### Re: Is water pressure on Europa greater or lesser than Earth?

Date: Wed Jul 4 08:53:14 2007
Posted By: John Link, Senior Staff Physicist
Area of science: Astronomy
ID: 1180322058.As
Message:

The question: "Is water pressure on Europa greater or lesser than Earth? Europa apparently has 5-10 miles of ice atop a large ocean of liquid water. Say that a probe managed to dig through all the ice and accessed an ocean, then dropped a pressure-measuring device. Now, upon reaching the water and sinking, would the device record a water pressure roughly equivalent to water pressure 5-10 miles under water or would the pressure be about as much as if you dropped the device into the water from the surface?"

The question actually has two separate paths that contribute to the answer! First, Europa is a less massive body than Earth and the gravity at its surface is less than Earth's. Second, water ice has a density less than liquid water and so the pressure under 5 miles of ice is less than under 5 miles of liquid.

For the first point: Europa has a mass of about 4.8x1022 kg (compared to Earth's 6.0x1024 kg) and a radius of about 1.56x106 m (compared to Earth's 6.37x106 m). The Wikipedia article gives a surface gravity for Europa of 1.314 m s-2 (compared to Earth's 9.8 m s-2), and we are going to try to calculate that value! There are many previous answers in our archives that discuss how to calculate the value for gravity's acceleration, which can be found by using our search engine to search on "calculate gravity Earth surface" (Use the right side of the search page and do not put the quotation marks around the search string.). Here is one such previous answer. The formula we want is
a = GM/r2
where G is the universal gravitational constant, M is the mass of the planet, and r is the surface radius of the planet. "G" has the value 6.67x10-11 Newton meters squared per kilogram squared (N m2 kg-2). So, if we put in the values above for Europa we obtain a = 1.32 m s-2. So Wikipedia is right! (The fourth significant digit of the calculation provides a = 1.316 m s-2, but the numbers we put in to the calculation are good to only two or three significant digits, so I am not worried about the difference at the third digit of our calculated answer.)

The pressure at a depth, d, in a liquid is
p = rho a d
where rho is the density of the liquid and a is the gravitational acceleration. The density of liquid water is about 1.0x103 kg m-3. So if the surface of Europa were liquid water instead of water ice the pressure at a depth of 5 miles (8.0x103 m) would be 1.05x107 kg m-1 s2 (which unit is called a Pascal, abbreviated "Pa"). (For reference, atmospheric pressure at the surface of Earth is about 1.013x105 Pa.) That pressure (at 5 miles depth of liquid water on Europa) is about 104 atmospheres (Earth atmospheres) of pressure. But, at 5 miles depth of water on Earth the pressure is about 7.84x107 Pa, or about 772 Earth atmospheres.

For the second point: Water ice is less dense than liquid water, with a density of about 0.92x103 kg m-3. So with 5 miles (8.0x103 m) of ice above a liquid ocean the pressure at the ice-liquid boundary would be only 92% of what it would be at 5 miles depth of liquid water on the same planet (or moon).

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