Magnitudes of elastic constants

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Problem 2.4

To illustrate the relationships and magnitudes of the elastic constants, complete Table 2.4a. Note that these values apply to specific specimens; the elastic constants for rocks range considerably, especially as porosity and pressure change.

Solution

We use the row-column notation to designate equations from Table 2.2a.

Water

Since water is a fluid we know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} =0=\beta } . Equation (4,1) shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E=0} also. From equation (4,5) we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} =k=2.1\times 10^{9}} Pa.

Table 2.4a. Magnitudes of elastic constants and velocities.
Constant Water Stiff mud Shale Sandstone Limestone Granite
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E\,\,\,({\times}10^9 \hbox{Pa})} 16 54 50
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\,\,\,({\times}10^9 \hbox{Pa})} 2.1
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}}\,\,\,({\times}10^9 \hbox{Pa})}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}}\,\,\,({\times}10^9 \hbox{Pa})}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}} 0.50 0.43 0.38 0.34 0.25 0.20
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}\,\,\,(\hbox{g/cm}^{3})} 1.0 1.5 1.8 1.9 2.5 2.7
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\alpha}\,\,\,(\hbox{km/s})} 1.5 1.6 3.2
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\beta}\,\,\,(\hbox{km/s})}

Stiff mud

Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}<0.5} , stiff mud is equivalent to a solid, hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} \ne 0} . From Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\alpha}} } we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)} using equation (9,6), while equation (9,2) expresses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}} in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} } , thus enabling us to find both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}}} .

We have:


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{\rho} {{\alpha} ^2} = \left( {{\lambda} + 2{\mu} } \right) = \left( {1.5\;{\rm{g/c}}{{\rm{m}}^3}} \right) {\times} {{(1.6 {\times} {{10}^5}\;{\rm{cm/s}})}^2}}\\ {{\qquad} = 3.8 {\times} {{10}^{10}}\;{\rm{dynes/c}}{{\rm{m}}^2} = 3.8 {\times} {{10}^9}\;{\rm{Pa}},}\\ {{\quad}{\sigma} = 0.43 = {\lambda} /2\left( {{\lambda} + {\mu} } \right),\;{\rm{i}}.{\rm{e}}.,\;0.86{\mu} = 0.14{\lambda} .} \end{align} }

Solving the two equations, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} =2.9{\times} 10^{9} } Pa, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} =0.47{\times} 10^{9} } Pa. Using equation (6,1),


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {E={\mathrm{\lambda}} \frac{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)}{{\mathrm{\sigma}} } =2.9{\times} 10^{9} {\times} 1.43{\times} 0.14/0.43}\\ {{\qquad}{\qquad}{\qquad}{\qquad}{\quad}=1.4{\times} 10^{9}\ \mathrm{Pa}.} \end{align} }

Equation (6,3) gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} k={\mathrm{\lambda}} \frac{\left(1+{\mathrm{\sigma}} \right)}{3{\mathrm{\sigma}} } =2.9{\times} 10^{9} {\times} 1.43/1.29=3.2{\times} 10^{9}\ \mathrm{Pa}. \end{align} }

Finally, to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\beta}} } we note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k} are in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N/m^{2} } , and so we must express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}} in appropriate units of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle kg/m^{3} } , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}=1.5{\times} 10^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle kg/m^{3}} . We now have


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta =({\mathrm{\mu}} /{\mathrm{\rho}})^{1/2} =\left(0.47{\times} 10^{9} /1.5{\times} 10^{3} \right)=0.56\ \mathrm{\hbox{km/s}}. }

Shale

As with stiff mud, we have been given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}} , so that again Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\alpha}} } give us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}} gives us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} /2\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)} so that we can solve these equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} } , then find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} using equations (9,3) and (9,1). Thus,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left(\lambda +2\mu \right)=\rho\alpha^{2} =\left(1.8\ \mathrm{g/cm}^{3} \right)\times (3.2\times 10^{5}\ \mathrm{cm/s})^{2} =18.4\times 10^{9}\ \mathrm{Pa}, \end{align} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sigma =0.38=\lambda /2\left(\lambda +\mu \right),\quad \mathrm{i.e.}, 0.76\mu =0.24\lambda. \end{align} }

Solving the two equations gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}} =11{\times} 10^{9} } Pa, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}} =3.6{\times} 10^{9} } Pa.

From equation (9,3), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k={\mathrm{\lambda}} +\frac{2}{3} {\mathrm{\mu}} =14{\times} 10^{9} } Pa, and equation (9,1) gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} E={\mathrm{\mu}} \frac{\left(3{\mathrm{\lambda}} +2{\mathrm{\mu}} \right)}{\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)} =9.9{\times} 10^{9}\ \mathrm{Pa}. \end{align} }

Finally,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \beta =({\mathrm{\mu}} /{\mathrm{\rho}})^{1/2} =(3.6{\times} 10^{9} /1.8{\times} 10^{3} )^{1/2} =1.4\ \mathrm{\hbox{km/s}}. \end{align} }

Sandstone

We are given the elastic constants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}} (plus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\rho}}} ), so we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\beta}} } using equations (1,3) to (1,7) in Table 2.2a. Thus


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {k=16{\times} 10^{9} /3{\times} 0.32=17{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\mu}} =16{\times} 10^{9} /2{\times} 1.34=6.0{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\lambda}} =16{\times} 10^{9} {\times} 0.34/1.34{\times} 0.32=13{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\alpha}} =[16{\times} 10^{9} {\times} 0.66/1.34{\times} 0.32{\times} 1.9{\times} 10^{3} ]^{1/2} =3.6\ \mathrm{\hbox{km/s}},}\\ {{\mathrm{\beta}} =[16{\times} 10^{9} /2{\times} 1.34{\times} 1.9{\times} 10^{3} ]^{1/2} =1.8\ \mathrm{\hbox{km/s}}.} \end{align} }

[We could also have obtained Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\alpha}} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\beta}} } by using equations (9,6) and (9,7).]

Limestone

We solve in the same way as with sandstone since we are given the same constants:


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {k =54{\times} 10^{9} /3{\times} 0.50=36{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\mu}} =54{\times} 10^{9} /2{\times} 1.25=22{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\lambda}} =54{\times} 10^{9} {\times} 0.25/1.25{\times} 0.50=22{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\alpha}} =[54{\times} 10^{9} {\times} 0.75/1.25{\times} 0.50{\times} 2.5{\times} 10^{3} ]^{1/2} =5.1\ \mathrm{\hbox{km/s}},}\\ {{\mathrm{\beta}} =[54{\times} 10^{9} /2{\times} 1.25{\times} 2.5{\times} 10^{3} ]^{1/2} =2.9\ \mathrm{\hbox{km/s}}.} \end{align} }

Granite

Again the solution is the same as for sandstone.


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {k=50{\times} 10^{9} /3{\times} 0.60=28{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\mu}} =50{\times} 10^{9} /2{\times} 1.2=21{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\lambda}} =50{\times} 10^{9} {\times} 0.20/1.2{\times} 0.60=14{\times} 10^{9}\ \mathrm{Pa},}\\ {{\mathrm{\alpha}} =[50{\times} 10^{9} {\times} 0.80/1.2{\times} 0.60{\times} 2.7{\times} 10^{3} ]^{1/2} =4.5\ \mathrm{\hbox{km/s}},}\\ {{\mathrm{\beta}} =[50{\times} 10^{9} /2{\times} 1.2{\times} 2.7{\times} 10^{3} ]^{1/2} =2.8\ \mathrm{\hbox{km/s}}.} \end{align} }

Table 2.4b summarizes the results.

Table 2.4b Magnitudes of elastic constants.
Constant Water Stiffmud Shale Sandstone Limestone Granite
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E\,\,\,({\times} 10^{9} \hbox{Pa})} 0 1.4 9.9 16 54 50
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K\,\,\,({\times} 10^{9} \hbox{Pa})} 2.1 3.2 13 17 36 28
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}}\,\,\,({\times} 10^{9} \hbox{Pa})} 0 0.47 3.6 6.0 22 21
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\lambda}}\,\,\,({\times} 10^{9} \hbox{Pa})} 2.1 2.9 11 13 22 14
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\sigma}}} 0.50 0.43 0.38 0.34 0.25 0.20
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\mu}}\,\,\,(\hbox{g/cm}^{3})} 1.0 1.5 1.8 1.9 2.5 2.7
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\alpha}}\,\,\,(\hbox{km/s})} 1.5 1.6 3.2 3.6 5.1 4.5
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathrm{\beta}}\,\,\,(\hbox{km/s})} 0 0.56 1.4 1.8 2.9 2.8

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