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Use a Hohmann transfer orbit. At the initial radius $r_0$, make a short prograde burn parallel to the existing velocity. The increased speed changes the circular Kepler orbit into an elliptic orbit whose periapsis is the burn point and whose apoapsis lies on the desired larger circle. Coast for half the transfer orbital period to apoapsis.

At that point the transfer velocity is again tangent to the target circle, but its magnitude is smaller than the circular speed there. A second short prograde burn raises it to the required circular speed. Thus both burns increase the instantaneous speed and preserve its direction at the burn, even though the final circular speed is lower than the original one. During the coast the spacecraft loses kinetic energy while gaining gravitational potential energy.

The figure illustrates $3r_0$ for the subsequent parts, but the same construction works for any larger target radius. The impulsive approximation neglects the distance travelled during each burn and keeps both circles and the transfer ellipse coplanar.

The transfer is a Hohmann transfer orbit. Write $r_0$ for the original circular Kepler orbit radius and $a_t$ for the transfer ellipse's semi-major axis. The diagram measures both radii from Earth's centre, giving $r_1=1.2r_0$, and henceThe Hohmann transfer time is half the transfer orbital period, taking the satellite to the opposite end of the elliptic orbit. By Kepler's third law, relative to the original geostationary orbit's orbital period $T_0$,Using one sidereal day, $T_0=86164\mathrm{s}$, gives $t=4.9703\times 10^4\mathrm{s}=13.806\mathrm{h}$; the usual $24\mathrm{h}$ approximation gives $13.844\mathrm{h}$. The arrival impulse changes the velocity into that of the new circular Kepler orbit without adding coast time in the impulsive model.

There is a real discrepancy between the wording and the diagram: an increase in altitude above Earth's surface is different from the radius increase drawn. If altitude is interpreted literally, let $R_E$ be Earth's radius. Thenusing $R_E\simeq 6371\mathrm{km}$ and $r_0\simeq 42164\mathrm{km}$. Thus $13.8\mathrm{h}$ answers the explicitly labelled geometry; $13.5\mathrm{h}$ answers the literal altitude wording.

At closest approach the velocity is tangential. The conserved specific orbital energy and specific angular momentum areThe negative energy gives an elliptic orbit. Its semi-major axis isSince $r_p+r_a=2a$,This treats the star as fixed, appropriate to $m\ll M$. Equivalently, equate the energy at the two turning points using $v_a=\ell /r_a$. The planet mass cancels; a circular-orbit relation would be inappropriate here.

In Newtonian gravity, the radial equation with conserved specific angular momentum is $\ddot{r}=-GM/r^2+h^2/r^3$. The substitution $u=1/r$ therefore gives the linear differential equationChoosing $\varphi =0$ along perihelion, its elliptic orbit solution isHere $p$ is the semilatus rectum. One can check the last equality directly from $r_{\mathrm{peri}}=p/(1+e)$, $r_{\mathrm{apo}}=p/(1-e)$, and $a_{\mathrm{orb}}=(r_{\mathrm{peri}}+r_{\mathrm{apo}})/2$.

To find Schwarzschild perihelion precession, write $u=u_0+\delta u$ in the relativistic equation and retain first order in $\mu /p$. The linear ordinary differential equation for the correction isThe constant and double-frequency forcing change the orbit's shape without accumulating an angular shift. The resonant $\cos \varphi$ term producessince $(d^2/d{\varphi }^2+1)(\varphi \sin \varphi )=2\cos \varphi$. The oscillating part of the perturbed orbit is consequently the first-order expansion ofSuccessive perihelia are separated by $2\pi /(1-3\mu /p)$, slightly more than a full revolution. Their forward angular advance is thusin radians per orbit. This leading expression applies when $\mu /p\ll 1$. For a perfectly circular orbit there is no uniquely located perihelion; the same expression describes the limiting mismatch of radial and azimuthal frequencies for a small nonzero orbital eccentricity.

Use the $+---$ metric signature and abbreviate $f(r)=1-2\mu /r$, with $\mu =GM/c^2$ a length. Dots initially mean differentiation with respect to an affine parameter $\lambda$. The geodesic Lagrangian in the equatorial plane isThe planarity of geodesics in spherical symmetry permits this plane to be chosen. More explicitly, the full angular Euler-Lagrange equation is $\ddot{\theta }+2\dot{r}\dot{\theta }/r-\sin \theta \cos \theta {\dot{\varphi }}^2=0$, so initial data $\theta =\pi /2$, $\dot{\theta }=0$ preserve it.

Both $t$ and $\varphi$ are cyclic coordinates. Their Euler-Lagrange equations give conserved conjugate momenta $p_t=c^{2}f\dot{t}$ and $p_{\varphi }=-r^2\dot{\varphi }$. Defining constants $k,h$ therefore givesThe corresponding second-order geodesic equations areand variation of $r$ givesThe geodesic Lagrangian has no explicit $\lambda$ dependence. Its conserved Lagrangian energy function is ${\dot{x}}^{\alpha }\partial \mathcal{L}/\partial {\dot{x}}^{\alpha }-\mathcal{L}=\mathcal{L}$, since it is quadratic in the tangent components. Thus twice this function is a constant:This is the constant squared tangent norm of an affinely parametrized geodesic.

For a massive particle choose $\lambda =\tau$, its proper time, and obtain $\kappa =c^2$. For a photon, the tangent is null and $\kappa =0$. A general rescaling of a massive particle's affine parameter rescales a positive $\kappa$; the numerical value $c^2$ requires the proper time choice. With that choice, $h=L_z/m$ is signed specific angular momentum, of units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $k=E/(m{c}^2)$ is dimensionless conserved specific Killing energy. The time coordinate is normalized to proper time at infinity, so this $E$ is the particle's conserved energy referred to infinity. A photon's constants scale with the normalization of its null tangent; it has no rest frame or proper time parametrization.

Substitution of $\dot{t}=k/f$ and $\dot{\varphi }=h/r^2$ into the norm gives the general radial first integralFor a massive particle this becomesAdopting the usual half-normalization for the effective potential for timelike Schwarzschild geodesics, defineAll terms have units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$, or energy per unit mass. The first two terms are the Newtonian gravitational and centrifugal contributions; the last is the relativistic correction.

A circular orbit requires both $\dot{r}=0$ and $V_{\mathrm{eff}}^{\prime }(R)=0$. Differentiating at fixed $h$ givesThe radial first integral also givesFor the future-directed circular orbit at $R=4\mu$, these constants are $|h|=4\mu c$, $k=1$, and $f=1/2$. ThereforeSince Schwarzschild time agrees with the clock time of a static observer at infinity, the intended infinity-referred tangential orbital speed isThis uses the circumferential areal radius $R$ with the clock at infinity; it is not a local measurement made at the orbit. A local static observer's clock instead has $d{\tau }_{\mathrm{stat}}=\sqrt{f}dt$ and measures $v_{\mathrm{local}}=v_{\infty }/\sqrt{f}=\boxed{c/\sqrt{2}}$. These two operationally different speeds explain the significance of the phrase “at infinity”.

The effective-potential stability of a circular orbit follows fromConsequently the $R=4\mu$ orbit exists but is unstable to radial perturbations. The original plot below shows its position at a maximum, for $h=4\mu c$.

For a weak-field elliptic orbit, the relativistic correction makes the direction of closest approach advance after each radial cycle. The derivations under i and ii below giveper orbit to first order, where $a_{\mathrm{orb}}$ is the semi-major axis and $e$ the orbital eccentricity. For Mercury, the additional Schwarzschild perihelion precession is approximately 43 arcseconds per century, over and above the much larger Newtonian precession produced primarily by other planets. It is the unexplained residual after those effects have been included that provides the classic quantitative test of general relativity; one should not identify the whole observed precession with this relativistic correction.^[ref]

The labels i and ii in the original paper introduce optional orbit-equation aids, rather than separate mandatory subquestions. They are retained here for derivations of both aids and of the perihelion result.

Assume a circular orbit and an orbital inclination $i\simeq 90^{\circ }$, as appropriate for a simple eclipsing binary estimate. At quadrature the two radial velocities are opposite and equal to the orbital speeds. The more massive component has the smaller amplitude. With the centre of mass as origin,Thus $a_1=0.1103\mathrm{AU}$, $a_2=0.2206\mathrm{AU}$ and Kepler's third law givesThe orbital plane contains the observer's line of sight; at quadrature the two velocities point towards and away from that observer, as labelled in the original schematic below. For an elliptic orbit, the measured amplitudes instead give $a\sin i=P(K_1+K_2)\sqrt{1-e^2}/(2\pi )$; the numerical answer requires the circular assumption.

The Scientific Revolution is a useful name for linked transformations in European natural philosophy during roughly the sixteenth and seventeenth centuries, rather than a single event with an agreed beginning or end. Its strongest scientific content is a change in how motion, the heavens and natural phenomena were represented and investigated: increasingly mathematical accounts, new instruments, controlled experiments and public argument altered what could count as a persuasive explanation. The conventional span from Nicolaus Copernicus in 1543 to Isaac Newton in 1687 captures important developments, but excludes relevant earlier work and later consolidation.

In astronomy, the heliocentric model changed the arrangement of the planets without initially discarding all older mathematical devices. Johannes Kepler's elliptic orbits and empirical Kepler's laws of planetary motion then displaced uniform circular motion as a universal premise. Telescopic observations associated with Galileo Galilei, including the phases of Venus and satellites of Jupiter, challenged specific inherited cosmological claims. Their significance depended on arguments that the telescope reliably revealed celestial phenomena, not simply on looking through an uncontroversial window.

In mechanics, idealized mathematical treatments of falling bodies and projectiles helped establish quantitative relations between motion and causes. Newton's laws of motion and Newton's law of universal gravitation linked terrestrial falling bodies with celestial orbits in a single framework. The importance of this synthesis lay in detailed quantitative problems, such as the relation between inverse-square force and planetary motion, not merely in announcing that mathematics applied to nature.^[ref]

Experimental practices changed alongside theory. Air-pump work associated with Robert Boyle made carefully described apparatus, witnessed trials and replication central to disputes about air and the vacuum. Francis Bacon promoted organized inquiry, while René Descartes pursued a mathematical and mechanical account of nature; their differences caution against inventing a single timeless “scientific method”. Scientific societies, correspondence, printed diagrams and journals helped stabilize results and reputations beyond an individual's laboratory.

The term remains selective. Mathematics, craft knowledge, instruments and observations had long histories, including substantial contributions beyond Europe. Older and newer explanatory styles coexisted; natural philosophers did not suddenly become present-day professional scientists. Religious, courtly and commercial settings often supported the same work later narrated as a victory over tradition. The Scientific Revolution therefore describes a consequential reorganization of scientific questions, practices and authority, while requiring care about geographical scope, continuity and the uneven pace of change.

Doubt was reasonable if it concerned the physical cause or unlimited extrapolation of Newton's law of universal gravitation; wholesale dismissal of the quantitative evidence was harder to justify. Early eighteenth-century investigators had to distinguish the success of an inverse-square force law in explaining observations from the claim that every material body directly attracts every other one without a mediating mechanism.

The strongest case for Isaac Newton's framework was its unification of phenomena previously treated separately. Kepler's third law and nearly elliptic orbits could be connected to an inverse-square central force. Comparing terrestrial gravity with the centripetal acceleration of the Moon supported continuity between falling bodies and celestial motion. The Philosophiae Naturalis Principia Mathematica by Isaac Newton also developed treatments of interacting bodies, tides and the Earth's figure. These were substantive tests and organized a research programme capable of producing further quantitative comparisons.^[ref]

There were nevertheless rational reservations. A law proportional to $m_1{m}_2/r^2$ inferred from accessible bodies and distances did not logically establish unrestricted universality. Individual celestial predictions involved uncertain observations and approximations to difficult many-body classical mechanics. The Moon's motion and the Earth's shape were demanding tests whose successful treatment was not complete at the beginning of the century. A dissenter could question auxiliary assumptions, numerical accuracy or the inference from special cases without denying every success.

A deeper objection concerned explanation. In traditions requiring contact mechanisms, unexplained attraction across empty space appeared to reintroduce an occult quality. René Descartes's vortices supplied a mechanical alternative, though they faced their own serious dynamical difficulties. Newton himself separated a mathematically established pattern from a claimed cause of gravity; the 1713 General Scholium by Isaac Newton declined to invent an unsupported physical mechanism. Treating every criticism of action at a distance as a failure to understand evidence would therefore misrepresent the available positions.^[ref]

Later confirmations cannot be counted as evidence already possessed by an early-eighteenth-century critic. Subsequent geodetic measurements, improved lunar theory and successful comet predictions helped shift the balance, but their timing matters. Nor did retaining a mechanical alternative remain equally reasonable after every failed prediction: responsible doubt required comparative assessment and willingness to revise it. The balanced conclusion is that selective, explicit scepticism about universality or cause was defensible, while the breadth of Newtonian mechanics' quantitative achievements already imposed a substantial explanatory burden on rival accounts.

Assume a circular orbit and an orbital inclination $i\simeq 90^{\circ }$, as appropriate for a simple eclipsing binary estimate. At quadrature the two radial velocities are opposite and equal to the orbital speeds. The more massive component has the smaller amplitude. With the centre of mass as origin,Thus $a_1=0.1103\mathrm{AU}$, $a_2=0.2206\mathrm{AU}$ and Kepler's third law givesThe orbital plane contains the observer's line of sight; at quadrature the two velocities point towards and away from that observer, as labelled in the original schematic below. For an elliptic orbit, the measured amplitudes instead give $a\sin i=P(K_1+K_2)\sqrt{1-e^2}/(2\pi )$; the numerical answer requires the circular assumption.

The pericentre distance is the minimum separation from the attracting centre on an orbit, measured to the periapsis. For an elliptic orbit of semi-major axis $a$ and orbital eccentricity $e$,For example, $a=10$ length units and $e=0.9$ give $q=1$ unit even though the orbit extends to apocentre distance $Q=19$.

This distinction matters for close encounters, tidal effects and heating: a large semi-major axis does not guarantee a large closest approach. The distance is measured from the centre, rather than as altitude above a physical surface. A parabolic orbit has a finite pericentre distance despite lacking a finite elliptic semi-major axis.

The periapsis of an Earth-centred orbit. For an elliptic orbit its distance from the Earth's centre is $a(1-e)$. Both this distance and the apogee distance are measured from the force centre, rather than as heights above Earth's surface.

With $A=\beta G{M}_{\star }/c$ constant, average the drag work $\dot{\epsilon }=-A(2{\dot{r}}^2+v_{\theta }^2)/r^2$ over a bound elliptic orbit. Using $dt=r^{2}df/h$ gives the integral $\pi (2+3e^2){\mu }^2/h^2$, where $\mu$ is the conservative central parameter. Combining with $\dot{a}=2a^2\dot{\epsilon }/\mu$ givesThis is a weak-drag secular average. It also holds when the conservative parameter is $(1-\beta )G{M}_{\star }$, provided a bound orbit exists; the conservative parameter cancels but the drag amplitude remains $A$. For a circular Kepler orbit, $a^2(t)=a_0^2-4At$.

The same initial distance can lead to strikingly different orbits. Let $s=v_0/\sqrt{\mu /r_0}\ge 1$ and launch tangentially in one orbital plane. Then periapsis is the launch point, specific angular momentum is $h=r_0{v}_0$, andwhere $f$ is true anomaly. These follow by combining the initial specific orbital energy with the orbital eccentricity relation and the Kepler orbit equation. With $\mu =8{\mathrm{m}}^3/{\mathrm{s}}^2$ and $r_0=2\mathrm{m}$, the initial speeds $2$, $\sqrt{6}$, $2\sqrt{2}$, and $2\sqrt{3}m/s$ give $e=0,1/2,1,2$: a circular orbit, an elliptic orbit with apoapsis distance $6\mathrm{m}$, a parabolic trajectory, and a hyperbolic Kepler orbit. The two open paths have asymptotic speeds zero and $2m/s$ respectively. Their different directions at large distance are part of the Kepler orbit, not an extra scalar escape threshold.