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Weighing the Milky Way

May 13, 2019

Scientists usually take extreme care in the design and execution of their experiments. The first scientific conference that I attended had chemical thermodynamics as its topic, and one of the sessions was on calorimetry. One presenter had done combustion calorimetry on some volatile organic liquids contained in small plastic pouches, so an accurate weight of the combusted materials was important to know. This experimenter went a step further to correct for the buoyancy of the entrapped gas volume in the pouches during weighing, since the vapor had a different density than air.

That may have been a step too far. "The best is the enemy of the good" is one aphorism that I've always followed. At times, extra effort leads to diminishing returns. I was first introduced to this principle during high school chemistry when I learned the concept of significant figures in calculating quantities of reactants and their products.

The same principle applies to the selection of laboratory instrumentation. I've often used 5-1/2 digit benchtop multimeters (the "1/2" means that the maximum count is 199999), but it's rare when the 3-1/2 digit multimeter I purchased for about $20 doesn't give the information I need. Digital pan balances with five or six digits are very convenient, but most of the weighing I've done in my career didn't need such accuracy. Such was true throughout history when simple balance scales were used in commerce and later in science.

The accurate measurement of mass has been important to humans since antiquity, and balance scales with their associated balancing weights have existed for more than 4,000 years. The balance scale was so important to human culture that it appears as the constellation, Libra, the Latin word for the balance scale (see figure).

The Constellation Libra

The constellation, Libra, as imagined in antiquity (left) and shown in yellow in a modern star chart near the Milky Way as rendered in light blue. The modern constellation boundaries are set by the International Astronomical Union (IAU). (Left image, "Libra," an illustration by Sidney Hall (1788–1831), plate 22 in Urania's Mirror (Jehoshaphat Aspin, London, 1825), United States Library of Congress digital ID cph.3g10068, via Wikimedia Commons. Right image from KStars. Click for larger image.)

The balance scale is a conceptually simple way to measure mass, but how does one measure the mass of the Earth? For that, some fundamental physics is required in the form of the law of universal gravitation, deduced by Isaac Newton (1642-1727) and published in Book 1 (De motu corporum), of his 1687 Philosophae Naturalis Principia Mathematica, known by the shorthand name Principia. This law is simply stated,
Newton's Law of Universal Gravitation

where F is the gravitational force between two masses, m1 and m2, and r is the distance between them. We can obtain the mass of the Earth by measuring the force it exerts on a test mass, but only if we know the value of G. The present CODATA value of G is 6.67408 x 10-11 m3 kg-1 s-2, but it was first measured experimentally by British scientist, Henry Cavendish (1731-1810) in the famous Cavendish experiment, as detailed in the figure.

Henry Cavendish and a torsion balance

Henry Cavendish (1731-1810), and a diagram of the torsion balance used in the 1798 Cavendish experiment. The larger masses (M) are stationary lead spheres, and the smaller spheres (m) are also made of lead. Gravitational force causes the masses to twist the wire with torsion coefficient κ, which allows a force measurement. The force is so small that the lead spheres were chosen to be huge. The larger spheres were each 350 pounds, while the smaller spheres were each 1.61 pounds. (Left image, an illustration by George Wilson, from "The Life of the Hon. Henry Cavendish," 1851, via Wikimedia Commons. Right image, a diagram by Chris Burks, via Wikimedia Commons.)

As I wrote in an earlier article (Measuring the Gravitational Constant, November 14, 2014), Cavendish led an interesting scientific life, having also discovered hydrogen, but a strange personal life. In today's terminology, Cavendish was a geek. He wore clothing that was out-of-fashion for his time, and he was exceptionally shy around women. According to The Cavendish page on Wikipedia, he had a back door added to his house so that he could leave without seeing the housekeeper.

With such knowledge it's possible to apply a conservation law to the orbit of the Earth to calculate the mass of the Sun. The gravitational force holding the Earth and Sun together must equal the centripetal force that keeps the Earth in its orbit around the Sun. For a circular orbit of radius r, the centripetal force is
Centripetal force equaltion

where ME is the mass of the Earth, and v is the orbital velocity, easily calculated from the orbital radius and the length of the year. Once we have the mass of the Sun, it's possible to calculate the mass of the planets through similar calculations. Planetary orbits are elliptical, rather than circular, so the centripetal force would vary, but so would the orbital radius r and the gravitational force. A circular orbit is just a special case of an elliptical orbit that makes such calculations easy.

As the following figure shows, our Solar System has a vast extent that goes beyond the orbit of the last planet, Neptune, (after Pluto's demotion from planet to dwarf planet) to include the Kuiper Belt, a circumstellar disk of small objects similar to the asteroid belt between Mars and Jupiter; and the Oort Cloud, an hypothesized cloud of planetesimals that serves as a reservoir for comets. The Oort Cloud extends to about 100,000 astronomical units (AU, the Earth-Sun distance), and it marks the physical boundary of the Solar System at about 2 light years. This is not surprisingly half the distance to the nearest star.

Extent of the Solar System

The huge extent of the Solar System (not to scale).

Jupiter is about five times further from the Sun than the Earth, while Neptune is about thirty. Beyond Neptune is the Kuiper Belt at about 30-50 astronomical units (AU, the Earth-Sun distance). Beyond the Kuiper Belt is the Oort Cloud that extends to the limits of the Solar System, about two light years.

(Created using Inkscape. Click for larger image.)

It's easy to sum the masses of the Sun and its planets, but it's harder to calculate the total mass of the Solar System, since the masses of the Kuiper Belt and Oort Cloud can only be estimated. The vast majority of Solar System mass is in the Sun, since the total mass of Kuiper Belt objects is estimated to be less that a tenth that of the Earth, and the Oort Cloud's mass is about five times Earth's mass.

The next step up on the cosmological scale is the Milky Way Galaxy, which may contain up to 400 billion stars. One way to estimate the mass of the Milky Way would be to assume that our Sun is an average star, and just multiply the mass of the Sun (333,000 Earth masses, or 1.9885 x 1030 kilograms) by 400 billion to obtain 2 x 1041 kilograms. That estimate would be wrong, since we've ignored its contained dark matter, and the supermassive black hole at its center. This estimate is a factor of ten too small.

Figure caption

Milky Way over Lassen Peak.

Lassen Peak is an active volcano in the Cascade Range in Northern California. Its activity was underscored by an eruption on May 22, 1915, that spread volcanic ash over a large area.

Lassen Volcanic National Park was created to preserve the area and its volcanic features.

(Photo by Alison Taggart-Barone, via Wikimedia Commons.)

In the past, a variety of observational techniques have been used to estimate the mass of the Milky Way, and these have produced a range of masses from 500 billion solar masses, a trifle above our back-of-the-envelope calculation above, to 3 trillion solar masses. However, the same technique for finding the mass of the Sun from the mass and motion of the Earth can be applied also to the Milky Way. A team of astronomers from the Space Telescope Science Institute (Baltimore, Maryland), Johns Hopkins University (Baltimore, Maryland), and the University of Cambridge, (Cambridge, United Kingdom) has produced a new estimate based on the motions of globular clusters of stars that orbit the Milky Way with data from NASA's Hubble Space Telescope and the European Space Agency's Gaia satellite.[1-5]

Hubble and Gaia were used to measure the movements in three dimensions of globular star clusters, isolated patches of hundreds of thousands of stars that orbit the Milky Way.[2-3] About fifty of these star clusters were formed before creation of the Milky Way's spiral disk, and they can be used to measure the Milky Way's mass.[2-4] Says Tony Sohn of the Space Telescope Science Institute, who led the Hubble measurements, "Because of their great distances, globular star clusters are some of the best tracers astronomers have to measure the mass of the vast envelope of dark matter surrounding our galaxy far beyond the spiral disk of stars."[2-3] The Hubble-Gaia study was based on a decade of observations of 34 globular clusters out to 65,000 light-years and one cluster out to 130,000 light-years.[2-3]

The analysis of the motions of the globular clusters gave a Milky Way mass of 1.28 (+0.97-0.48) x 1012 (about a trillion) solar masses,[1] a mass that makes our galaxy somewhat of a heavyweight, since the lightest galaxies are about a billion solar masses. It's still lighter than some, since the heaviest galaxies are about 30 trillion solar masses.[2-3] In any case, it's assumed that most of its mass is in the form of dark matter.

Animation of Milky Way rotation

A computer simulation of the Milky Way Galaxy.

(Portion of a Creative Commons Attribution licensed YouTube Video by Hubble and the European Southern Observatory.)[4]

(Click for animated image of its rotation.)


  1. Laura L. Watkins, Roeland P. van der Marel, Sangmo Tony Sohn, and N. Wyn Evans, "Evidence for an Intermediate-mass Milky Way from Gaia DR2 Halo Globular Cluster Motions," The Astrophysical Journal, vol. 873, no. 2 (March 12, 2019), DOI: https://doi.org/10.3847/1538-4357/ab089f; also at arXiv.
  2. What Does the Milky Way Weigh? Hubble and Gaia Investigate, NASA Press Release, March 7, 2019.
  3. What Does the Milky Way Weigh? Hubble and Gaia Investigate, Hubblesite Press Release no. 2019-16, March 7, 2019.
  4. Hubblecast 117 Light: Hubble & Gaia weigh the Milky Way, YouTube Video by Hubble and the European Southern Observatory, March 7, 2019; also at spacetelescope.org.

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