# How was the mass of Venus determined?

The mass of Venus seems rather complicated to determine to me:

• Venus doesn't have any satellites, so you can't just apply Kepler's third law (like you would with Jupiter or Saturn for instance) to determine its mass.
• The gravitational tug of Venus on the Sun is very small compared to that of other Jupiter or Saturn, so it seems like it would be difficult to extract what part of the Sun's proper motion is caused by Venus.
• There are few asteroids with orbits close to that of Venus, so not many objects who might have their trajectory modified by Venus.

Knowing Venus' radius, and assuming it has the same density as the Earth, you can get a pretty close estimate of its mass (85% of Earth's mass with this assumption, when the actual value is 82%). But that's a pretty strong assumption (the density of Earth and Venus only happen to be close by chance) and a rather unsatisfactory "guesstimate".

Nowdays, there are a few probes that have flown by Venus, so by looking at their trajectory, you can infer what Venus' gravitational field looks like. But those fly-bys are pretty recent. Did we know about Venus' mass before those fly-bys?

How was the mass of Venus measured for the first time?

• Mercury's mean density is actually very close to Earth's, for what it's worth - 5.4 g/cm$^3$ as opposed to 5.5 g/cm$^3$. Sep 27, 2020 at 14:23
• @HDE226868 Wolfram|Alpha says 5.52 g/cm³ versus 5.24 g/cm³. Sep 27, 2020 at 22:32
• @gen-ℤreadytoperish though handy, WA isn't necessarily a good source for this kind of thing. Why doesn't Wolfram Alpha show low gravitational acceleration for the Hudson Bay?
– uhoh
Sep 28, 2020 at 2:06
• @gen-ℤreadytoperish check the planets, Mercury has a density very similar to Earth's, Venus doesn't (well, 5.24 is not that far off, but the paragraph in the question makes it sound like Mercury is far denser than either Earth or Venus) Sep 28, 2020 at 10:26

How was the mass of Venus measured for the first time?

In the mid 19th century, Urbain Le Verrier's predicted of the existence of a then unknown planet beyond the orbit of Uranus. He even predicted this planet's orbit. The discovery of Neptune based on his predictions was perhaps his greatest accomplishment.

Le Verrier then went on to investigate Mercury. He used observations of Mercury, Venus, the Sun (as a stand-in for the Earth) and Mars and calculated that Mercury should precess by 532 arc seconds per century based on Newtonian mechanics. Along the way, he had to (and did) estimate the mass of Venus. There was a problem here; the observed precession of Mercury's orbit is 575 arc seconds per century, 43 arc seconds per century greater than his calculated value. This led Le Verrier to conjecture that there was a planet even closer to the Sun than Mercury.

Despite the failure to discover the non-existent planet tentatively named Vulcan, Le Verrier's estimate for the mass of Venus was fairly close to the correct figure, within a couple of percent. Once the cause of this 43 arc second per century discrepancy was discovered by Einstein, the mass of Venus was determined with even greater accuracy. Of course, once probes were sent into orbit about Venus, its mass was determined with greater accuracy yet.

• Voted up your answer, however, was Le Verrier's remarkable work on Mercury the first attempt to determine the mass of Venus using observations of the orbit of the planets ? That is, are we reasonably sure that there was no earlier attempt ? Sep 27, 2020 at 21:25
• This doesn't make sense to me. Le Verrier would first have to estimate Venus's mass by assuming some density, presumably one equal to that of the earth. Then if the precession of Mercury came out about right, but a little off, couldn't he just adjust Venus's mass to reproduce the precession? Why would he invent a new planet instead?
– user15381
Sep 27, 2020 at 23:43
• @BenCrowell - The method of least square estimation was developed in the early 19th century. This is the technique Le Verrier used to discover "a planet with the point of his pen." This is the technique that the Jet Propulsion Laboratory uses to this day to compute its planetary ephemerides. Le Verrier made the mass of Venus one of the unknowns to be solved for in a least squares sense. Sep 28, 2020 at 2:04
• @StephenG As far as I can tell, Le Verrier was the first. While there is a priority dispute between Le Verrier and John Couch Adams regarding the mathematical prediction of Neptune's existence, I could not find anyone who claimed priority on the discovery of the 43 arc second per century discrepancy between the predicted and observed precession of Mercury. Getting there required estimating the mass of Venus as Venus is the largest perturber of Mercury's orbit. Sep 28, 2020 at 2:13
• This answer describes that Le Verrier (and later, others) estimated the mass of Venus, but not how any of them did it. Sep 28, 2020 at 17:29

The mass of Venus was determined by weighing the Earth, or more precisely, by determining the ratio of the density of the Earth to the density of Schiehallion, and assuming Schiehallion to be typical rock of 2500 kg per cubic meter.

Prior to that, Jérôme Lalande had worked out the relative masses of the major bodies of the Solar System as a byproduct of improving planetary tables. The mutual attraction of planets is small, but by the mid-1700s, astronomical measurements were good enough to spot the errors caused by ignoring them. In the case of Venus, the influence is greatest on Mercury, the Earth, and the Moon.

• That last paragraph is pretty crucial. Thanks for actually answering the question. Sep 30, 2020 at 14:12