

It's hosted at: http://thestarsbetween.com/maker/builder?stellar=o&rand=1233

And the code is shared open source at: https://github.com/jaycrossler/procyon

## Main Sequence Star

Main sequence stars are the most common type in the Universe. Main sequence stars are stable. They fuse hydrogen nuclei together to form helium nuclei, releasing energy and emitting light.

A main sequence star is a star in the stable part of its life cycle. They are the most common type of star in the universe. Our star, the Sun, is in the main sequence phase. It is about halfway through this stage, and ultimately will become a red giant in roughly five billion years.

All main sequence stars are in equilibrium, meaning the outward pressure caused by the fusion reactions is balanced by the force of gravity pulling the star together. The pressure and temperature of a main sequence star increase as you get closer to its center. The length of time that a star spends at this stage in its life depends on how much mass the star has. Counterintuitively, massive stars have a shorter lifespan than smaller stars. Large, massive stars use up their nuclear fuel at a much faster rate than smaller stars. Stars can range in size from about a tenth of the size of our sun all the way up to hundreds of times as big. The color of a star also varies depending on its size. Larger stars are hotter and they emit more blue light smaller stars are smaller and emit more red light.

The main sequence stage occurs after a stellar nebula collapses due to the force of gravity. As the nebula collapses, the internal temperature increases. When the core of the newly-formed protostar reaches a certain temperature, nuclear fusion starts. Nuclear fusion is a nuclear reaction that releases energy by fusing together smaller, lighter nuclei into a larger, heavier nucleus. This process releases photons of energy. These photons are absorbed and reabsorbed multiple times before leaving the star. The amount of energy that is released can be calculated using Einstein’s famous equation, E=mc 2 , where E is the amount of energy, m is the change in mass and c is the speed of light.

Most main sequence stars are nearly completely composed of hydrogen and helium. Some have a small percentage of heavier elements, such as carbon or oxygen. Scientists can analyze the composition of a main sequence star by studying the light that they emit.

### Stages in a Star’s Life Cycle

(a star with a similar mass to our Sun)

1. Stellar Nebula
2. Main Sequence Star
3. Red Giant Star
4. Planetary Nebula
5. White Dwarf
6. Black Dwarf

### How Do I Use This?

The picture encyclopedia storyboards have easily digestible information with a visual to stimulate understanding and retention. Storyboard That is passionate about student agency, and we want everyone to be storytellers. Storyboards provide an excellent medium to showcase what students have learned, and to teach to others.

Use these encyclopedias as a springboard for individual and class-wide projects!

• Assign a term/person/event to each student to complete their own storyboard
• Create your own picture encyclopedia of a topic you are studying
• Create a picture encyclopedia to the people in your class or school
• Post storyboards to class and school social media channels
• Copy and edit these storyboards and use as references or visuals

• Temperature: mean speeds of the nuclei
• Density: number of nuclei per cubic centimeter
• The remaining nuclei must move faster to maintain the same high Pressure as before.
• The gas at the center gets slowly hotter.
• This causes fusion to run faster as the temperature rises.

As we saw in Lecture 11, M-S stars obey a strong Mass-Luminosity Relation: (In words: High-mass M-S stars are more luminous than low-mass M-S stars proportional to the 4th power of their Mass.)

## Accurate way to programmatically generate a main sequence star? - Astronomy

Science_afficionado writes: By measuring the flicker pattern of light from distant stars, astronomers have developed a new and improved method for measuring the masses of millions of solitary stars, especially those hosting exoplanets. Stevenson Professor of Physics and Astronomy Keivan Stassun says, "First, we use the total light from the star and its parallax to infer its diameter. Next, we analyze the way in which the light from the star flickers, which provides us with a measure of its surface gravity. Then we combine the two to get the star's total mass." Stassun and his colleagues describe the method and demonstrate its accuracy using 675 stars of known mass in an article titled "Empirical, accurate masses and radii of single stars with TESS and GAIA" accepted for publication in the Astronomical Journal.

David Salisbury via Vanderbilt University explains the other methods of determining the mass of distant stars, and why they aren't always the most accurate: "Traditionally, the most accurate method for determining the mass of distant stars is to measure the orbits of double star systems, called binaries. Newton's laws of motion allow astronomers to calculate the masses of both stars by measuring their orbits with considerable accuracy. However, fewer than half of the star systems in the galaxy are binaries, and binaries make up only about one-fifth of red dwarf stars that have become prized hunting grounds for exoplanets, so astronomers have come up with a variety of other methods for estimating the masses of solitary stars. The photometric method that classifies stars by color and brightness is the most general, but it isn't very accurate. Asteroseismology, which measures light fluctuations caused by sound pulses that travel through a star's interior, is highly accurate but only works on several thousand of the closest, brightest stars." Stassun says his method "can measure the mass of a large number of stars with an accuracy of 10 to 25 percent," which is "far more accurate than is possible with other available methods, and importantly it can be applied to solitary stars so we aren't limited to binaries."

## Spectral Standard Stars [ edit | edit source ]

The revised Yerkes Atlas system (Johnson & Morgan 1953) listed a dense grid of F-type dwarf spectral standard stars however, not all of these have survived to this day as standards. The anchor points of the MK spectral classification system among the F-type main-sequence dwarf stars, i.e. those standard stars that have remained unchanged over years and can be used to define the system, are considered to be 78 Ursae Majoris (F2 V) and pi3 Orionis (F6 V). In addition to those two standards, Morgan & Keenan (1973) considered the following stars to be dagger standards: HR 1279 (F3 V), HD 27524 (F5 V), HD 27808 (F8 V), HD 27383 (F9 V), and Beta Virginis (F9 V). Other primary MK standard stars include HD 23585 (F0 V), HD 26015 (F3 V), and HD 27534 (F5 V). Note that two Hyades members with almost identical HD names (HD 27524 and HD 27534) are both considered strong F5 V standard stars, and indeed they share nearly identical colors and magnitudes. Gray & Garrison (1989) provide a modern table of dwarf standards for the hotter F-type stars. F1 and F7 dwarf standards stars are rarely listed, but have changed slightly over the years among expert classifiers. Often-used standard stars include 37 Ursae Majoris (F1 V) and Iota Piscium (F7 V). No F4 V standard stars have been published. Unfortunately F9 V defines the boundary between the hot stars classified by Morgan, and the cooler stars classified by Keenan, and there are discrepancies in the literature on which stars define the F/G dwarf boundary. Morgan & Keenan (1973) listed Beta Virginis and HD 27383 as F9 V standards, but Keenan & McNeil (1989) listed HD 10647 as their F9 V standard. Eta Cassiopeiae A should probably be avoided as a standard star because it was often considered F9 V in Keenan's publications, but G0 V in Morgan's publications.

As has been said, this is probably a very subjective question/answer. Not only that, but the composition of galaxies, and even regions within a galaxy, varies a great deal. Then there is the question of what constitutes as being part of the galaxy as opposed to perhaps a small orbiting dwarf galaxy. The answer you got from the Quora seems to be pretty comprehensive.

The volume of an area of interest, divided by the number of stars in that area seems to be the one that most people take as the approach. Which may not get a very accurate result, but smoothed out over said volume. Although, I will note that the first technique given on the quora site gives an answer that is close to the accepted "average" in the Milky Way, so at least there doesn't seem to be a large disagreement there. Of course, that assumes that the same initial starting conditions are used in both problems, which is highly unlikely since they aren't totally agreed upon anyway.

EDIT TO ADD: For more examples of similar math, here Dr. Plait calculates the number of habitable planets (where he shows the calculation for the volume of the galaxy). Making some assumptions of our own (like 200,000,000,000 stars which is LOW in my opinion), we come out to an average distance of about 5 light years. Doubling the number of stars gives an average of about 4 light years though, so again, we are not off by factors.

## How is a main-sequence star like the sun able to maintain a stable size?

During the main sequence stage the Star burns it's primary fuel that is hydrogen and converts it into helium, this process provides a balance against the inward acting gravity.

In a Star there is always gravitational force acting towards the inside of the Star. The only reason that the Star is not collapsing or even changing it's size is because it is generating enough force to balance that gravity so it has a Stable size.

Once the hydrogen has been consumed the Star will transform into another form but will not collapse as the fusion reactions will still be going on but with a different element.

The only way a Star collapses is when there is no outward acting pressure to balance gravity. This will only happen when the fusion reactions stop. For example when a Stars core contains only Iron as a product of previous fusion reactions. Iron is the most stable element so Star will not be able to fuse Iron into another element and hence will collapse on it's core.

## Contents

### Astronomical Unit Edit

The astronomical unit is the mean (average) distance of the Earth from the Sun. This we know quite accurately. Kepler's Laws tell the ratios of the distances of planets, and radar tells the absolute distance to inner planets and artificial satellites in orbit around them.

### Parallax Edit

Parallax is the use of trigonometry to discover the distances of objects near to the solar system.

As the Earth orbits around the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in a right triangle, with 2 AU making the short leg of the triangle and the distance to the star being the long leg. The amount of shift is quite small, measuring 1 arcsecond for an object at a distance of 1 parsec (3.26 light-years)

This method works for distances up to a few hundred parsecs.

Objects of known brightness are called standard candles. Most physical distance indicators are standard candles. These are objects which belong to a class that has a known brightness. By comparing the known luminosity of the latter to its observed brightness, the distance to the object can be computed using the inverse-square law.

In astronomy, the brightness of an object is given in terms of its absolute magnitude. This quantity is derived from the logarithm of its luminosity as seen from a distance of 10 parsecs. The apparent magnitude is the magnitude as seen by the observer. It can be used to determine the distance D to the object in kiloparsecs (kiloparsec = 1,000 parsecs) as follows:

where m the apparent magnitude and M the absolute magnitude. For this to be accurate, both magnitudes must be in the same frequency band and there can be no relative motion in the radial direction.

Some means of accounting for interstellar extinction, which also makes objects appear fainter and more red, is also needed. The difference between absolute and apparent magnitudes is called the distance modulus, and astronomical distances, especially intergalactic ones, are sometimes tabulated in this way.

### Problems Edit

Two problems exist for any class of standard candle. The principal one is calibration, finding out exactly what the absolute magnitude of the candle is.

The second lies in recognizing members of the class. The standard candle calibration does not work unless the object belongs to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.

A significant issue with standard candles is the question of how standard they are. For example, all observations seem to indicate that Type Ia supernovae that are of known distance have the same brightness, but it's possible that distant Type Ia supernovae have different properties than nearby Type Ia supernovae. [1]

With few exceptions, distances based on direct measurements are available only up to about a thousand parsecs, which is a modest portion of our own Galaxy. For distances beyond that, measures depend upon physical assumptions, that is, the assertion that one recognizes the object in question, and the class of objects is homogeneous enough that its members can be used for meaningful estimation of distance.

Physical distance indicators, used on progressively larger distance scales, include:

— In the last decade, measurement of eclipsing binaries offers a way to gauge the distance to galaxies. Accuracy at the 5% level up to a distance of around 3 million parsecs. [2] — are periodic variable stars, commonly found in globular clusters, and often used as standard candles to measure galactic distances. These red giants are used for measuring distances within the galaxy and in nearby globular clusters.
• In galactic astronomy, X-ray bursts (thermonuclear flashes on the surface of a neutron star) are used as standard candles. Observations of X-ray burst sometimes show X-ray spectra indicating radius expansion. Therefore, the X-ray flux at the peak of the burst should correspond to Eddington luminosity, [3] which can be calculated once the mass of the neutron star is known (1.5 solar masses is a commonly used assumption). and novae
• Cepheids are a class of very luminous variable stars. The strong direct relationship between a Cepheid variable's luminosity and pulsation period, secures for Cepheids their status as important standard candles for establishing the Galactic and extragalactic distance scales. [4]
• Novae have some promise for use as standard candles. For instance, the distribution of their absolute magnitude is bimodal, with a main peak at magnitude −8.8, and a lesser one at −7.5. Novae also have roughly the same absolute magnitude 15 days after their peak (−5.5). This method is about as accurate as the Cepheid variable stars method. [5]
. Because the white dwarf stars which become supernovae have a uniform mass, Type Ia supernovae produce consistent peak luminosity. The stability of this value allows these explosions to be used as standard candles to measure the distance to their host galaxies, because the visual magnitude of the supernovae depends primarily on the distance. [6] and Hubble's Law By using Hubble's law, which relates redshift to distance, one can estimate the distance of any particular galaxy.
• ### Main sequence fitting Edit

In a Hertzsprung-Russell diagram the absolute magnitude for a group of stars is plotted against the spectral classification of the stars. Evolutionary patterns are found that relate to the mass, age and composition of the star. In particular, during their hydrogen burning period, stars lie along a curve in the diagram called the main sequence.

By measuring the properties from a star's spectrum, the position of a main sequence star on the H-R diagram can be found. From this the star's absolute magnitude is estimated. A comparison of this value with the apparent magnitude allows the approximate distance to be determined, after correcting for interstellar extinction of the luminosity because of gas and dust.

In a gravitationally-bound star cluster such as the Hyades, the stars formed at approximately the same age and lie at the same distance. This allows relatively accurate main sequence fitting, providing both age and distance determination.

This is not a complete list of methods, but it does show the ways astronomers go about estimating the distance of astronomical objects.

## Lessons from Kepler and friends

Inspired by research performed with Kepler data, J. Bean and collaborators propose the statistical comparative planetology approach. Its premise lies in utilising the diversity of planets in large samples to obtain information about their nature based on comparative studies. This approach has successfully been applied to identify false positive rates in Kepler data, as well as to study the water content of transiting planets. These results are robust in that they only depend on simple physical models, and allow the identification of outliers, which reveal potential model weaknesses.

One of the examples the authors describe in today’s paper aims to test the habitable zone concept using carbon dioxide abundances. Planets that receive more irradiation from their star need less CO2 to maintain accommodating temperatures via the greenhouse effect, while those that receive less irradiation need more carbon dioxide. The idea of the test is to measure CO2 content on planets inside habitability zones and seeing if they are compatible with the amount expected for planets with accommodating temperatures (see the figure below). These measurements don’t need to be extremely precise given that the uncertainties in the physical properties of the planets is offset by the large sample size.

The habitable zone concept (blue curve) assumes a decreasing amount of CO2 as stellar irradiation increases. The black points represent hypothetical planets scrambled away from the blue curve based on the expected uncertainties of a statistical inference. The habitable zone concept can be tested if we are able to perform this kind of bulk analysis of a large sample of planets.

A similar test can be performed for water content on potentially habitable planets: if they are closer to the inner edge of the habitability zone, they should have lost all their water due to a runaway greenhouse effect while those near the outer edge of the zone should have little water because it has frozen out. Any deviation from this means we are getting habitability wrong. Again, this test does not require a lot of precision, but does need a large number of planets.

The authors make it clear that the statistical approach is not the single best way to study habitability. For instance, this method still requires detailed analysis of a few planets to identify key diagnostics for habitability, and it is limited to planets around M dwarf stars, whose habitability zones are the easiest to observe. However, comparative planetology still looks like a very pragmatic way to circumvent the unknowns and progress towards answering The Big Question.

Disclaimer: the leading author of today’s paper, Jacob L. Bean, was my research supervisor in 2016, but I am not in any form involved with this particular publication.