What is the Minimum Mass Necessary to make a Celestial Body

Round?Julieanna Bacon

About MeHow many people can say their first word was a celestial

object? Probably not a lot, and when your first word is something like star, chances are you have already been hooked by astronomy. Since that day about 15 years ago, I have found out that I won’t be happy doing anything else but astrophysics. I am now in the 10th grade at Cooper City High School, taking Spanish 3 here at FLVS. This is my first course at the school & the freedom it grants is extremely gratifying. I have attended the SPLASH program at MIT, completed college-level astronomy courses, completed biology and chemistry courses at my high school, and have served as treasurer for the Science Club at Cooper City High School. I was elected president for the upcoming 2012-2013 school year as well. Science has been my passion for as long as I can remember.

Me, on the left, at a DECA leadership conference

Project IntroductionThe purpose of this study was to find the minimum required mass for

gravity to make an object spherical. The experiment’s goal was to find a variety of masses for celestial bodies (both spherical & non-spherical) using data from various sources & calculate the masses. After much initial research, 0.0006% of Earth’s mass was hypothesized to be the critical minimum mass. First a telescope was used that was capable of measuring the length of time an object takes to orbit another object. Second, the distance between the two objects was recorded. Next, Kepler’s 3rd Law was used to simply Newton’s law of universal gravitation, and using that allowed the final mass of each celestial object to be calculated. Images from Hubble, Cassini, and other spacecrafts photographing the solar system were used to determine curvature, as well as descriptions in scientific publications. Astronomers and physicists struggle with determining characteristics of exosolar planets, often only knowing the mass and orbit of the celestial object. They could apply the minimum mass found here to determine if the body is spherical or not large enough to attain a spherical shape. The subjects of the experiment were the 7 celestial bodies whose mass was tested.

HypothesisThe hypothesis was that 0.0006% of Earth’s

mass is the minimum mass required to shape the object into a sphere by gravity.

The exact figure predicted was 3.58452 * 10²°

Initial Research and Works CitedAs you know, Pluto was stripped of its status of planet & given the title

“Dwarf Planet”, at the IAU meeting of 2006, which created a new classification of objects big enough to be round, yet too small to have cleared their orbit of stray asteroids. When I first heard of this major event, I was intrigued by this. It inspired me to inquire further. This new classification also helped narrow down the objects that were to be subjected to the experiment. The objects being tested had to have a variety of masses, but none too great otherwise it would not help to know their masses. The reason the planets in our solar system were not tested & used to determine the minimum mass, rather than the hypothesis, was because they are so much bigger than the minimum mass. Researching various dwarf planets helped find bodies on the cusp of being round or not round, which were the ones perfect to test. The 5 IAU recognized dwarf planets are Ceres, Eris, Makemake, Haumea, & Pluto. Once these dwarf planets were research, it was found that Ceres is the smallest, but still clearly round.

Initial Research and Works CitedThe implications of this study are many in number. If this study has

not been done before, then imagine knowing the exact figure that determines if a celestial body is round or not. This would be most useful to astronomer & astrophysicists. Unless the body was in a special circumstance, you could know if a body in space was round or not, simply by knowing it’s mass! This could mean a great deal of time saved, because previously telescopes had to be booked many months earlier to check whether or not a body in space was round, if you were classifying asteroids, moons, or comets in the solar system. Instead, this could mean that astronomers no longer need to waste time & money booking expensive telescopes to simply check if an object was round. They can book telescopes for more important things now! For example, measuring the mass & rotation speed of neighboring galaxies to see how much dark matter they contain & furthering the study of this mystery.

Initial Research and Works CitedWhen studying exosolar planets, astrophysicists, studying

only the tiny “wobble” effect the planet has its sun, can determine the mass of the planet. Since they are much too far away to be seen with a telescope, characteristics of the planet must be deduced. The minimum mass figure can indicate whether or not the planet is spherical. As technology improves, astrophysicists will be able to detect exosolar planets much farther than the 3,000 light years they have currently reached. As this occurs, they will be looking farther and farther into the past. Astronomers that study the early formation of solar systems will find this data especially useful in determining the history of a celestial object.

MaterialsTelescope (Celestron – 8”)Calculator (Ti-89)PaperPencilComputerChartNewton’s law of universal gravitation,

simplified by Kepler’s 3rd law of planetary motion - M=4 pi2*r3/T2G

Materials

Ti- 89 used in calculations Celestron NexStar 8 inch Telescope

MethodThere are two basic ways to detect a natural satellite’s mass:

the effect the satellite has on other satellites’ orbits and if the body has a satellite itself. Newton’s law of universal gravitation can then be used to determine the mass of the object.

The independent variables of the experiment were amount of distance between object A (the object whose mass is being measured) and object B, and amount of time it takes object B to orbit object A. The independent variables of this experiment were chosen as such, because they are the ones that can be changed to come to a definite mass. They control the experiment part of this study. The formula used to find the mass of an object using another object orbiting it is M=4 pi2*r3/T2G (Tom Young 2001) ( see procedures #4 to see what each variable in the equation stands for).

MethodThe dependent variable in this experiment is

the mass of the celestial body, or simply, the outcome of the equation after you have done your math & found your figures via telescope.

The control of this experiment was the masses of the celestial objects that can be found online. After the experiment, the masses found were compared online with the masses found. The masses found had a large propagated error, since equipment usually used to find the masses of celestial objects are telescopes of over 15”.

Procedures1. To find the mass of an object such as Pluto (dwarf planet with

moons), you must first find a telescope capable of measuring the length of time that an object far into space takes to orbit another celestial object, such as Pluto & Pluto’s moon.

2. Once you have done that, record the amount of time it takes the moon (let’s call it object B) to orbit the body (Pluto in this case, object A).

3. Find the distance between object A & object B (astronomical records are the best resources).

4. Take the equation M=4 pi2*r3/T2G, M being the mass of the orbited object, T the amount of time it takes the orbiting object to orbit the planet or sun you want the mass of, pi is 3.14159, R is the distance between the centers of Objects A&B in meters, and G is the gravitational constant (6.672 x 10-11 Nm2/kg2).

5. Plug the numbers in & solve for the mass of object A.

ResultsThe final result of the project showed that an object

with a diameter of 400km, give or take 10km, has just barely enough mass to be spherical. Its mass should be roughly 1.3 * 10⁸kg. The table below shows the data collected in this experiment. 9 Metis was included in the table because since it is close to being spherical, but not really so, it shows almost the minimum mass require for gravity to shape a celestial object into a sphere. The minimum mass is very close to 9 Metis’s mass. In fact, 10 Hygiea just qualifies as “spherical”, meaning from any point in its center, the surface is the same distance within 20km.

DataObject Mass (kilograms) Moon ClassificationPluto 1.3 *1022 3 Dwarf planetHaumea 4.1 * 1021 2 Dwarf planetMakemake 4.341 * 1021 0 (even though it needs a moon to be

calculated my way, the mass was found online)

Dwarf planet

Eris 1.1*10²² 1 Dwarf planetCeres 9.43 * 1020 0 (even though it needs a moon to be

calculated my way, the mass was found online)

Dwarf planet

10 Hygiea 1 * 10⁸ 1 satellite Asteroid

9 Metis 1.13 * 10⁷ 0 (even though it needs a moon to be calculated my way, the mass was found online)

Asteroid (NON-SPHEREICAL)

Object Mass Found Mass Online

Pluto 1 *1022 1.3 *1022

Haumea 4 * 1021 4.1 * 1021

Makemake n/a 4.341 * 1021

Eris 1 * 10²² 1.1* 10²²

Ceres n/a 9.43 * 1020

10 Hygiea 1 * 10⁸ 1.3 * 10⁸

ConclusionThe hypothesis was that 3.58452 * 10²° is the

minimum mass requires for gravity to shape the object into a sphere, or 0.0006 of Earth’s mass. The actually minimum mass turned out to be 1.3 * 10⁸.

The data showed that while 9 Metis was just on the cusp of being round with a mass of 1.13 * 10⁷, it was 10 Hygiea that just qualified as being round with a mass of 1.3 * 10⁸kg. The results for this research are versatile.

ConclusionThe implications of this study were mentioned earlier in the

introduction, but to summarize, the conclusion of this study would most help astronomers & people who study space. They could save time that is normally used to check whether an asteroid, moon, comet, etc is round by a very expensive telescope by simply using the minimum mass to determine if it is round. So, if the IAU needs to find out if a large asteroid qualifies as a dwarf planet, they simply need to know the mass of the object to find out if it is round. If the mass of the object is above the minimum, then it is round, unless it is very young or in a special circumstance & if it is below the minimum, it is not round. This method can save time & money, both of which can instead be used for more important things.

ConclusionAfter comparing the controls & the data that was

found in the experiment, conclusions were drawn about what the minimum mass was. Since 10 Hygiea was just barely round (meeting the requirements as being “round” by just a smidgeon) its mass seemed the perfect minimum mass requirement. With 9 Metis trailing so closely behind, only 1 zero smaller, and yet definitely not meeting the “round” requirements, it proved that the remarkable 10 Hygiea was the minimum, almost exactly.

DiscussionSome limitations of this study might have been that

what qualified an object as being round was not clear enough or was hard to determine. The minimum mass may vary from what was deduced. Also, my way of finding the masses of the celestial bodies was limited by my resources. Perhaps if a professional telescope was used, it may have wielded different and more accurate results. But since controls, the actual masses of the objects found online from studies done by NASA and other astronomical organizations, were used, most error was eliminated from that limitation.

DiscussionA question raised by this study was, is it

possible to determine using this data if an exosolar planet thousands of light-years away is round? Currently technology is still too crude to find any exosolar planets smaller than Earth, so they are all round for sure, but in 10 years or so from now, when the objects are so small we have doubts about whether they qualify as exosolar planets or just exosolar asteroids or dwarf planets or whatever it may be, will this data help determine if they are?

12 Candidates for Planets, credit (IAU/NASA) Martin Kornmesser

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