Sections · Lensing

Vol. Science · Astronomy · The bend

Light, bent
by a mass

A point mass bends a light ray by α = 4GM / (c² b). b is the impact parameter, the closest the undeflected ray would come. c = 299,792,458 m/s exactly. One nominal solar mass uses the IAU 2015 value GM☉ = 1.3271244×10²⁰ m³ s⁻² and the nominal solar radius 6.957×10⁸ m, which is the Sun Fact Sheet’s 695,700 km. That limb ray computes 1.751″. Dyson, Eddington, and Davidson 1920 quote Einstein’s limb value as 1.75″ and the Newtonian half as 0.87″. Their Sobral 4-inch result is 1.98″, probable error about 0.12″. The May 29, 1919 eclipse was the test. If the source, the mass, and the observer line up, the image is a ring. The angular Einstein radius in this page’s flat small-angle formula is θ_E = √(4GM/c² × D_LS / (D_L D_S)), with D_LS = D_S − D_L. Off the line, there are two images, θ_± = (β ± √(β² + 4 θ_E²)) / 2. Their total magnification is A = (u² + 2) / (u √(u² + 4)), where u = β / θ_E. A straight track past the mass is the Paczyński curve. A horizon is the black holes desk. The weak-field clock is time dilation. The geometry, without these angles, is gravity.

Point mass. Weak field. Small angle. D_LS on this page is D_S minus D_L, in a flat line. A galaxy or a cluster is not that mass, and a cosmological lens uses angular-diameter distances, which are not this subtraction. The Einstein Cross and SDSS J1004+4112 are cited below as published separations. Their redshifts are not loaded into the calculator. The light curve is the Paczyński point-source track. No parallax, no finite source, no planet. Not a fit to the photometry.

Interactive · Deflection, images, and a light curve

Impact parameter · the bend on the drawing is exaggerated

Mass
Ray

Limb presets · formula on cited GM and radius

Mass M☉

Impact km

Opens on the Sun’s limb. The sketch exaggerates the angle. 1.751″ is the formula, not the 1919 measurement.

Drag the source · zero is the Einstein ring

Lens
Einstein ring
Source
Images

Opening scale · 1 M☉, 4 kpc, 8 kpc · not a catalog lens

Mass M☉

Lens kpc

Source kpc

Offset θ_E

The ring radius is θ_E. Distances are a flat line, D_LS = D_S − D_L. 4 kpc and 8 kpc are round opening values, not a measured lens. Left and right arrows move the source. Home centers it.

Flat space · small angle

1.009 mas

θ_E · 1 M☉ · 4 kpc · 8 kpc

Arcsec0.001009″
Milliarcsec1.009 mas
Microarcsec1,009 μas
Source β0.5045 mas
Image θ+1.292 mas
Image θ−−0.7878 mas
Magnification A2.183
Same side |μ|1.591
Opposite |μ|0.5914

θ_E = sqrt(4GM/c² × D_LS/(D_L D_S)), with D_LS = D_S − D_L. Mass 1 M☉, lens 4 kpc, source 8 kpc. The angle is 1.009 mas (0.001009″, 1.009 mas, 1,009 μas). Flat space. Small angle. Not an angular-diameter distance.

Source offset u = 0.5, so β = 0.5045 mas. θ+ = 1.292 mas and θ− = −0.7878 mas. Total magnification A = 2.183. The image on the source’s side, at 1.292 mas, is magnified by 1.591. The opposite image, at −0.7878 mas, is 0.5914 and has opposite parity. A = |μ+| + |μ−|, and |μ+| − |μ−| = 1.

Magnification against time · point lens, point source, straight track

A(t)
A = 1

Single-lens fit · OGLE-2005-BLG-390 · not the planetary model

u0 θ_E

t_E days

u(t) = √(u0² + (t/t_E)²). The peak is at closest approach. OGLE’s printed Amax is 3.136 ± 0.044. This curve recomputes A. It does not include the planetary deviation.

The sliders need JavaScript. The opening arithmetic is still this. Sun limb: α = 1.751″, half 0.8756″. Jupiter limb: 16.64 mas. One solar mass at 4 kpc, source at 8 kpc: θ_E = 1.009 mas. At u = 0.5, A = 2.183. OGLE-2005-BLG-390 single-lens u_min = 0.332 and t_E = 11.735 days give A = 3.135. Jupiter’s fact-sheet GM, 126.687 × 10^6 km³/s², and volumetric mean radius, 69,911 km. The same formula gives 16.64 mas (0.01664″, 16.64 mas, 16,640 μas). That is the limb deflection of a point mass with Jupiter’s GM. It is not a quoted measurement of a ray that grazed Jupiter. The source sits on the line of sight. The two solutions fall on the Einstein ring, at +1.009 mas and −1.009 mas. A point source on the axis is the whole ring, not two dots. The point-source magnification is unbounded.

Published image separations · not inputs to the point-mass calculator
System	What was published	Why it is not a preset here
Q2237+0305 · Einstein Cross	Galaxy z = 0.0394, quasar z = 1.695 (Huchra et al. 1985). Four components, largest separation 1.8″ (Yee 1988). NASA: 1.6″ between the upper and lower images.	A galaxy, four images. Not a point mass. The redshifts are not Euclidean distances.
SDSS J1004+4112	Maximum separation 14.62″ (Inada et al. 2003). Quasar z = 1.734, cluster z = 0.68 (Oguri et al. 2004).	A cluster. The flat subtraction D_S − D_L is the wrong distance.

A point, not a cluster

The ring on this page is one mass.

The Einstein Cross is four images of one quasar around the nucleus of a nearby galaxy. Yee’s largest separation is 1.8″. NASA’s Hubble note gives 1.6″ between the upper and lower images, and says a faint fifth image is expected near the center. SDSS J1004+4112 is a quasar split by a cluster, maximum separation 14.62″. Neither system is loaded into the mass and distance sliders. A cluster’s Einstein radius uses angular-diameter distances, and the mass is not a point.

OGLE-2005-BLG-390 is on the light curve as the survey’s single-lens numbers, u_min = 0.332 and t_E = 11.735 days. Beaulieu et al. 2006 found a planet in a short deviation. Their binary-lens u0 is 0.359 ± 0.005 and their t_E is 11.03 ± 0.11 days. The median lens in that paper is about 0.22 solar masses at 6.6 ± 1.0 kpc. The source distance there is a prior, not a single measured distance, so this page does not turn those medians into a θ_E preset. The curve above is only the point-lens formula.

Literacy · the words on the readout

Deflection: α = 4GM/(c² b), in radians, then converted with 648,000/π arcseconds in a radian. The 1919 paper’s 1.75″ is the limb value they attribute to Einstein. This page recomputes the formula with the IAU nominal GM☉ and the nominal solar radius. The half, α/2, is the 1911 result the same paper calls 0.87″ at the limb. This page’s half is 0.8756″.
Einstein radius: θ_E = √(4GM/c² × D_LS/(D_L D_S)). On this page D_LS is D_S − D_L. That is a flat line and a small angle. It is the right shape for a Galactic microlens. It is not a cosmological angular-diameter distance.
Two images: θ± = (β ± √(β² + 4 θ_E²))/2. They and the source lie on one line through the lens. At β = 0 the two solutions sit on the ring, and the image of a point source is the ring itself.
Magnification: u = β/θ_E. A = (u² + 2)/(u √(u² + 4)). The same-side image carries (A+1)/2. The opposite image carries (A−1)/2 and the opposite parity. Their absolute values add to A and differ by 1. At u = 0 the point-source formula is unbounded.
Light curve: u(t) = √(u0² + (t/t_E)²), with t measured from closest approach. t_E is the time to cross one Einstein radius. The plot is that formula. Parallax, the finite size of the source, and a planet are not in it.
Opening numbers: The image panel opens at 1 nominal solar mass, a lens at 4 kpc, and a source at 8 kpc. Those two distances are round values so the angle is easy to read. They are not a measured lens. The deflection panel opens on the Sun’s limb. The light curve opens on the OGLE single-lens fit.

Sources · the constants and the lenses

Cited, not invented

IAU 2015 Resolution B3 (Prša et al. 2016, AJ 152, 41) — nominal solar mass parameter GM☉ = 1.3271244 × 10²⁰ m³ s⁻² and nominal solar radius 6.957 × 10⁸ m. Conversion constants, not a new measurement of the Sun. The resolution text is also at the IAU archive.
IAU 2012 Resolution B2 — the astronomical unit is exactly 149,597,870,700 m. A parsec on this page is (648,000/π) of those units.
NIST — speed of light in vacuum — c = 299,792,458 m/s exactly.
NASA NSSDCA — Sun Fact Sheet — volumetric mean radius 695,700 km, the same length as the IAU nominal solar radius. The limb preset uses that radius with the IAU GM☉, not the sheet’s rounded GM.
NASA NSSDCA — Jupiter Fact Sheet — GM 126.687 × 10⁶ km³/s². Volumetric mean radius 69,911 km. The Jupiter limb button is the formula on those two numbers.
Dyson, Eddington, and Davidson 1920, Philosophical Transactions of the Royal Society A 220, 291 — the expeditions to Sobral and Príncipe for the eclipse of May 29, 1919. The paper states Einstein’s limb deflection as 1″.75 and the Newtonian value as 0″.87. The Sobral 4-inch mean is 1″.98 with a probable error of about 0″.12. The displacement is inversely proportional to distance from the Sun’s centre.
Longair 2015, Philosophical Transactions of the Royal Society A 373, 20140287 — reading of the 1920 reductions: Príncipe 1.61 ± 0.31 arcsec, and the Sobral 4-inch result 1.98 ± 0.12 arcsec. This page quotes the Príncipe figure from Longair because the 1920 summary’s digits are the ones he prints. The Sobral probable error is also in the 1920 paper itself.
Paczyński 1986, ApJ 304, 1 — gravitational microlensing. The magnification and the straight-line track on this page are that point-lens, point-source picture.
OGLE Early Warning System — OGLE-2005-BLG-390 — single-lens fit printed on the event page: u_min = 0.332 ± 0.005, tau = 11.735 ± 0.116 days, Amax = 3.136 ± 0.044. This page uses 0.332 and 11.735 and recomputes A.
Beaulieu et al. 2006, Nature 439, 437 — the planet OGLE-2005-BLG-390Lb. Table 1 of the paper: u0 = 0.359 ± 0.005 and t_E = 11.03 ± 0.11 days for the binary-lens model. The median lens is a 0.22 solar-mass star at D_L = 6.6 ± 1.0 kpc. The source distance in that analysis is a prior. Those medians are not a θ_E preset here. Also arXiv:astro-ph/0601563.
Huchra et al. 1985, AJ 90, 691 — 2237+0305, a quasar at z = 1.695 nearly centered on a galaxy at z = 0.0394.
Yee 1988, AJ 95, 1331 — four components of 2237+030 in a cross. The largest separation of the components is 1.8″.
NASA Science — The gravitational lens G2237+0305 — Hubble’s Faint Object Camera image of the Einstein Cross. The angular separation between the upper and lower images is 1.6 arcseconds. The note says a faint fifth image is expected near the galaxy’s center. The quasar distance is given as approximately 8 billion light-years and the galaxy as 400 million light-years. Those approximations are not converted into the calculator.
Inada et al. 2003, Nature 426, 810 — SDSS J1004+4112, maximum separation 14.62 arcsec. Also arXiv:astro-ph/0312427.
Oguri et al. 2004, ApJ 605, 78 — the same system. Quasar redshift z = 1.734. Cluster redshift z = 0.68. Four images, maximum separation 14.62″.

Classroom point mass. α = 4GM/(c² b) with the IAU 2015 nominal GM☉, the nominal solar radius, and c = 299,792,458 m/s. The Sun’s limb computes 1.751″. The 1919 paper quotes 1.75″. θ_E uses D_LS = D_S − D_L. The two images and A = (u²+2)/(u √(u²+4)) are the small-angle point lens. The light curve is the Paczyński track for a chosen u0 and t_E. OGLE-2005-BLG-390 uses the survey single-lens fit, not the planetary model. Galaxy and cluster separations are cited and not computed. Not a cosmological lens model and not a photometric fit. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The horizon

Sun, limb	Mass 1 nominal solar mass. Impact parameter 695,700 km. The readout is α = 4GM/(c² b).
Jupiter, limb	Fact-sheet GM and volumetric radius. The same formula. Not a measured graze.
Mass and impact	Mass from 0.0001 to 10¹⁵ solar masses, logarithmic. Impact parameter from 10,000 km to 1,000,000,000 km, logarithmic.
Einstein radius	Mass, lens distance, and source distance. Distances are kiloparsecs, from 0.001 to 1,000,000. The source has to be farther than the lens. D_LS = D_S − D_L.
Source offset	From −3 to 3 Einstein radii. Zero is the ring. Drag the image diagram, or use Left and Right. Shift steps farther. Home centers the source.
Light curve	u0 from 0.01 to 2. t_E from 0.5 to 500 days, logarithmic. The OGLE button sets 0.332 and 11.735 days.

Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Light, bent
by a mass

Cited, not invented

Black holes

Gravity

Angular size

Redshift

Cosmology

Telescope

Gravitational waves

Solar map

Lensing desk

Controls

Readouts

Keyboard and touch

Deflection	Arcseconds, milliarcseconds, and microarcseconds. The large figure is the sensible unit. The half is α/2.
Images	θ+ and θ− from the quadratic. A is the total magnification. The same-side image is the larger absolute magnification.
Light curve	A(t) for a straight track. The horizontal line is A = 1. The survey’s printed Amax is quoted beside the recomputed peak.

Light, bentby a mass

Cited, not invented

Black holes

Gravity

Angular size

Redshift

Cosmology

Telescope

Gravitational waves

Solar map

Light, bent
by a mass