5.6 - Astrophysics A2 and Cosmology

A gravitational field is a region in which a mass experiences a force. The field describes the influence of the source mass throughout the surrounding space, whether or not a test mass is currently placed there.

At any point, the field direction is the direction of the force on a small positive test mass. Around an isolated spherical mass, this is radially inward because gravity is attractive.

If several masses are present, their gravitational forces add as vectors. Equivalently, add their field contributions vectorially at the point of interest.

Gravitational field strength $g$ at a point is the gravitational force $F_g$ per unit mass $m$ placed at that point. It is a vector and points in the direction of the force.

If a $2.5\,\mathrm{kg}$ object experiences a gravitational force of $20\,\mathrm{N}$, then $g=20/2.5=8.0\,\mathrm{N\,kg^{-1}}$. A different small test mass at the same point has the same $g$ but a different force $F_g=mg$.

Here $m_1$ and $m_2$ are the interacting masses, $r$ is the centre-to-centre separation, and $G=6.67 imes10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$. Each mass experiences an equal-magnitude force directed towards the other.

Choose SI units, find the centre separation, square it, then substitute. For a uniform spherical body, its external gravitational effect can be modelled as if all its mass were concentrated at its centre.

Place a test mass $m$ a distance $r$ from a point or spherical source mass $M$. Newton's law gives $F_g=GMm/r^2$; dividing by the test mass in $g=F_g/m$ removes $m$.

The magnitude falls as $1/r^2$, while the vector direction is radially inward. Outside a spherical planet or star, measure $r$ from its centre, so at altitude $h$ use $r=R+h$.

At a distance $2R$ from the centre of a planet, $g=GM/(2R)^2=g_{surface}/4$. This comparison avoids recalculating $G$ and $M$.

Gravitational potential $V_{grav}$ is potential energy per unit mass, measured in $\mathrm{J\,kg^{-1}}$. The conventional zero is at infinity. Because gravity is attractive, a finite point in the field has negative potential.

As $r$ increases, $V$ rises towards zero: it becomes less negative. The gradient is steep close to the source and shallow far away. Field strength and potential are linked by $g=-\mathrm{d}V/\mathrm{d}r$ in the radial direction.

Both are long-range force fields, obey superposition, and have inverse-square radial fields for point sources. Their field lines show force direction, while their potentials are scalar quantities.

Gravity is normally much weaker than electrostatic interaction at particle scale, but astronomical bodies contain enormous mass and are often nearly electrically neutral, so gravity dominates their large-scale motion.

For a mass $m$ in a circular orbit of radius $r$ around a much larger central mass $M$, the inward gravitational force supplies the required centripetal force.

The orbiting mass cancels from the equations. For the same central mass, a larger circular orbit has a lower speed but a longer period. The velocity is tangential while acceleration and force point inward.

A black body is an ideal object that absorbs all incident electromagnetic radiation and is also a perfect emitter. Its emission spectrum depends only on its absolute temperature.

To compare two curves, check that the axes and scales match. Locate each $\lambda_{max}$ for temperature information and compare areas for total emitted power per unit area.

A star can be approximated as a black-body radiator. Its colour does not mean it emits one wavelength: it emits a distribution whose peak and total area change with temperature.

Luminosity $L$ is the total power radiated, $A$ is emitting surface area, and $T$ is absolute temperature. For a spherical star, $A=4\pi R^2$.

At fixed radius, doubling $T$ multiplies $L$ by $2^4=16$. At fixed temperature, doubling radius multiplies surface area and luminosity by $2^2=4$. Ratio methods often remove $\sigma$ and $4\pi$.

Read or calculate the peak wavelength, convert it to metres, then rearrange to $T=(2.898 imes10^{-3})/\lambda_{max}$. The relation is inverse: hotter black bodies peak at shorter wavelengths.

If $\lambda_{max}=5.00 imes10^{-7}\,\mathrm{m}$, then $T=2.898 imes10^{-3}/(5.00 imes10^{-7})=5.80 imes10^3\,\mathrm{K}$.

If a source radiates uniformly in all directions, its luminosity $L$ passes through a spherical surface of area $4\pi d^2$ at distance $d$. Intensity is power received per unit area.

At twice the distance, the same luminosity is spread over four times the area, so intensity is one quarter. Rearranging gives $L=4\pi d^2I$ or $d=\sqrt{L/(4\pi I)}$.

Observe a nearby star against very distant background stars, then repeat six months later from the opposite side of Earth's orbit. The nearby star appears to shift relative to the fixed background.

Half the total angular shift is the parallax angle $p$. With the known radius of Earth's orbit as the baseline, right-triangle geometry gives the distance.

Measure the angular displacement carefully, halve it to obtain $p$, select one consistent unit relation, and calculate $d$. More distant stars have smaller parallax angles.

A standard candle is an astronomical object whose luminosity $L$ is known. Find such an object in a distant cluster or galaxy and measure the intensity $I$ received at Earth.

Nearby distance methods can calibrate the luminosity of a standard-candle class; that calibrated class can then extend the astronomical distance scale to much greater distances.

A simple Hertzsprung-Russell diagram plots stellar luminosity vertically, increasing upward, against surface temperature horizontally. Temperature conventionally decreases from left to right.

Luminosity depends on both surface temperature and radius through $L=4\pi R^2\sigma T^4$. A cool star can therefore be highly luminous if it has a very large radius; a hot star can be dim if it is very small.

Label both axes and their directions first, then place the three regions. If luminosity is expressed relative to the Sun, it is commonly shown on a logarithmic scale.

A main-sequence star is stable while hydrogen fusion in its core supplies energy and pressure balances gravitational contraction. Its mass determines how rapidly it evolves and what later stages are possible.

On the H-R diagram, a main-sequence star moves toward the giant region as its surface cools and its radius and luminosity change. A white dwarf lies in the hot-but-dim region because its surface is hot but its area is small.

The sequence is driven by fuel depletion: reduced core fusion permits contraction, contraction raises core temperature, and new fusion stages or collapse change the star's radius, temperature and luminosity.

When a wave source moves relative to an observer, successive wavefronts are emitted from different positions. Motion towards the observer compresses their spacing; motion away spreads them out.

Astronomers compare known spectral lines with their observed wavelengths. A periodic change between redshift and blueshift can reveal a star moving back and forth because of an orbiting companion.

The effect depends on radial relative motion—the component along the observer's line of sight. Sideways motion alone does not produce the same first-order wavelength shift.

Use $\Delta\lambda=\lambda_{observed}-\lambda_{rest}$. A receding source has a longer observed wavelength, so $z>0$. Its frequency falls, so $\Delta f=f_{observed}-f_{rest}<0$ and the minus sign makes the same positive $z$. The speed approximation applies when $v\ll c$.

For galaxies at cosmological distances, recession speed $v$ is proportional to distance $d$. A graph of $v$ against $d$ has gradient $H_0$; the widespread redshift-distance relation is evidence that the universe is expanding.

If expansion is approximated as steady, distance divided by recession speed gives $d/v=1/H_0$, a characteristic age. A larger measured $H_0$ gives a smaller age estimate. Convert $H_0$ to $\mathrm{s^{-1}}$ before taking its reciprocal in seconds.

Galaxy motions and gravitational effects can indicate more mass than is directly visible. This proposed dark matter increases the total gravitational influence and therefore affects models of how expansion may change.

The ultimate fate depends on the competition represented in the model between cosmic expansion and the gravitational effect of the universe's matter. Because $H_0$ and unseen mass are inferred from observations with assumptions and uncertainties, conclusions about age and fate have been controversial.