Area density

This article is about the property of two-dimensional objects. For other uses, see Density (disambiguation).

The area density (also known as areal density, surface density, superficial density, areic density, mass thickness, column density, or density thickness) of a two-dimensional object is calculated as the mass per unit area. The SI derived unit is the "kilogram per square metre" (kg·m⁻²).

Area density
Common symbols	${\displaystyle \rho _{A}}$
SI unit	kg/m2
In SI base units	m−2⋅kg
Derivations from other quantities	${\displaystyle \rho _{A}={\frac {m}{A}}}$ ${\displaystyle \rho _{A}=\rho l}$
Dimension	${\displaystyle {\mathsf {L}}^{-2}{\mathsf {M}}}$

In the paper and fabric industries, it is called grammage and is expressed in grams per square meter (g/m²); for paper in particular, it may be expressed as pounds per ream of standard sizes ("basis ream").

A related area number density can be defined by replacing mass by number of particles or other countable quantity, with resulting units of m⁻².

Formulation

Area density can be calculated as: $${\displaystyle \rho _{A}={\frac {m}{A}}}$$ or $${\displaystyle \rho _{A}=\rho \cdot l,}$$ where ρ_A is the average area density, m is the total mass of the object, A is the total area of the object, ρ is the average density, and l is the average thickness of the object.

Column density

A special type of area density is called column density (also columnar mass density or simply column density), denoted ρ_A or σ. It is the mass of substance per unit area integrated along a path;^[1] It is obtained integrating volumetric density ${\displaystyle \rho }$ over a column:^[2] $${\displaystyle \sigma =\int \rho \,\mathrm {d} s.}$$

In general the integration path can be slant or oblique incidence (as in, for example, line of sight propagation in atmospheric physics). A common special case is a vertical path, from the bottom to the top of the medium: $${\displaystyle \sigma =\int \rho \,\mathrm {d} z,}$$ where ${\displaystyle z}$ denotes the vertical coordinate (e.g., height or depth).

Columnar density ${\displaystyle \rho _{A}}$ is closely related to the vertically averaged volumetric density ${\displaystyle {\bar {\rho }}}$ as $${\displaystyle {\bar {\rho }}={\frac {\rho _{A}}{\Delta z}},}$$ where ${\textstyle \Delta z=\int 1\,\mathrm {d} z}$; ${\displaystyle {\bar {\rho }}}$, ${\displaystyle \rho _{A}}$, and ${\displaystyle \Delta z}$ have units of, for example, grams per cubic metre, grams per square metre, and metres, respectively.

Usage

Atmospheric physics

It is a quantity commonly retrieved by remote sensing instruments, for instance the Total Ozone Mapping Spectrometer (TOMS) which retrieves ozone columns around the globe. Columns are also returned by the differential optical absorption spectroscopy (DOAS) method^[3] and are a common retrieval product from nadir-looking microwave radiometers.^[4][5]

A closely related concept is that of ice or liquid water path, which specifies the volume per unit area or depth instead of mass per unit area, thus the two are related: $${\displaystyle P={\frac {\sigma }{\rho _{0}}}.}$$

Another closely related concept is optical depth.

Astronomy

In astronomy, the column density is generally used to indicate the number of atoms or molecules per square cm (cm²) along the line of sight in a particular direction, as derived from observations of e.g. the 21-cm hydrogen line or from observations of a certain molecular species. Also the interstellar extinction can be related to the column density of H or H₂.^[6]

The concept of area density can be useful when analysing accretion disks. In the case of a disk seen face-on, area density for a given area of the disk is defined as column density: that is, either as the mass of substance per unit area integrated along the vertical path that goes through the disk (line-of-sight), from the bottom to the top of the medium:

$${\displaystyle \sigma =\int \rho \,\mathrm {d} z,}$$

where ${\displaystyle z}$ denotes the vertical coordinate (e.g., height or depth), or as the number or count of a substance—rather than the mass—per unit area integrated along a path (column number density):

$${\displaystyle N=\int n\,\mathrm {d} z.}$$

Data storage media

Main article: Areal density (computer storage)

Areal density is used to quantify and compare different types media used in data storage devices such as hard disk drives, optical disc drives and tape drives. The current unit of measure is typically gigabits per square inch.^[7]

Paper

Main article: grammage

The area density is often used to describe the thickness of paper; e.g., 80 g/m² is very common.

Fabric

Main article: grammage

Fabric "weight" is often specified as mass per unit area, grams per square meter (gsm) or ounces per square yard. It is also sometimes specified in ounces per yard in a standard width for the particular cloth. One gram per square meter equals 0.0295 ounces per square yard; one ounce per square yard equals 33.9 grams per square meter.

Other

It is also an important quantity for the absorption of radiation.

When studying bodies falling through air, area density is important because resistance depends on area, and gravitational force is dependent on mass.

Bone density is often expressed in grams per square centimeter (g·cm⁻²) as measured by x-ray absorptiometry, as a proxy for the actual density.

The body mass index is expressed in units of kilograms per square meter, though the area figure is nominal, being the square of the height.

The total electron content in the ionosphere is a quantity of type columnar number density.

Snow water equivalent is a quantity of type columnar mass density.

See also

• Areic quantity
• Linear density
• Paper density

References

1. Egbert Boeker; Rienk van Grondelle (2000). Environmental Physics (2nd ed.). Wiley.
2. Visconti, Guido (2001). Fundamentals of physics and chemistry of the atmosphere. Berlin: Springer. p. 470. ISBN 978-3-540-67420-7.
3. R. Sinreich; U. Frieß; T. Wagner; S. Yilmaz; U. Platt (2008). "Retrieval of Aerosol Distributions by Multi-Axis Differential Absorption Spectroscopy (MAX-DOAS)". Nucleation and Atmospheric Aerosols. pp. 1145–1149. doi:10.1007/978-1-4020-6475-3_227. ISBN 978-1-4020-6474-6.
4. C. Melsheimer; G. Heygster (2008). "Improved retrieval of total water vapor over polar regions from AMSU-B microwave radiometer data". IEEE Trans. Geosci. Remote Sens. 46 (8): 2307–2322. Bibcode:2008ITGRS..46.2307M. doi:10.1109/TGRS.2008.918013. S2CID 20910677.
5. C. Melsheimer; G. Heygster; N. Mathew; L. Toudal Pedersen (2009). "Retrieval of Sea Ice Emissivity and Integrated Retrieval of Surface and Atmospheric Parameters over the Arctic from AMSR-E data". Journal of the Remote Sensing Society of Japan. Vol. 29, no. 1. pp. 236–241.
6. "Column Density | COSMOS".
7. "Areal Density". Webopedia. 3 March 1997. Retrieved April 9, 2014.